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e.classList.remove("mermaid"),Zr||(Zr=gs().pipe(w(()=>mermaid.initialize({startOnLoad:!1,themeCSS:ci,sequence:{actorFontSize:"16px",messageFontSize:"16px",noteFontSize:"16px"}})),m(()=>{}),X(1))),Zr.subscribe(()=>{e.classList.add("mermaid");let t=`__mermaid_${vs++}`,r=M("div",{class:"mermaid"});mermaid.mermaidAPI.render(t,e.textContent,n=>{let o=r.attachShadow({mode:"closed"});o.innerHTML=n,e.replaceWith(r)})}),Zr.pipe(m(()=>({ref:e})))}function ys(e,{target$:t,print$:r}){let n=!0;return A(t.pipe(m(o=>o.closest("details:not([open])")),x(o=>e===o),m(()=>({action:"open",reveal:!0}))),r.pipe(x(o=>o||!n),w(()=>n=e.open),m(o=>({action:o?"open":"close"}))))}function ui(e,t){return P(()=>{let r=new E;return r.subscribe(({action:n,reveal:o})=>{e.toggleAttribute("open",n==="open"),o&&e.scrollIntoView()}),ys(e,t).pipe(w(n=>r.next(n)),C(()=>r.complete()),m(n=>H({ref:e},n)))})}var pi=M("table");function li(e){return e.replaceWith(pi),pi.replaceWith(Zo(e)),I({ref:e})}function xs(e){let t=Q(":scope 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P(()=>Y([ve(e),Es(t)])).pipe(m(([{height:r},n])=>({height:r,hidden:n})),B((r,n)=>r.height===n.height&&r.hidden===n.hidden),X(1))}function vi(e,{header$:t,main$:r}){return P(()=>{let n=new E,o=n.pipe(de(1));return n.pipe(J("active"),Ze(t)).subscribe(([{active:i},{hidden:s}])=>{e.classList.toggle("md-header--shadow",i&&!s),e.hidden=s}),r.subscribe(n),t.pipe(ee(o),m(i=>H({ref:e},i)))})}function Ss(e,{viewport$:t,header$:r}){return dr(e,{viewport$:t,header$:r}).pipe(m(({offset:{y:n}})=>{let{height:o}=he(e);return{active:n>=o}}),J("active"))}function gi(e,t){return P(()=>{let r=new E;r.subscribe(({active:o})=>{e.classList.toggle("md-header__title--active",o)});let n=pe("article h1");return typeof n=="undefined"?R:Ss(n,t).pipe(w(o=>r.next(o)),C(()=>r.complete()),m(o=>H({ref:e},o)))})}function yi(e,{viewport$:t,header$:r}){let n=r.pipe(m(({height:i})=>i),B()),o=n.pipe(S(()=>ve(e).pipe(m(({height:i})=>({top:e.offsetTop,bottom:e.offsetTop+i})),J("bottom"))));return Y([n,o,t]).pipe(m(([i,{top:s,bottom:a},{offset:{y:c},size:{height:f}}])=>(f=Math.max(0,f-Math.max(0,s-c,i)-Math.max(0,f+c-a)),{offset:s-i,height:f,active:s-i<=c})),B((i,s)=>i.offset===s.offset&&i.height===s.height&&i.active===s.active))}function Os(e){let t=__md_get("__palette")||{index:e.findIndex(r=>matchMedia(r.getAttribute("data-md-color-media")).matches)};return I(...e).pipe(se(r=>v(r,"change").pipe(m(()=>r))),N(e[Math.max(0,t.index)]),m(r=>({index:e.indexOf(r),color:{scheme:r.getAttribute("data-md-color-scheme"),primary:r.getAttribute("data-md-color-primary"),accent:r.getAttribute("data-md-color-accent")}})),X(1))}function xi(e){return P(()=>{let t=new E;t.subscribe(n=>{document.body.setAttribute("data-md-color-switching","");for(let[o,i]of Object.entries(n.color))document.body.setAttribute(`data-md-color-${o}`,i);for(let o=0;o{document.body.removeAttribute("data-md-color-switching")});let r=Q("input",e);return 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o=t.pipe(x(bt),oe(1),m(()=>n.get("q")||""));dt("search").pipe(x(a=>!a),oe(1)).subscribe(()=>{let a=new URL(location.href);a.searchParams.delete("q"),history.replaceState({},"",`${a}`)}),o.subscribe(a=>{a&&(e.value=a,e.focus())});let i=nr(e),s=A(v(e,"keyup"),v(e,"focus").pipe(ke(1)),o).pipe(m(()=>r(e.value)),N(""),B());return Y([s,i]).pipe(m(([a,c])=>({value:a,focus:c})),X(1))}function Ai(e,{tx$:t,rx$:r}){let n=new E,o=n.pipe(de(1));return n.pipe(J("value"),m(({value:i})=>({type:2,data:i}))).subscribe(t.next.bind(t)),n.pipe(J("focus")).subscribe(({focus:i})=>{i?(Ke("search",i),e.placeholder=""):e.placeholder=re("search.placeholder")}),v(e.form,"reset").pipe(ee(o)).subscribe(()=>e.focus()),ks(e,{tx$:t,rx$:r}).pipe(w(i=>n.next(i)),C(()=>n.complete()),m(i=>H({ref:e},i)),ie())}function Ci(e,{rx$:t},{query$:r}){let n=new E,o=Ao(e.parentElement).pipe(x(Boolean)),i=K(":scope > :first-child",e),s=K(":scope > :last-child",e),a=t.pipe(x(bt),oe(1));return n.pipe(ae(r),Mt(a)).subscribe(([{items:f},{value:u}])=>{if(u)switch(f.length){case 0:i.textContent=re("search.result.none");break;case 1:i.textContent=re("search.result.one");break;default:i.textContent=re("search.result.other",lr(f.length))}else i.textContent=re("search.result.placeholder")}),n.pipe(w(()=>s.innerHTML=""),S(({items:f})=>A(I(...f.slice(0,10)),I(...f.slice(10)).pipe(Ce(4),Nr(o),S(([u])=>u))))).subscribe(f=>s.appendChild(Jo(f))),t.pipe(x(vt),m(({data:f})=>f)).pipe(w(f=>n.next(f)),C(()=>n.complete()),m(f=>H({ref:e},f)))}function Hs(e,{query$:t}){return t.pipe(m(({value:r})=>{let n=Oe();return n.hash="",n.searchParams.delete("h"),n.searchParams.set("q",r),{url:n}}))}function Ri(e,t){let r=new E;return r.subscribe(({url:n})=>{e.setAttribute("data-clipboard-text",e.href),e.href=`${n}`}),v(e,"click").subscribe(n=>n.preventDefault()),Hs(e,t).pipe(w(n=>r.next(n)),C(()=>r.complete()),m(n=>H({ref:e},n)))}function ki(e,{rx$:t},{keyboard$:r}){let n=new E,o=_e("search-query"),i=A(v(o,"keydown"),v(o,"focus")).pipe(Le(fe),m(()=>o.value),B());return n.pipe(Ze(i),m(([{suggestions:a},c])=>{let f=c.split(/([\s-]+)/);if((a==null?void 0:a.length)&&f[f.length-1]){let u=a[a.length-1];u.startsWith(f[f.length-1])&&(f[f.length-1]=u)}else f.length=0;return f})).subscribe(a=>e.innerHTML=a.join("").replace(/\s/g," ")),r.pipe(x(({mode:a})=>a==="search")).subscribe(a=>{switch(a.type){case"ArrowRight":e.innerText.length&&o.selectionStart===o.value.length&&(o.value=e.innerText);break}}),t.pipe(x(vt),m(({data:a})=>a)).pipe(w(a=>n.next(a)),C(()=>n.complete()),m(()=>({ref:e})))}function Hi(e,{index$:t,keyboard$:r}){let n=le();try{let o=(__search==null?void 0:__search.worker)||n.search,i=Mi(o,t),s=_e("search-query",e),a=_e("search-result",e),{tx$:c,rx$:f}=i;c.pipe(x(Ti),ut(f.pipe(x(bt))),oe(1)).subscribe(c.next.bind(c)),r.pipe(x(({mode:l})=>l==="search")).subscribe(l=>{let d=Ie();switch(l.type){case"Enter":if(d===s){let h=new Map;for(let b of Q(":first-child 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"src/assets/javascripts/components/search/result/index.ts", "src/assets/javascripts/components/search/share/index.ts", "src/assets/javascripts/components/search/suggest/index.ts", "src/assets/javascripts/components/search/_/index.ts", "src/assets/javascripts/components/search/highlight/index.ts", "src/assets/javascripts/components/sidebar/index.ts", "src/assets/javascripts/components/source/facts/github/index.ts", "src/assets/javascripts/components/source/facts/gitlab/index.ts", "src/assets/javascripts/components/source/facts/_/index.ts", "src/assets/javascripts/components/source/_/index.ts", "src/assets/javascripts/components/tabs/index.ts", "src/assets/javascripts/components/toc/index.ts", "src/assets/javascripts/components/top/index.ts", "src/assets/javascripts/patches/indeterminate/index.ts", "src/assets/javascripts/patches/scrollfix/index.ts", "src/assets/javascripts/patches/scrolllock/index.ts", "src/assets/javascripts/polyfills/index.ts"], + "sourceRoot": "../../../..", + "sourcesContent": ["(function (global, factory) {\n typeof exports === 'object' && typeof module !== 'undefined' ? factory() :\n typeof define === 'function' && define.amd ? define(factory) :\n (factory());\n}(this, (function () { 'use strict';\n\n /**\n * Applies the :focus-visible polyfill at the given scope.\n * A scope in this case is either the top-level Document or a Shadow Root.\n *\n * @param {(Document|ShadowRoot)} scope\n * @see https://github.com/WICG/focus-visible\n */\n function applyFocusVisiblePolyfill(scope) {\n var hadKeyboardEvent = true;\n var hadFocusVisibleRecently = false;\n var hadFocusVisibleRecentlyTimeout = null;\n\n var inputTypesAllowlist = {\n text: true,\n search: true,\n url: true,\n tel: true,\n email: true,\n password: true,\n number: true,\n date: true,\n month: true,\n week: true,\n time: true,\n datetime: true,\n 'datetime-local': true\n };\n\n /**\n * Helper function for legacy browsers and iframes which sometimes focus\n * elements like document, body, and non-interactive SVG.\n * @param {Element} el\n */\n function isValidFocusTarget(el) {\n if (\n el &&\n el !== document &&\n el.nodeName !== 'HTML' &&\n el.nodeName !== 'BODY' &&\n 'classList' in el &&\n 'contains' in el.classList\n ) {\n return true;\n }\n return false;\n }\n\n /**\n * Computes whether the given element should automatically trigger the\n * `focus-visible` class being added, i.e. whether it should always match\n * `:focus-visible` when focused.\n * @param {Element} el\n * @return {boolean}\n */\n function focusTriggersKeyboardModality(el) {\n var type = el.type;\n var tagName = el.tagName;\n\n if (tagName === 'INPUT' && inputTypesAllowlist[type] && !el.readOnly) {\n return true;\n }\n\n if (tagName === 'TEXTAREA' && !el.readOnly) {\n return true;\n }\n\n if (el.isContentEditable) {\n return true;\n }\n\n return false;\n }\n\n /**\n * Add the `focus-visible` class to the given element if it was not added by\n * the author.\n * @param {Element} el\n */\n function addFocusVisibleClass(el) {\n if (el.classList.contains('focus-visible')) {\n return;\n }\n el.classList.add('focus-visible');\n el.setAttribute('data-focus-visible-added', '');\n }\n\n /**\n * Remove the `focus-visible` class from the given element if it was not\n * originally added by the author.\n * @param {Element} el\n */\n function removeFocusVisibleClass(el) {\n if (!el.hasAttribute('data-focus-visible-added')) {\n return;\n }\n el.classList.remove('focus-visible');\n el.removeAttribute('data-focus-visible-added');\n }\n\n /**\n * If the most recent user interaction was via the keyboard;\n * and the key press did not include a meta, alt/option, or control key;\n * then the modality is keyboard. Otherwise, the modality is not keyboard.\n * Apply `focus-visible` to any current active element and keep track\n * of our keyboard modality state with `hadKeyboardEvent`.\n * @param {KeyboardEvent} e\n */\n function onKeyDown(e) {\n if (e.metaKey || e.altKey || e.ctrlKey) {\n return;\n }\n\n if (isValidFocusTarget(scope.activeElement)) {\n addFocusVisibleClass(scope.activeElement);\n }\n\n hadKeyboardEvent = true;\n }\n\n /**\n * If at any point a user clicks with a pointing device, ensure that we change\n * the modality away from keyboard.\n * This avoids the situation where a user presses a key on an already focused\n * element, and then clicks on a different element, focusing it with a\n * pointing device, while we still think we're in keyboard modality.\n * @param {Event} e\n */\n function onPointerDown(e) {\n hadKeyboardEvent = false;\n }\n\n /**\n * On `focus`, add the `focus-visible` class to the target if:\n * - the target received focus as a result of keyboard navigation, or\n * - the event target is an element that will likely require interaction\n * via the keyboard (e.g. a text box)\n * @param {Event} e\n */\n function onFocus(e) {\n // Prevent IE from focusing the document or HTML element.\n if (!isValidFocusTarget(e.target)) {\n return;\n }\n\n if (hadKeyboardEvent || focusTriggersKeyboardModality(e.target)) {\n addFocusVisibleClass(e.target);\n }\n }\n\n /**\n * On `blur`, remove the `focus-visible` class from the target.\n * @param {Event} e\n */\n function onBlur(e) {\n if (!isValidFocusTarget(e.target)) {\n return;\n }\n\n if (\n e.target.classList.contains('focus-visible') ||\n e.target.hasAttribute('data-focus-visible-added')\n ) {\n // To detect a tab/window switch, we look for a blur event followed\n // rapidly by a visibility change.\n // If we don't see a visibility change within 100ms, it's probably a\n // regular focus change.\n hadFocusVisibleRecently = true;\n window.clearTimeout(hadFocusVisibleRecentlyTimeout);\n hadFocusVisibleRecentlyTimeout = window.setTimeout(function() {\n hadFocusVisibleRecently = false;\n }, 100);\n removeFocusVisibleClass(e.target);\n }\n }\n\n /**\n * If the user changes tabs, keep track of whether or not the previously\n * focused element had .focus-visible.\n * @param {Event} e\n */\n function onVisibilityChange(e) {\n if (document.visibilityState === 'hidden') {\n // If the tab becomes active again, the browser will handle calling focus\n // on the element (Safari actually calls it twice).\n // If this tab change caused a blur on an element with focus-visible,\n // re-apply the class when the user switches back to the tab.\n if (hadFocusVisibleRecently) {\n hadKeyboardEvent = true;\n }\n addInitialPointerMoveListeners();\n }\n }\n\n /**\n * Add a group of listeners to detect usage of any pointing devices.\n * These listeners will be added when the polyfill first loads, and anytime\n * the window is blurred, so that they are active when the window regains\n * focus.\n */\n function addInitialPointerMoveListeners() {\n document.addEventListener('mousemove', onInitialPointerMove);\n document.addEventListener('mousedown', onInitialPointerMove);\n document.addEventListener('mouseup', onInitialPointerMove);\n document.addEventListener('pointermove', onInitialPointerMove);\n document.addEventListener('pointerdown', onInitialPointerMove);\n document.addEventListener('pointerup', onInitialPointerMove);\n document.addEventListener('touchmove', onInitialPointerMove);\n document.addEventListener('touchstart', onInitialPointerMove);\n document.addEventListener('touchend', onInitialPointerMove);\n }\n\n function removeInitialPointerMoveListeners() {\n document.removeEventListener('mousemove', onInitialPointerMove);\n document.removeEventListener('mousedown', onInitialPointerMove);\n document.removeEventListener('mouseup', onInitialPointerMove);\n document.removeEventListener('pointermove', onInitialPointerMove);\n document.removeEventListener('pointerdown', onInitialPointerMove);\n document.removeEventListener('pointerup', onInitialPointerMove);\n document.removeEventListener('touchmove', onInitialPointerMove);\n document.removeEventListener('touchstart', onInitialPointerMove);\n document.removeEventListener('touchend', onInitialPointerMove);\n }\n\n /**\n * When the polfyill first loads, assume the user is in keyboard modality.\n * If any event is received from a pointing device (e.g. mouse, pointer,\n * touch), turn off keyboard modality.\n * This accounts for situations where focus enters the page from the URL bar.\n * @param {Event} e\n */\n function onInitialPointerMove(e) {\n // Work around a Safari quirk that fires a mousemove on whenever the\n // window blurs, even if you're tabbing out of the page. \u00AF\\_(\u30C4)_/\u00AF\n if (e.target.nodeName && e.target.nodeName.toLowerCase() === 'html') {\n return;\n }\n\n hadKeyboardEvent = false;\n removeInitialPointerMoveListeners();\n }\n\n // For some kinds of state, we are interested in changes at the global scope\n // only. For example, global pointer input, global key presses and global\n // visibility change should affect the state at every scope:\n document.addEventListener('keydown', onKeyDown, true);\n document.addEventListener('mousedown', onPointerDown, true);\n document.addEventListener('pointerdown', onPointerDown, true);\n document.addEventListener('touchstart', onPointerDown, true);\n document.addEventListener('visibilitychange', onVisibilityChange, true);\n\n addInitialPointerMoveListeners();\n\n // For focus and blur, we specifically care about state changes in the local\n // scope. This is because focus / blur events that originate from within a\n // shadow root are not re-dispatched from the host element if it was already\n // the active element in its own scope:\n scope.addEventListener('focus', onFocus, true);\n scope.addEventListener('blur', onBlur, true);\n\n // We detect that a node is a ShadowRoot by ensuring that it is a\n // DocumentFragment and also has a host property. This check covers native\n // implementation and polyfill implementation transparently. If we only cared\n // about the native implementation, we could just check if the scope was\n // an instance of a ShadowRoot.\n if (scope.nodeType === Node.DOCUMENT_FRAGMENT_NODE && scope.host) {\n // Since a ShadowRoot is a special kind of DocumentFragment, it does not\n // have a root element to add a class to. So, we add this attribute to the\n // host element instead:\n scope.host.setAttribute('data-js-focus-visible', '');\n } else if (scope.nodeType === Node.DOCUMENT_NODE) {\n document.documentElement.classList.add('js-focus-visible');\n document.documentElement.setAttribute('data-js-focus-visible', '');\n }\n }\n\n // It is important to wrap all references to global window and document in\n // these checks to support server-side rendering use cases\n // @see https://github.com/WICG/focus-visible/issues/199\n if (typeof window !== 'undefined' && typeof document !== 'undefined') {\n // Make the polyfill helper globally available. This can be used as a signal\n // to interested libraries that wish to coordinate with the polyfill for e.g.,\n // applying the polyfill to a shadow root:\n window.applyFocusVisiblePolyfill = applyFocusVisiblePolyfill;\n\n // Notify interested libraries of the polyfill's presence, in case the\n // polyfill was loaded lazily:\n var event;\n\n try {\n event = new CustomEvent('focus-visible-polyfill-ready');\n } catch (error) {\n // IE11 does not support using CustomEvent as a constructor directly:\n event = document.createEvent('CustomEvent');\n event.initCustomEvent('focus-visible-polyfill-ready', false, false, {});\n }\n\n window.dispatchEvent(event);\n }\n\n if (typeof document !== 'undefined') {\n // Apply the polyfill to the global document, so that no JavaScript\n // coordination is required to use the polyfill in the top-level document:\n applyFocusVisiblePolyfill(document);\n }\n\n})));\n", "(function(global) {\r\n /**\r\n * Polyfill URLSearchParams\r\n *\r\n * Inspired from : https://github.com/WebReflection/url-search-params/blob/master/src/url-search-params.js\r\n */\r\n\r\n var checkIfIteratorIsSupported = function() {\r\n try {\r\n return !!Symbol.iterator;\r\n } catch (error) {\r\n return false;\r\n }\r\n };\r\n\r\n\r\n var iteratorSupported = checkIfIteratorIsSupported();\r\n\r\n var createIterator = function(items) {\r\n var iterator = {\r\n next: function() {\r\n var value = items.shift();\r\n return { done: value === void 0, value: value };\r\n }\r\n };\r\n\r\n if (iteratorSupported) {\r\n iterator[Symbol.iterator] = function() {\r\n return iterator;\r\n };\r\n }\r\n\r\n return iterator;\r\n };\r\n\r\n /**\r\n * Search param name and values should be encoded according to https://url.spec.whatwg.org/#urlencoded-serializing\r\n * encodeURIComponent() produces the same result except encoding spaces as `%20` instead of `+`.\r\n */\r\n var serializeParam = function(value) {\r\n return encodeURIComponent(value).replace(/%20/g, '+');\r\n };\r\n\r\n var deserializeParam = function(value) {\r\n return decodeURIComponent(String(value).replace(/\\+/g, ' '));\r\n };\r\n\r\n var polyfillURLSearchParams = function() {\r\n\r\n var URLSearchParams = function(searchString) {\r\n Object.defineProperty(this, '_entries', { writable: true, value: {} });\r\n var typeofSearchString = typeof searchString;\r\n\r\n if (typeofSearchString === 'undefined') {\r\n // do nothing\r\n } else if (typeofSearchString === 'string') {\r\n if (searchString !== '') {\r\n this._fromString(searchString);\r\n }\r\n } else if (searchString instanceof URLSearchParams) {\r\n var _this = this;\r\n searchString.forEach(function(value, name) {\r\n _this.append(name, value);\r\n });\r\n } else if ((searchString !== null) && (typeofSearchString === 'object')) {\r\n if (Object.prototype.toString.call(searchString) === '[object Array]') {\r\n for (var i = 0; i < searchString.length; i++) {\r\n var entry = searchString[i];\r\n if ((Object.prototype.toString.call(entry) === '[object Array]') || (entry.length !== 2)) {\r\n this.append(entry[0], entry[1]);\r\n } else {\r\n throw new TypeError('Expected [string, any] as entry at index ' + i + ' of URLSearchParams\\'s input');\r\n }\r\n }\r\n } else {\r\n for (var key in searchString) {\r\n if (searchString.hasOwnProperty(key)) {\r\n this.append(key, searchString[key]);\r\n }\r\n }\r\n }\r\n } else {\r\n throw new TypeError('Unsupported input\\'s type for URLSearchParams');\r\n }\r\n };\r\n\r\n var proto = URLSearchParams.prototype;\r\n\r\n proto.append = function(name, value) {\r\n if (name in this._entries) {\r\n this._entries[name].push(String(value));\r\n } else {\r\n this._entries[name] = [String(value)];\r\n }\r\n };\r\n\r\n proto.delete = function(name) {\r\n delete this._entries[name];\r\n };\r\n\r\n proto.get = function(name) {\r\n return (name in this._entries) ? this._entries[name][0] : null;\r\n };\r\n\r\n proto.getAll = function(name) {\r\n return (name in this._entries) ? this._entries[name].slice(0) : [];\r\n };\r\n\r\n proto.has = function(name) {\r\n return (name in this._entries);\r\n };\r\n\r\n proto.set = function(name, value) {\r\n this._entries[name] = [String(value)];\r\n };\r\n\r\n proto.forEach = function(callback, thisArg) {\r\n var entries;\r\n for (var name in this._entries) {\r\n if (this._entries.hasOwnProperty(name)) {\r\n entries = this._entries[name];\r\n for (var i = 0; i < entries.length; i++) {\r\n callback.call(thisArg, entries[i], name, this);\r\n }\r\n }\r\n }\r\n };\r\n\r\n proto.keys = function() {\r\n var items = [];\r\n this.forEach(function(value, name) {\r\n items.push(name);\r\n });\r\n return createIterator(items);\r\n };\r\n\r\n proto.values = function() {\r\n var items = [];\r\n this.forEach(function(value) {\r\n items.push(value);\r\n });\r\n return createIterator(items);\r\n };\r\n\r\n proto.entries = function() {\r\n var items = [];\r\n this.forEach(function(value, name) {\r\n items.push([name, value]);\r\n });\r\n return createIterator(items);\r\n };\r\n\r\n if (iteratorSupported) {\r\n proto[Symbol.iterator] = proto.entries;\r\n }\r\n\r\n proto.toString = function() {\r\n var searchArray = [];\r\n this.forEach(function(value, name) {\r\n searchArray.push(serializeParam(name) + '=' + serializeParam(value));\r\n });\r\n return searchArray.join('&');\r\n };\r\n\r\n\r\n global.URLSearchParams = URLSearchParams;\r\n };\r\n\r\n var checkIfURLSearchParamsSupported = function() {\r\n try {\r\n var URLSearchParams = global.URLSearchParams;\r\n\r\n return (\r\n (new URLSearchParams('?a=1').toString() === 'a=1') &&\r\n (typeof URLSearchParams.prototype.set === 'function') &&\r\n (typeof URLSearchParams.prototype.entries === 'function')\r\n );\r\n } catch (e) {\r\n return false;\r\n }\r\n };\r\n\r\n if (!checkIfURLSearchParamsSupported()) {\r\n polyfillURLSearchParams();\r\n }\r\n\r\n var proto = global.URLSearchParams.prototype;\r\n\r\n if (typeof proto.sort !== 'function') {\r\n proto.sort = function() {\r\n var _this = this;\r\n var items = [];\r\n this.forEach(function(value, name) {\r\n items.push([name, value]);\r\n if (!_this._entries) {\r\n _this.delete(name);\r\n }\r\n });\r\n items.sort(function(a, b) {\r\n if (a[0] < b[0]) {\r\n return -1;\r\n } else if (a[0] > b[0]) {\r\n return +1;\r\n } else {\r\n return 0;\r\n }\r\n });\r\n if (_this._entries) { // force reset because IE keeps keys index\r\n _this._entries = {};\r\n }\r\n for (var i = 0; i < items.length; i++) {\r\n this.append(items[i][0], items[i][1]);\r\n }\r\n };\r\n }\r\n\r\n if (typeof proto._fromString !== 'function') {\r\n Object.defineProperty(proto, '_fromString', {\r\n enumerable: false,\r\n configurable: false,\r\n writable: false,\r\n value: function(searchString) {\r\n if (this._entries) {\r\n this._entries = {};\r\n } else {\r\n var keys = [];\r\n this.forEach(function(value, name) {\r\n keys.push(name);\r\n });\r\n for (var i = 0; i < keys.length; i++) {\r\n this.delete(keys[i]);\r\n }\r\n }\r\n\r\n searchString = searchString.replace(/^\\?/, '');\r\n var attributes = searchString.split('&');\r\n var attribute;\r\n for (var i = 0; i < attributes.length; i++) {\r\n attribute = attributes[i].split('=');\r\n this.append(\r\n deserializeParam(attribute[0]),\r\n (attribute.length > 1) ? deserializeParam(attribute[1]) : ''\r\n );\r\n }\r\n }\r\n });\r\n }\r\n\r\n // HTMLAnchorElement\r\n\r\n})(\r\n (typeof global !== 'undefined') ? global\r\n : ((typeof window !== 'undefined') ? window\r\n : ((typeof self !== 'undefined') ? self : this))\r\n);\r\n\r\n(function(global) {\r\n /**\r\n * Polyfill URL\r\n *\r\n * Inspired from : https://github.com/arv/DOM-URL-Polyfill/blob/master/src/url.js\r\n */\r\n\r\n var checkIfURLIsSupported = function() {\r\n try {\r\n var u = new global.URL('b', 'http://a');\r\n u.pathname = 'c d';\r\n return (u.href === 'http://a/c%20d') && u.searchParams;\r\n } catch (e) {\r\n return false;\r\n }\r\n };\r\n\r\n\r\n var polyfillURL = function() {\r\n var _URL = global.URL;\r\n\r\n var URL = function(url, base) {\r\n if (typeof url !== 'string') url = String(url);\r\n if (base && typeof base !== 'string') base = String(base);\r\n\r\n // Only create another document if the base is different from current location.\r\n var doc = document, baseElement;\r\n if (base && (global.location === void 0 || base !== global.location.href)) {\r\n base = base.toLowerCase();\r\n doc = document.implementation.createHTMLDocument('');\r\n baseElement = doc.createElement('base');\r\n baseElement.href = base;\r\n doc.head.appendChild(baseElement);\r\n try {\r\n if (baseElement.href.indexOf(base) !== 0) throw new Error(baseElement.href);\r\n } catch (err) {\r\n throw new Error('URL unable to set base ' + base + ' due to ' + err);\r\n }\r\n }\r\n\r\n var anchorElement = doc.createElement('a');\r\n anchorElement.href = url;\r\n if (baseElement) {\r\n doc.body.appendChild(anchorElement);\r\n anchorElement.href = anchorElement.href; // force href to refresh\r\n }\r\n\r\n var inputElement = doc.createElement('input');\r\n inputElement.type = 'url';\r\n inputElement.value = url;\r\n\r\n if (anchorElement.protocol === ':' || !/:/.test(anchorElement.href) || (!inputElement.checkValidity() && !base)) {\r\n throw new TypeError('Invalid URL');\r\n }\r\n\r\n Object.defineProperty(this, '_anchorElement', {\r\n value: anchorElement\r\n });\r\n\r\n\r\n // create a linked searchParams which reflect its changes on URL\r\n var searchParams = new global.URLSearchParams(this.search);\r\n var enableSearchUpdate = true;\r\n var enableSearchParamsUpdate = true;\r\n var _this = this;\r\n ['append', 'delete', 'set'].forEach(function(methodName) {\r\n var method = searchParams[methodName];\r\n searchParams[methodName] = function() {\r\n method.apply(searchParams, arguments);\r\n if (enableSearchUpdate) {\r\n enableSearchParamsUpdate = false;\r\n _this.search = searchParams.toString();\r\n enableSearchParamsUpdate = true;\r\n }\r\n };\r\n });\r\n\r\n Object.defineProperty(this, 'searchParams', {\r\n value: searchParams,\r\n enumerable: true\r\n });\r\n\r\n var search = void 0;\r\n Object.defineProperty(this, '_updateSearchParams', {\r\n enumerable: false,\r\n configurable: false,\r\n writable: false,\r\n value: function() {\r\n if (this.search !== search) {\r\n search = this.search;\r\n if (enableSearchParamsUpdate) {\r\n enableSearchUpdate = false;\r\n this.searchParams._fromString(this.search);\r\n enableSearchUpdate = true;\r\n }\r\n }\r\n }\r\n });\r\n };\r\n\r\n var proto = URL.prototype;\r\n\r\n var linkURLWithAnchorAttribute = function(attributeName) {\r\n Object.defineProperty(proto, attributeName, {\r\n get: function() {\r\n return this._anchorElement[attributeName];\r\n },\r\n set: function(value) {\r\n this._anchorElement[attributeName] = value;\r\n },\r\n enumerable: true\r\n });\r\n };\r\n\r\n ['hash', 'host', 'hostname', 'port', 'protocol']\r\n .forEach(function(attributeName) {\r\n linkURLWithAnchorAttribute(attributeName);\r\n });\r\n\r\n Object.defineProperty(proto, 'search', {\r\n get: function() {\r\n return this._anchorElement['search'];\r\n },\r\n set: function(value) {\r\n this._anchorElement['search'] = value;\r\n this._updateSearchParams();\r\n },\r\n enumerable: true\r\n });\r\n\r\n Object.defineProperties(proto, {\r\n\r\n 'toString': {\r\n get: function() {\r\n var _this = this;\r\n return function() {\r\n return _this.href;\r\n };\r\n }\r\n },\r\n\r\n 'href': {\r\n get: function() {\r\n return this._anchorElement.href.replace(/\\?$/, '');\r\n },\r\n set: function(value) {\r\n this._anchorElement.href = value;\r\n this._updateSearchParams();\r\n },\r\n enumerable: true\r\n },\r\n\r\n 'pathname': {\r\n get: function() {\r\n return this._anchorElement.pathname.replace(/(^\\/?)/, '/');\r\n },\r\n set: function(value) {\r\n this._anchorElement.pathname = value;\r\n },\r\n enumerable: true\r\n },\r\n\r\n 'origin': {\r\n get: function() {\r\n // get expected port from protocol\r\n var expectedPort = { 'http:': 80, 'https:': 443, 'ftp:': 21 }[this._anchorElement.protocol];\r\n // add port to origin if, expected port is different than actual port\r\n // and it is not empty f.e http://foo:8080\r\n // 8080 != 80 && 8080 != ''\r\n var addPortToOrigin = this._anchorElement.port != expectedPort &&\r\n this._anchorElement.port !== '';\r\n\r\n return this._anchorElement.protocol +\r\n '//' +\r\n this._anchorElement.hostname +\r\n (addPortToOrigin ? (':' + this._anchorElement.port) : '');\r\n },\r\n enumerable: true\r\n },\r\n\r\n 'password': { // TODO\r\n get: function() {\r\n return '';\r\n },\r\n set: function(value) {\r\n },\r\n enumerable: true\r\n },\r\n\r\n 'username': { // TODO\r\n get: function() {\r\n return '';\r\n },\r\n set: function(value) {\r\n },\r\n enumerable: true\r\n },\r\n });\r\n\r\n URL.createObjectURL = function(blob) {\r\n return _URL.createObjectURL.apply(_URL, arguments);\r\n };\r\n\r\n URL.revokeObjectURL = function(url) {\r\n return _URL.revokeObjectURL.apply(_URL, arguments);\r\n };\r\n\r\n global.URL = URL;\r\n\r\n };\r\n\r\n if (!checkIfURLIsSupported()) {\r\n polyfillURL();\r\n }\r\n\r\n if ((global.location !== void 0) && !('origin' in global.location)) {\r\n var getOrigin = function() {\r\n return global.location.protocol + '//' + global.location.hostname + (global.location.port ? 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IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT,\r\nINDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM\r\nLOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR\r\nOTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR\r\nPERFORMANCE OF THIS SOFTWARE.\r\n***************************************************************************** */\r\n/* global global, define, System, Reflect, Promise */\r\nvar __extends;\r\nvar __assign;\r\nvar __rest;\r\nvar __decorate;\r\nvar __param;\r\nvar __metadata;\r\nvar __awaiter;\r\nvar __generator;\r\nvar __exportStar;\r\nvar __values;\r\nvar __read;\r\nvar __spread;\r\nvar __spreadArrays;\r\nvar __spreadArray;\r\nvar __await;\r\nvar __asyncGenerator;\r\nvar __asyncDelegator;\r\nvar __asyncValues;\r\nvar __makeTemplateObject;\r\nvar __importStar;\r\nvar __importDefault;\r\nvar __classPrivateFieldGet;\r\nvar __classPrivateFieldSet;\r\nvar __createBinding;\r\n(function (factory) {\r\n var root = typeof global === \"object\" ? global : typeof self === \"object\" ? self : typeof this === \"object\" ? this : {};\r\n if (typeof define === \"function\" && define.amd) {\r\n define(\"tslib\", [\"exports\"], function (exports) { factory(createExporter(root, createExporter(exports))); });\r\n }\r\n else if (typeof module === \"object\" && typeof module.exports === \"object\") {\r\n factory(createExporter(root, createExporter(module.exports)));\r\n }\r\n else {\r\n factory(createExporter(root));\r\n }\r\n function createExporter(exports, previous) {\r\n if (exports !== root) {\r\n if (typeof Object.create === \"function\") {\r\n Object.defineProperty(exports, \"__esModule\", { value: true });\r\n }\r\n else {\r\n exports.__esModule = true;\r\n }\r\n }\r\n return function (id, v) { return exports[id] = previous ? previous(id, v) : v; };\r\n }\r\n})\r\n(function (exporter) {\r\n var extendStatics = Object.setPrototypeOf ||\r\n ({ __proto__: [] } instanceof Array && function (d, b) { d.__proto__ = b; }) ||\r\n function (d, b) { for (var p in b) if (Object.prototype.hasOwnProperty.call(b, p)) d[p] = b[p]; };\r\n\r\n __extends = function (d, b) {\r\n if (typeof b !== \"function\" && b !== null)\r\n throw new TypeError(\"Class extends value \" + String(b) + \" is not a constructor or null\");\r\n extendStatics(d, b);\r\n function __() { this.constructor = d; }\r\n d.prototype = b === null ? 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'right' : 'left'] = '-9999px'; // Move element to the same position vertically\n\n var yPosition = window.pageYOffset || document.documentElement.scrollTop;\n fakeElement.style.top = \"\".concat(yPosition, \"px\");\n fakeElement.setAttribute('readonly', '');\n fakeElement.value = value;\n return fakeElement;\n}\n;// CONCATENATED MODULE: ./src/actions/copy.js\n\n\n\n/**\n * Create fake copy action wrapper using a fake element.\n * @param {String} target\n * @param {Object} options\n * @return {String}\n */\n\nvar fakeCopyAction = function fakeCopyAction(value, options) {\n var fakeElement = createFakeElement(value);\n options.container.appendChild(fakeElement);\n var selectedText = select_default()(fakeElement);\n command('copy');\n fakeElement.remove();\n return selectedText;\n};\n/**\n * Copy action wrapper.\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @return {String}\n */\n\n\nvar ClipboardActionCopy = function ClipboardActionCopy(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n container: document.body\n };\n var selectedText = '';\n\n if (typeof target === 'string') {\n selectedText = fakeCopyAction(target, options);\n } else if (target instanceof HTMLInputElement && !['text', 'search', 'url', 'tel', 'password'].includes(target === null || target === void 0 ? void 0 : target.type)) {\n // If input type doesn't support `setSelectionRange`. Simulate it. https://developer.mozilla.org/en-US/docs/Web/API/HTMLInputElement/setSelectionRange\n selectedText = fakeCopyAction(target.value, options);\n } else {\n selectedText = select_default()(target);\n command('copy');\n }\n\n return selectedText;\n};\n\n/* harmony default export */ var actions_copy = (ClipboardActionCopy);\n;// CONCATENATED MODULE: ./src/actions/default.js\nfunction _typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { _typeof = function _typeof(obj) { return typeof obj; }; } else { _typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return _typeof(obj); }\n\n\n\n/**\n * Inner function which performs selection from either `text` or `target`\n * properties and then executes copy or cut operations.\n * @param {Object} options\n */\n\nvar ClipboardActionDefault = function ClipboardActionDefault() {\n var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n // Defines base properties passed from constructor.\n var _options$action = options.action,\n action = _options$action === void 0 ? 'copy' : _options$action,\n container = options.container,\n target = options.target,\n text = options.text; // Sets the `action` to be performed which can be either 'copy' or 'cut'.\n\n if (action !== 'copy' && action !== 'cut') {\n throw new Error('Invalid \"action\" value, use either \"copy\" or \"cut\"');\n } // Sets the `target` property using an element that will be have its content copied.\n\n\n if (target !== undefined) {\n if (target && _typeof(target) === 'object' && target.nodeType === 1) {\n if (action === 'copy' && target.hasAttribute('disabled')) {\n throw new Error('Invalid \"target\" attribute. Please use \"readonly\" instead of \"disabled\" attribute');\n }\n\n if (action === 'cut' && (target.hasAttribute('readonly') || target.hasAttribute('disabled'))) {\n throw new Error('Invalid \"target\" attribute. You can\\'t cut text from elements with \"readonly\" or \"disabled\" attributes');\n }\n } else {\n throw new Error('Invalid \"target\" value, use a valid Element');\n }\n } // Define selection strategy based on `text` property.\n\n\n if (text) {\n return actions_copy(text, {\n container: container\n });\n } // Defines which selection strategy based on `target` property.\n\n\n if (target) {\n return action === 'cut' ? actions_cut(target) : actions_copy(target, {\n container: container\n });\n }\n};\n\n/* harmony default export */ var actions_default = (ClipboardActionDefault);\n;// CONCATENATED MODULE: ./src/clipboard.js\nfunction clipboard_typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { clipboard_typeof = function _typeof(obj) { return typeof obj; }; } else { clipboard_typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return clipboard_typeof(obj); }\n\nfunction _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError(\"Cannot call a class as a function\"); } }\n\nfunction _defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if (\"value\" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } }\n\nfunction _createClass(Constructor, protoProps, staticProps) { if (protoProps) _defineProperties(Constructor.prototype, protoProps); if (staticProps) _defineProperties(Constructor, staticProps); return Constructor; }\n\nfunction _inherits(subClass, superClass) { if (typeof superClass !== \"function\" && superClass !== null) { throw new TypeError(\"Super expression must either be null or a function\"); } subClass.prototype = Object.create(superClass && superClass.prototype, { constructor: { value: subClass, writable: true, configurable: true } }); if (superClass) _setPrototypeOf(subClass, superClass); }\n\nfunction _setPrototypeOf(o, p) { _setPrototypeOf = Object.setPrototypeOf || function _setPrototypeOf(o, p) { o.__proto__ = p; return o; }; return _setPrototypeOf(o, p); }\n\nfunction _createSuper(Derived) { var hasNativeReflectConstruct = _isNativeReflectConstruct(); return function _createSuperInternal() { var Super = _getPrototypeOf(Derived), result; if (hasNativeReflectConstruct) { var NewTarget = _getPrototypeOf(this).constructor; result = Reflect.construct(Super, arguments, NewTarget); } else { result = Super.apply(this, arguments); } return _possibleConstructorReturn(this, result); }; }\n\nfunction _possibleConstructorReturn(self, call) { if (call && (clipboard_typeof(call) === \"object\" || typeof call === \"function\")) { return call; } return _assertThisInitialized(self); }\n\nfunction _assertThisInitialized(self) { if (self === void 0) { throw new ReferenceError(\"this hasn't been initialised - super() hasn't been called\"); } return self; }\n\nfunction _isNativeReflectConstruct() { if (typeof Reflect === \"undefined\" || !Reflect.construct) return false; if (Reflect.construct.sham) return false; if (typeof Proxy === \"function\") return true; try { Date.prototype.toString.call(Reflect.construct(Date, [], function () {})); return true; } catch (e) { return false; } }\n\nfunction _getPrototypeOf(o) { _getPrototypeOf = Object.setPrototypeOf ? Object.getPrototypeOf : function _getPrototypeOf(o) { return o.__proto__ || Object.getPrototypeOf(o); }; return _getPrototypeOf(o); }\n\n\n\n\n\n\n/**\n * Helper function to retrieve attribute value.\n * @param {String} suffix\n * @param {Element} element\n */\n\nfunction getAttributeValue(suffix, element) {\n var attribute = \"data-clipboard-\".concat(suffix);\n\n if (!element.hasAttribute(attribute)) {\n return;\n }\n\n return element.getAttribute(attribute);\n}\n/**\n * Base class which takes one or more elements, adds event listeners to them,\n * and instantiates a new `ClipboardAction` on each click.\n */\n\n\nvar Clipboard = /*#__PURE__*/function (_Emitter) {\n _inherits(Clipboard, _Emitter);\n\n var _super = _createSuper(Clipboard);\n\n /**\n * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n * @param {Object} options\n */\n function Clipboard(trigger, options) {\n var _this;\n\n _classCallCheck(this, Clipboard);\n\n _this = _super.call(this);\n\n _this.resolveOptions(options);\n\n _this.listenClick(trigger);\n\n return _this;\n }\n /**\n * Defines if attributes would be resolved using internal setter functions\n * or custom functions that were passed in the constructor.\n * @param {Object} options\n */\n\n\n _createClass(Clipboard, [{\n key: \"resolveOptions\",\n value: function resolveOptions() {\n var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n this.action = typeof options.action === 'function' ? options.action : this.defaultAction;\n this.target = typeof options.target === 'function' ? options.target : this.defaultTarget;\n this.text = typeof options.text === 'function' ? options.text : this.defaultText;\n this.container = clipboard_typeof(options.container) === 'object' ? options.container : document.body;\n }\n /**\n * Adds a click event listener to the passed trigger.\n * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n */\n\n }, {\n key: \"listenClick\",\n value: function listenClick(trigger) {\n var _this2 = this;\n\n this.listener = listen_default()(trigger, 'click', function (e) {\n return _this2.onClick(e);\n });\n }\n /**\n * Defines a new `ClipboardAction` on each click event.\n * @param {Event} e\n */\n\n }, {\n key: \"onClick\",\n value: function onClick(e) {\n var trigger = e.delegateTarget || e.currentTarget;\n var action = this.action(trigger) || 'copy';\n var text = actions_default({\n action: action,\n container: this.container,\n target: this.target(trigger),\n text: this.text(trigger)\n }); // Fires an event based on the copy operation result.\n\n this.emit(text ? 'success' : 'error', {\n action: action,\n text: text,\n trigger: trigger,\n clearSelection: function clearSelection() {\n if (trigger) {\n trigger.focus();\n }\n\n window.getSelection().removeAllRanges();\n }\n });\n }\n /**\n * Default `action` lookup function.\n * @param {Element} trigger\n */\n\n }, {\n key: \"defaultAction\",\n value: function defaultAction(trigger) {\n return getAttributeValue('action', trigger);\n }\n /**\n * Default `target` lookup function.\n * @param {Element} trigger\n */\n\n }, {\n key: \"defaultTarget\",\n value: function defaultTarget(trigger) {\n var selector = getAttributeValue('target', trigger);\n\n if (selector) {\n return document.querySelector(selector);\n }\n }\n /**\n * Allow fire programmatically a copy action\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @returns Text copied.\n */\n\n }, {\n key: \"defaultText\",\n\n /**\n * Default `text` lookup function.\n * @param {Element} trigger\n */\n value: function defaultText(trigger) {\n return getAttributeValue('text', trigger);\n }\n /**\n * Destroy lifecycle.\n */\n\n }, {\n key: \"destroy\",\n value: function destroy() {\n this.listener.destroy();\n }\n }], [{\n key: \"copy\",\n value: function copy(target) {\n var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n container: document.body\n };\n return actions_copy(target, options);\n }\n /**\n * Allow fire programmatically a cut action\n * @param {String|HTMLElement} target\n * @returns Text cutted.\n */\n\n }, {\n key: \"cut\",\n value: function cut(target) {\n return actions_cut(target);\n }\n /**\n * Returns the support of the given action, or all actions if no action is\n * given.\n * @param {String} [action]\n */\n\n }, {\n key: \"isSupported\",\n value: function isSupported() {\n var action = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : ['copy', 'cut'];\n var actions = typeof action === 'string' ? [action] : action;\n var support = !!document.queryCommandSupported;\n actions.forEach(function (action) {\n support = support && !!document.queryCommandSupported(action);\n });\n return support;\n }\n }]);\n\n return Clipboard;\n}((tiny_emitter_default()));\n\n/* harmony default export */ var clipboard = (Clipboard);\n\n/***/ }),\n\n/***/ 828:\n/***/ (function(module) {\n\nvar DOCUMENT_NODE_TYPE = 9;\n\n/**\n * A polyfill for Element.matches()\n */\nif (typeof Element !== 'undefined' && !Element.prototype.matches) {\n var proto = Element.prototype;\n\n proto.matches = proto.matchesSelector ||\n proto.mozMatchesSelector ||\n proto.msMatchesSelector ||\n proto.oMatchesSelector ||\n proto.webkitMatchesSelector;\n}\n\n/**\n * Finds the closest parent that matches a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @return {Function}\n */\nfunction closest (element, selector) {\n while (element && element.nodeType !== DOCUMENT_NODE_TYPE) {\n if (typeof element.matches === 'function' &&\n element.matches(selector)) {\n return element;\n }\n element = element.parentNode;\n }\n}\n\nmodule.exports = closest;\n\n\n/***/ }),\n\n/***/ 438:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar closest = __webpack_require__(828);\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction _delegate(element, selector, type, callback, useCapture) {\n var listenerFn = listener.apply(this, arguments);\n\n element.addEventListener(type, listenerFn, useCapture);\n\n return {\n destroy: function() {\n element.removeEventListener(type, listenerFn, useCapture);\n }\n }\n}\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element|String|Array} [elements]\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction delegate(elements, selector, type, callback, useCapture) {\n // Handle the regular Element usage\n if (typeof elements.addEventListener === 'function') {\n return _delegate.apply(null, arguments);\n }\n\n // Handle Element-less usage, it defaults to global delegation\n if (typeof type === 'function') {\n // Use `document` as the first parameter, then apply arguments\n // This is a short way to .unshift `arguments` without running into deoptimizations\n return _delegate.bind(null, document).apply(null, arguments);\n }\n\n // Handle Selector-based usage\n if (typeof elements === 'string') {\n elements = document.querySelectorAll(elements);\n }\n\n // Handle Array-like based usage\n return Array.prototype.map.call(elements, function (element) {\n return _delegate(element, selector, type, callback, useCapture);\n });\n}\n\n/**\n * Finds closest match and invokes callback.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Function}\n */\nfunction listener(element, selector, type, callback) {\n return function(e) {\n e.delegateTarget = closest(e.target, selector);\n\n if (e.delegateTarget) {\n callback.call(element, e);\n }\n }\n}\n\nmodule.exports = delegate;\n\n\n/***/ }),\n\n/***/ 879:\n/***/ (function(__unused_webpack_module, exports) {\n\n/**\n * Check if argument is a HTML element.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.node = function(value) {\n return value !== undefined\n && value instanceof HTMLElement\n && value.nodeType === 1;\n};\n\n/**\n * Check if argument is a list of HTML elements.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.nodeList = function(value) {\n var type = Object.prototype.toString.call(value);\n\n return value !== undefined\n && (type === '[object NodeList]' || type === '[object HTMLCollection]')\n && ('length' in value)\n && (value.length === 0 || exports.node(value[0]));\n};\n\n/**\n * Check if argument is a string.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.string = function(value) {\n return typeof value === 'string'\n || value instanceof String;\n};\n\n/**\n * Check if argument is a function.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.fn = function(value) {\n var type = Object.prototype.toString.call(value);\n\n return type === '[object Function]';\n};\n\n\n/***/ }),\n\n/***/ 370:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar is = __webpack_require__(879);\nvar delegate = __webpack_require__(438);\n\n/**\n * Validates all params and calls the right\n * listener function based on its target type.\n *\n * @param {String|HTMLElement|HTMLCollection|NodeList} target\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listen(target, type, callback) {\n if (!target && !type && !callback) {\n throw new Error('Missing required arguments');\n }\n\n if (!is.string(type)) {\n throw new TypeError('Second argument must be a String');\n }\n\n if (!is.fn(callback)) {\n throw new TypeError('Third argument must be a Function');\n }\n\n if (is.node(target)) {\n return listenNode(target, type, callback);\n }\n else if (is.nodeList(target)) {\n return listenNodeList(target, type, callback);\n }\n else if (is.string(target)) {\n return listenSelector(target, type, callback);\n }\n else {\n throw new TypeError('First argument must be a String, HTMLElement, HTMLCollection, or NodeList');\n }\n}\n\n/**\n * Adds an event listener to a HTML element\n * and returns a remove listener function.\n *\n * @param {HTMLElement} node\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNode(node, type, callback) {\n node.addEventListener(type, callback);\n\n return {\n destroy: function() {\n node.removeEventListener(type, callback);\n }\n }\n}\n\n/**\n * Add an event listener to a list of HTML elements\n * and returns a remove listener function.\n *\n * @param {NodeList|HTMLCollection} nodeList\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNodeList(nodeList, type, callback) {\n Array.prototype.forEach.call(nodeList, function(node) {\n node.addEventListener(type, callback);\n });\n\n return {\n destroy: function() {\n Array.prototype.forEach.call(nodeList, function(node) {\n node.removeEventListener(type, callback);\n });\n }\n }\n}\n\n/**\n * Add an event listener to a selector\n * and returns a remove listener function.\n *\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenSelector(selector, type, callback) {\n return delegate(document.body, selector, type, callback);\n}\n\nmodule.exports = listen;\n\n\n/***/ }),\n\n/***/ 817:\n/***/ (function(module) {\n\nfunction select(element) {\n var selectedText;\n\n if (element.nodeName === 'SELECT') {\n element.focus();\n\n selectedText = element.value;\n }\n else if (element.nodeName === 'INPUT' || element.nodeName === 'TEXTAREA') {\n var isReadOnly = element.hasAttribute('readonly');\n\n if (!isReadOnly) {\n element.setAttribute('readonly', '');\n }\n\n element.select();\n element.setSelectionRange(0, element.value.length);\n\n if (!isReadOnly) {\n element.removeAttribute('readonly');\n }\n\n selectedText = element.value;\n }\n else {\n if (element.hasAttribute('contenteditable')) {\n element.focus();\n }\n\n var selection = window.getSelection();\n var range = document.createRange();\n\n range.selectNodeContents(element);\n selection.removeAllRanges();\n selection.addRange(range);\n\n selectedText = selection.toString();\n }\n\n return selectedText;\n}\n\nmodule.exports = select;\n\n\n/***/ }),\n\n/***/ 279:\n/***/ (function(module) {\n\nfunction E () {\n // Keep this empty so it's easier to inherit from\n // (via https://github.com/lipsmack from https://github.com/scottcorgan/tiny-emitter/issues/3)\n}\n\nE.prototype = {\n on: function (name, callback, ctx) {\n var e = this.e || (this.e = {});\n\n (e[name] || (e[name] = [])).push({\n fn: callback,\n ctx: ctx\n });\n\n return this;\n },\n\n once: function (name, callback, ctx) {\n var self = this;\n function listener () {\n self.off(name, listener);\n callback.apply(ctx, arguments);\n };\n\n listener._ = callback\n return this.on(name, listener, ctx);\n },\n\n emit: function (name) {\n var data = [].slice.call(arguments, 1);\n var evtArr = ((this.e || (this.e = {}))[name] || []).slice();\n var i = 0;\n var len = evtArr.length;\n\n for (i; i < len; i++) {\n evtArr[i].fn.apply(evtArr[i].ctx, data);\n }\n\n return this;\n },\n\n off: function (name, callback) {\n var e = this.e || (this.e = {});\n var evts = e[name];\n var liveEvents = [];\n\n if (evts && callback) {\n for (var i = 0, len = evts.length; i < len; i++) {\n if (evts[i].fn !== callback && evts[i].fn._ !== callback)\n liveEvents.push(evts[i]);\n }\n }\n\n // Remove event from queue to prevent memory leak\n // Suggested by https://github.com/lazd\n // Ref: https://github.com/scottcorgan/tiny-emitter/commit/c6ebfaa9bc973b33d110a84a307742b7cf94c953#commitcomment-5024910\n\n (liveEvents.length)\n ? e[name] = liveEvents\n : delete e[name];\n\n return this;\n }\n};\n\nmodule.exports = E;\nmodule.exports.TinyEmitter = E;\n\n\n/***/ })\n\n/******/ \t});\n/************************************************************************/\n/******/ \t// The module cache\n/******/ \tvar __webpack_module_cache__ = {};\n/******/ \t\n/******/ \t// The require function\n/******/ \tfunction __webpack_require__(moduleId) {\n/******/ \t\t// Check if module is in cache\n/******/ \t\tif(__webpack_module_cache__[moduleId]) {\n/******/ \t\t\treturn __webpack_module_cache__[moduleId].exports;\n/******/ \t\t}\n/******/ \t\t// Create a new module (and put it into the cache)\n/******/ \t\tvar module = __webpack_module_cache__[moduleId] = {\n/******/ \t\t\t// no module.id needed\n/******/ \t\t\t// no module.loaded needed\n/******/ \t\t\texports: {}\n/******/ \t\t};\n/******/ \t\n/******/ \t\t// Execute the module function\n/******/ \t\t__webpack_modules__[moduleId](module, module.exports, __webpack_require__);\n/******/ \t\n/******/ \t\t// Return the exports of the module\n/******/ \t\treturn module.exports;\n/******/ \t}\n/******/ \t\n/************************************************************************/\n/******/ \t/* webpack/runtime/compat get default export */\n/******/ \t!function() {\n/******/ \t\t// getDefaultExport function for compatibility with non-harmony modules\n/******/ \t\t__webpack_require__.n = function(module) {\n/******/ \t\t\tvar getter = module && module.__esModule ?\n/******/ \t\t\t\tfunction() { return module['default']; } :\n/******/ \t\t\t\tfunction() { return module; };\n/******/ \t\t\t__webpack_require__.d(getter, { a: getter });\n/******/ \t\t\treturn getter;\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/define property getters */\n/******/ \t!function() {\n/******/ \t\t// define getter functions for harmony exports\n/******/ \t\t__webpack_require__.d = function(exports, definition) {\n/******/ \t\t\tfor(var key in definition) {\n/******/ \t\t\t\tif(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {\n/******/ \t\t\t\t\tObject.defineProperty(exports, key, { enumerable: true, get: definition[key] });\n/******/ \t\t\t\t}\n/******/ \t\t\t}\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/hasOwnProperty shorthand */\n/******/ \t!function() {\n/******/ \t\t__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }\n/******/ \t}();\n/******/ \t\n/************************************************************************/\n/******/ \t// module exports must be returned from runtime so entry inlining is disabled\n/******/ \t// startup\n/******/ \t// Load entry module and return exports\n/******/ \treturn __webpack_require__(686);\n/******/ })()\n.default;\n});", "/*!\n * escape-html\n * Copyright(c) 2012-2013 TJ Holowaychuk\n * Copyright(c) 2015 Andreas Lubbe\n * Copyright(c) 2015 Tiancheng \"Timothy\" Gu\n * MIT Licensed\n */\n\n'use strict';\n\n/**\n * Module variables.\n * @private\n */\n\nvar matchHtmlRegExp = /[\"'&<>]/;\n\n/**\n * Module exports.\n * @public\n */\n\nmodule.exports = escapeHtml;\n\n/**\n * Escape special characters in the given string of html.\n *\n * @param {string} string The string to escape for inserting into HTML\n * @return {string}\n * @public\n */\n\nfunction escapeHtml(string) {\n var str = '' + string;\n var match = matchHtmlRegExp.exec(str);\n\n if (!match) {\n return str;\n }\n\n var escape;\n var html = '';\n var index = 0;\n var lastIndex = 0;\n\n for (index = match.index; index < str.length; index++) {\n switch (str.charCodeAt(index)) {\n case 34: // \"\n escape = '"';\n break;\n case 38: // &\n escape = '&';\n break;\n case 39: // '\n escape = ''';\n break;\n case 60: // <\n escape = '<';\n break;\n case 62: // >\n escape = '>';\n break;\n default:\n continue;\n }\n\n if (lastIndex !== index) {\n html += str.substring(lastIndex, index);\n }\n\n lastIndex = index + 1;\n html += escape;\n }\n\n return lastIndex !== index\n ? html + str.substring(lastIndex, index)\n : html;\n}\n", "Array.prototype.flat||Object.defineProperty(Array.prototype,\"flat\",{configurable:!0,value:function r(){var t=isNaN(arguments[0])?1:Number(arguments[0]);return t?Array.prototype.reduce.call(this,function(a,e){return Array.isArray(e)?a.push.apply(a,r.call(e,t-1)):a.push(e),a},[]):Array.prototype.slice.call(this)},writable:!0}),Array.prototype.flatMap||Object.defineProperty(Array.prototype,\"flatMap\",{configurable:!0,value:function(r){return Array.prototype.map.apply(this,arguments).flat()},writable:!0})\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport \"array-flat-polyfill\"\nimport \"focus-visible\"\nimport \"unfetch/polyfill\"\nimport \"url-polyfill\"\n\nimport {\n EMPTY,\n NEVER,\n Subject,\n defer,\n delay,\n filter,\n map,\n merge,\n mergeWith,\n shareReplay,\n switchMap\n} from \"rxjs\"\n\nimport { configuration, feature } from \"./_\"\nimport {\n at,\n getOptionalElement,\n requestJSON,\n setToggle,\n watchDocument,\n watchKeyboard,\n watchLocation,\n watchLocationTarget,\n watchMedia,\n watchPrint,\n watchViewport\n} from \"./browser\"\nimport {\n getComponentElement,\n getComponentElements,\n mountAnnounce,\n mountBackToTop,\n mountConsent,\n mountContent,\n mountDialog,\n mountHeader,\n mountHeaderTitle,\n mountPalette,\n mountSearch,\n mountSearchHiglight,\n mountSidebar,\n mountSource,\n mountTableOfContents,\n mountTabs,\n watchHeader,\n watchMain\n} from \"./components\"\nimport {\n SearchIndex,\n setupClipboardJS,\n setupInstantLoading,\n setupVersionSelector\n} from \"./integrations\"\nimport {\n patchIndeterminate,\n patchScrollfix,\n patchScrolllock\n} from \"./patches\"\nimport \"./polyfills\"\n\n/* ----------------------------------------------------------------------------\n * Application\n * ------------------------------------------------------------------------- */\n\n/* Yay, JavaScript is available */\ndocument.documentElement.classList.remove(\"no-js\")\ndocument.documentElement.classList.add(\"js\")\n\n/* Set up navigation observables and subjects */\nconst document$ = watchDocument()\nconst location$ = watchLocation()\nconst target$ = watchLocationTarget()\nconst keyboard$ = watchKeyboard()\n\n/* Set up media observables */\nconst viewport$ = watchViewport()\nconst tablet$ = watchMedia(\"(min-width: 960px)\")\nconst screen$ = watchMedia(\"(min-width: 1220px)\")\nconst print$ = watchPrint()\n\n/* Retrieve search index, if search is enabled */\nconst config = configuration()\nconst index$ = document.forms.namedItem(\"search\")\n ? __search?.index || requestJSON(\n new URL(\"search/search_index.json\", config.base)\n )\n : NEVER\n\n/* Set up Clipboard.js integration */\nconst alert$ = new Subject()\nsetupClipboardJS({ alert$ })\n\n/* Set up instant loading, if enabled */\nif (feature(\"navigation.instant\"))\n setupInstantLoading({ document$, location$, viewport$ })\n\n/* Set up version selector */\nif (config.version?.provider === \"mike\")\n setupVersionSelector({ document$ })\n\n/* Always close drawer and search on navigation */\nmerge(location$, target$)\n .pipe(\n delay(125)\n )\n .subscribe(() => {\n setToggle(\"drawer\", false)\n setToggle(\"search\", false)\n })\n\n/* Set up global keyboard handlers */\nkeyboard$\n .pipe(\n filter(({ mode }) => mode === \"global\")\n )\n .subscribe(key => {\n switch (key.type) {\n\n /* Go to previous page */\n case \"p\":\n case \",\":\n const prev = getOptionalElement(\"[href][rel=prev]\")\n if (typeof prev !== \"undefined\")\n prev.click()\n break\n\n /* Go to next page */\n case \"n\":\n case \".\":\n const next = getOptionalElement(\"[href][rel=next]\")\n if (typeof next !== \"undefined\")\n next.click()\n break\n }\n })\n\n/* Set up patches */\npatchIndeterminate({ document$, tablet$ })\npatchScrollfix({ document$ })\npatchScrolllock({ viewport$, tablet$ })\n\n/* Set up header and main area observable */\nconst header$ = watchHeader(getComponentElement(\"header\"), { viewport$ })\nconst main$ = document$\n .pipe(\n map(() => getComponentElement(\"main\")),\n switchMap(el => watchMain(el, { viewport$, header$ })),\n shareReplay(1)\n )\n\n/* Set up control component observables */\nconst control$ = merge(\n\n /* Consent */\n ...getComponentElements(\"consent\")\n .map(el => mountConsent(el, { target$ })),\n\n /* Dialog */\n ...getComponentElements(\"dialog\")\n .map(el => mountDialog(el, { alert$ })),\n\n /* Header */\n ...getComponentElements(\"header\")\n .map(el => mountHeader(el, { viewport$, header$, main$ })),\n\n /* Color palette */\n ...getComponentElements(\"palette\")\n .map(el => mountPalette(el)),\n\n /* Search */\n ...getComponentElements(\"search\")\n .map(el => mountSearch(el, { index$, keyboard$ })),\n\n /* Repository information */\n ...getComponentElements(\"source\")\n .map(el => mountSource(el))\n)\n\n/* Set up content component observables */\nconst content$ = defer(() => merge(\n\n /* Announcement bar */\n ...getComponentElements(\"announce\")\n .map(el => mountAnnounce(el)),\n\n /* Content */\n ...getComponentElements(\"content\")\n .map(el => mountContent(el, { viewport$, target$, print$ })),\n\n /* Search highlighting */\n ...getComponentElements(\"content\")\n .map(el => feature(\"search.highlight\")\n ? mountSearchHiglight(el, { index$, location$ })\n : EMPTY\n ),\n\n /* Header title */\n ...getComponentElements(\"header-title\")\n .map(el => mountHeaderTitle(el, { viewport$, header$ })),\n\n /* Sidebar */\n ...getComponentElements(\"sidebar\")\n .map(el => el.getAttribute(\"data-md-type\") === \"navigation\"\n ? at(screen$, () => mountSidebar(el, { viewport$, header$, main$ }))\n : at(tablet$, () => mountSidebar(el, { viewport$, header$, main$ }))\n ),\n\n /* Navigation tabs */\n ...getComponentElements(\"tabs\")\n .map(el => mountTabs(el, { viewport$, header$ })),\n\n /* Table of contents */\n ...getComponentElements(\"toc\")\n .map(el => mountTableOfContents(el, { viewport$, header$, target$ })),\n\n /* Back-to-top button */\n ...getComponentElements(\"top\")\n .map(el => mountBackToTop(el, { viewport$, header$, main$, target$ }))\n))\n\n/* Set up component observables */\nconst component$ = document$\n .pipe(\n switchMap(() => content$),\n mergeWith(control$),\n shareReplay(1)\n )\n\n/* Subscribe to all components */\ncomponent$.subscribe()\n\n/* ----------------------------------------------------------------------------\n * Exports\n * ------------------------------------------------------------------------- */\n\nwindow.document$ = document$ /* Document observable */\nwindow.location$ = location$ /* Location subject */\nwindow.target$ = target$ /* Location target observable */\nwindow.keyboard$ = keyboard$ /* Keyboard observable */\nwindow.viewport$ = viewport$ /* Viewport observable */\nwindow.tablet$ = tablet$ /* Media tablet observable */\nwindow.screen$ = screen$ /* Media screen observable */\nwindow.print$ = print$ /* Media print observable */\nwindow.alert$ = alert$ /* Alert subject */\nwindow.component$ = component$ /* Component observable */\n", "self.fetch||(self.fetch=function(e,n){return n=n||{},new Promise(function(t,s){var r=new XMLHttpRequest,o=[],u=[],i={},a=function(){return{ok:2==(r.status/100|0),statusText:r.statusText,status:r.status,url:r.responseURL,text:function(){return Promise.resolve(r.responseText)},json:function(){return Promise.resolve(r.responseText).then(JSON.parse)},blob:function(){return Promise.resolve(new Blob([r.response]))},clone:a,headers:{keys:function(){return o},entries:function(){return u},get:function(e){return i[e.toLowerCase()]},has:function(e){return e.toLowerCase()in i}}}};for(var c in r.open(n.method||\"get\",e,!0),r.onload=function(){r.getAllResponseHeaders().replace(/^(.*?):[^\\S\\n]*([\\s\\S]*?)$/gm,function(e,n,t){o.push(n=n.toLowerCase()),u.push([n,t]),i[n]=i[n]?i[n]+\",\"+t:t}),t(a())},r.onerror=s,r.withCredentials=\"include\"==n.credentials,n.headers)r.setRequestHeader(c,n.headers[c]);r.send(n.body||null)})});\n", "import tslib from '../tslib.js';\r\nconst {\r\n __extends,\r\n __assign,\r\n __rest,\r\n __decorate,\r\n __param,\r\n __metadata,\r\n __awaiter,\r\n __generator,\r\n __exportStar,\r\n __createBinding,\r\n __values,\r\n __read,\r\n __spread,\r\n __spreadArrays,\r\n __spreadArray,\r\n __await,\r\n __asyncGenerator,\r\n __asyncDelegator,\r\n __asyncValues,\r\n __makeTemplateObject,\r\n __importStar,\r\n __importDefault,\r\n __classPrivateFieldGet,\r\n __classPrivateFieldSet,\r\n} = tslib;\r\nexport {\r\n __extends,\r\n __assign,\r\n __rest,\r\n __decorate,\r\n __param,\r\n __metadata,\r\n __awaiter,\r\n __generator,\r\n __exportStar,\r\n __createBinding,\r\n __values,\r\n __read,\r\n __spread,\r\n __spreadArrays,\r\n __spreadArray,\r\n __await,\r\n __asyncGenerator,\r\n __asyncDelegator,\r\n __asyncValues,\r\n __makeTemplateObject,\r\n __importStar,\r\n __importDefault,\r\n __classPrivateFieldGet,\r\n __classPrivateFieldSet,\r\n};\r\n", null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n ReplaySubject,\n Subject,\n fromEvent\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch document\n *\n * Documents are implemented as subjects, so all downstream observables are\n * automatically updated when a new document is emitted.\n *\n * @returns Document subject\n */\nexport function watchDocument(): Subject {\n const document$ = new ReplaySubject(1)\n fromEvent(document, \"DOMContentLoaded\", { once: true })\n .subscribe(() => document$.next(document))\n\n /* Return document */\n return document$\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve all elements matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getElements(\n selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T][]\n\nexport function getElements(\n selector: string, node?: ParentNode\n): T[]\n\nexport function getElements(\n selector: string, node: ParentNode = document\n): T[] {\n return Array.from(node.querySelectorAll(selector))\n}\n\n/**\n * Retrieve an element matching a query selector or throw a reference error\n *\n * Note that this function assumes that the element is present. If unsure if an\n * element is existent, use the `getOptionalElement` function instead.\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getElement(\n selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T]\n\nexport function getElement(\n selector: string, node?: ParentNode\n): T\n\nexport function getElement(\n selector: string, node: ParentNode = document\n): T {\n const el = getOptionalElement(selector, node)\n if (typeof el === \"undefined\")\n throw new ReferenceError(\n `Missing element: expected \"${selector}\" to be present`\n )\n\n /* Return element */\n return el\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Retrieve an optional element matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element or nothing\n */\nexport function getOptionalElement(\n selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T] | undefined\n\nexport function getOptionalElement(\n selector: string, node?: ParentNode\n): T | undefined\n\nexport function getOptionalElement(\n selector: string, node: ParentNode = document\n): T | undefined {\n return node.querySelector(selector) || undefined\n}\n\n/**\n * Retrieve the currently active element\n *\n * @returns Element or nothing\n */\nexport function getActiveElement(): HTMLElement | undefined {\n return document.activeElement instanceof HTMLElement\n ? document.activeElement || undefined\n : undefined\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n debounceTime,\n distinctUntilChanged,\n fromEvent,\n map,\n merge,\n startWith\n} from \"rxjs\"\n\nimport { getActiveElement } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element focus\n *\n * Previously, this function used `focus` and `blur` events to determine whether\n * an element is focused, but this doesn't work if there are focusable elements\n * within the elements itself. A better solutions are `focusin` and `focusout`\n * events, which bubble up the tree and allow for more fine-grained control.\n *\n * `debounceTime` is necessary, because when a focus change happens inside an\n * element, the observable would first emit `false` and then `true` again.\n *\n * @param el - Element\n *\n * @returns Element focus observable\n */\nexport function watchElementFocus(\n el: HTMLElement\n): Observable {\n return merge(\n fromEvent(document.body, \"focusin\"),\n fromEvent(document.body, \"focusout\")\n )\n .pipe(\n debounceTime(1),\n map(() => {\n const active = getActiveElement()\n return typeof active !== \"undefined\"\n ? el.contains(active)\n : false\n }),\n startWith(el === getActiveElement()),\n distinctUntilChanged()\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n animationFrameScheduler,\n auditTime,\n fromEvent,\n map,\n merge,\n startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Element offset\n */\nexport interface ElementOffset {\n x: number /* Horizontal offset */\n y: number /* Vertical offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element offset\n *\n * @param el - Element\n *\n * @returns Element offset\n */\nexport function getElementOffset(\n el: HTMLElement\n): ElementOffset {\n return {\n x: el.offsetLeft,\n y: el.offsetTop\n }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element offset\n *\n * @param el - Element\n *\n * @returns Element offset observable\n */\nexport function watchElementOffset(\n el: HTMLElement\n): Observable {\n return merge(\n fromEvent(window, \"load\"),\n fromEvent(window, \"resize\")\n )\n .pipe(\n auditTime(0, animationFrameScheduler),\n map(() => getElementOffset(el)),\n startWith(getElementOffset(el))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n animationFrameScheduler,\n auditTime,\n fromEvent,\n map,\n merge,\n startWith\n} from \"rxjs\"\n\nimport { ElementOffset } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element content offset (= scroll offset)\n *\n * @param el - Element\n *\n * @returns Element content offset\n */\nexport function getElementContentOffset(\n el: HTMLElement\n): ElementOffset {\n return {\n x: el.scrollLeft,\n y: el.scrollTop\n }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element content offset\n *\n * @param el - Element\n *\n * @returns Element content offset observable\n */\nexport function watchElementContentOffset(\n el: HTMLElement\n): Observable {\n return merge(\n fromEvent(el, \"scroll\"),\n fromEvent(window, \"resize\")\n )\n .pipe(\n auditTime(0, animationFrameScheduler),\n map(() => getElementContentOffset(el)),\n startWith(getElementContentOffset(el))\n )\n}\n", "/**\r\n * A collection of shims that provide minimal functionality of the ES6 collections.\r\n *\r\n * These implementations are not meant to be used outside of the ResizeObserver\r\n * modules as they cover only a limited range of use cases.\r\n */\r\n/* eslint-disable require-jsdoc, valid-jsdoc */\r\nvar MapShim = (function () {\r\n if (typeof Map !== 'undefined') {\r\n return Map;\r\n }\r\n /**\r\n * Returns index in provided array that matches the specified key.\r\n *\r\n * @param {Array} arr\r\n * @param {*} key\r\n * @returns {number}\r\n */\r\n function getIndex(arr, key) {\r\n var result = -1;\r\n arr.some(function (entry, index) {\r\n if (entry[0] === key) {\r\n result = index;\r\n return true;\r\n }\r\n return false;\r\n });\r\n return result;\r\n }\r\n return /** @class */ (function () {\r\n function class_1() {\r\n this.__entries__ = [];\r\n }\r\n Object.defineProperty(class_1.prototype, \"size\", {\r\n /**\r\n * @returns {boolean}\r\n */\r\n get: function () {\r\n return this.__entries__.length;\r\n },\r\n enumerable: true,\r\n configurable: true\r\n });\r\n /**\r\n * @param {*} key\r\n * @returns {*}\r\n */\r\n class_1.prototype.get = function (key) {\r\n var index = getIndex(this.__entries__, key);\r\n var entry = this.__entries__[index];\r\n return entry && entry[1];\r\n };\r\n /**\r\n * @param {*} key\r\n * @param {*} value\r\n * @returns {void}\r\n */\r\n class_1.prototype.set = function (key, value) {\r\n var index = getIndex(this.__entries__, key);\r\n if (~index) {\r\n this.__entries__[index][1] = value;\r\n }\r\n else {\r\n this.__entries__.push([key, value]);\r\n }\r\n };\r\n /**\r\n * @param {*} key\r\n * @returns {void}\r\n */\r\n class_1.prototype.delete = function (key) {\r\n var entries = this.__entries__;\r\n var index = getIndex(entries, key);\r\n if (~index) {\r\n entries.splice(index, 1);\r\n }\r\n };\r\n /**\r\n * @param {*} key\r\n * @returns {void}\r\n */\r\n class_1.prototype.has = function (key) {\r\n return !!~getIndex(this.__entries__, key);\r\n };\r\n /**\r\n * @returns {void}\r\n */\r\n class_1.prototype.clear = function () {\r\n this.__entries__.splice(0);\r\n };\r\n /**\r\n * @param {Function} callback\r\n * @param {*} [ctx=null]\r\n * @returns {void}\r\n */\r\n class_1.prototype.forEach = function (callback, ctx) {\r\n if (ctx === void 0) { ctx = null; }\r\n for (var _i = 0, _a = this.__entries__; _i < _a.length; _i++) {\r\n var entry = _a[_i];\r\n callback.call(ctx, entry[1], entry[0]);\r\n }\r\n };\r\n return class_1;\r\n }());\r\n})();\n\n/**\r\n * Detects whether window and document objects are available in current environment.\r\n */\r\nvar isBrowser = typeof window !== 'undefined' && typeof document !== 'undefined' && window.document === document;\n\n// Returns global object of a current environment.\r\nvar global$1 = (function () {\r\n if (typeof global !== 'undefined' && global.Math === Math) {\r\n return global;\r\n }\r\n if (typeof self !== 'undefined' && self.Math === Math) {\r\n return self;\r\n }\r\n if (typeof window !== 'undefined' && window.Math === Math) {\r\n return window;\r\n }\r\n // eslint-disable-next-line no-new-func\r\n return Function('return this')();\r\n})();\n\n/**\r\n * A shim for the requestAnimationFrame which falls back to the setTimeout if\r\n * first one is not supported.\r\n *\r\n * @returns {number} Requests' identifier.\r\n */\r\nvar requestAnimationFrame$1 = (function () {\r\n if (typeof requestAnimationFrame === 'function') {\r\n // It's required to use a bounded function because IE sometimes throws\r\n // an \"Invalid calling object\" error if rAF is invoked without the global\r\n // object on the left hand side.\r\n return requestAnimationFrame.bind(global$1);\r\n }\r\n return function (callback) { return setTimeout(function () { return callback(Date.now()); }, 1000 / 60); };\r\n})();\n\n// Defines minimum timeout before adding a trailing call.\r\nvar trailingTimeout = 2;\r\n/**\r\n * Creates a wrapper function which ensures that provided callback will be\r\n * invoked only once during the specified delay period.\r\n *\r\n * @param {Function} callback - Function to be invoked after the delay period.\r\n * @param {number} delay - Delay after which to invoke callback.\r\n * @returns {Function}\r\n */\r\nfunction throttle (callback, delay) {\r\n var leadingCall = false, trailingCall = false, lastCallTime = 0;\r\n /**\r\n * Invokes the original callback function and schedules new invocation if\r\n * the \"proxy\" was called during current request.\r\n *\r\n * @returns {void}\r\n */\r\n function resolvePending() {\r\n if (leadingCall) {\r\n leadingCall = false;\r\n callback();\r\n }\r\n if (trailingCall) {\r\n proxy();\r\n }\r\n }\r\n /**\r\n * Callback invoked after the specified delay. It will further postpone\r\n * invocation of the original function delegating it to the\r\n * requestAnimationFrame.\r\n *\r\n * @returns {void}\r\n */\r\n function timeoutCallback() {\r\n requestAnimationFrame$1(resolvePending);\r\n }\r\n /**\r\n * Schedules invocation of the original function.\r\n *\r\n * @returns {void}\r\n */\r\n function proxy() {\r\n var timeStamp = Date.now();\r\n if (leadingCall) {\r\n // Reject immediately following calls.\r\n if (timeStamp - lastCallTime < trailingTimeout) {\r\n return;\r\n }\r\n // Schedule new call to be in invoked when the pending one is resolved.\r\n // This is important for \"transitions\" which never actually start\r\n // immediately so there is a chance that we might miss one if change\r\n // happens amids the pending invocation.\r\n trailingCall = true;\r\n }\r\n else {\r\n leadingCall = true;\r\n trailingCall = false;\r\n setTimeout(timeoutCallback, delay);\r\n }\r\n lastCallTime = timeStamp;\r\n }\r\n return proxy;\r\n}\n\n// Minimum delay before invoking the update of observers.\r\nvar REFRESH_DELAY = 20;\r\n// A list of substrings of CSS properties used to find transition events that\r\n// might affect dimensions of observed elements.\r\nvar transitionKeys = ['top', 'right', 'bottom', 'left', 'width', 'height', 'size', 'weight'];\r\n// Check if MutationObserver is available.\r\nvar mutationObserverSupported = typeof MutationObserver !== 'undefined';\r\n/**\r\n * Singleton controller class which handles updates of ResizeObserver instances.\r\n */\r\nvar ResizeObserverController = /** @class */ (function () {\r\n /**\r\n * Creates a new instance of ResizeObserverController.\r\n *\r\n * @private\r\n */\r\n function ResizeObserverController() {\r\n /**\r\n * Indicates whether DOM listeners have been added.\r\n *\r\n * @private {boolean}\r\n */\r\n this.connected_ = false;\r\n /**\r\n * Tells that controller has subscribed for Mutation Events.\r\n *\r\n * @private {boolean}\r\n */\r\n this.mutationEventsAdded_ = false;\r\n /**\r\n * Keeps reference to the instance of MutationObserver.\r\n *\r\n * @private {MutationObserver}\r\n */\r\n this.mutationsObserver_ = null;\r\n /**\r\n * A list of connected observers.\r\n *\r\n * @private {Array}\r\n */\r\n this.observers_ = [];\r\n this.onTransitionEnd_ = this.onTransitionEnd_.bind(this);\r\n this.refresh = throttle(this.refresh.bind(this), REFRESH_DELAY);\r\n }\r\n /**\r\n * Adds observer to observers list.\r\n *\r\n * @param {ResizeObserverSPI} observer - Observer to be added.\r\n * @returns {void}\r\n */\r\n ResizeObserverController.prototype.addObserver = function (observer) {\r\n if (!~this.observers_.indexOf(observer)) {\r\n this.observers_.push(observer);\r\n }\r\n // Add listeners if they haven't been added yet.\r\n if (!this.connected_) {\r\n this.connect_();\r\n }\r\n };\r\n /**\r\n * Removes observer from observers list.\r\n *\r\n * @param {ResizeObserverSPI} observer - Observer to be removed.\r\n * @returns {void}\r\n */\r\n ResizeObserverController.prototype.removeObserver = function (observer) {\r\n var observers = this.observers_;\r\n var index = observers.indexOf(observer);\r\n // Remove observer if it's present in registry.\r\n if (~index) {\r\n observers.splice(index, 1);\r\n }\r\n // Remove listeners if controller has no connected observers.\r\n if (!observers.length && this.connected_) {\r\n this.disconnect_();\r\n }\r\n };\r\n /**\r\n * Invokes the update of observers. It will continue running updates insofar\r\n * it detects changes.\r\n *\r\n * @returns {void}\r\n */\r\n ResizeObserverController.prototype.refresh = function () {\r\n var changesDetected = this.updateObservers_();\r\n // Continue running updates if changes have been detected as there might\r\n // be future ones caused by CSS transitions.\r\n if (changesDetected) {\r\n this.refresh();\r\n }\r\n };\r\n /**\r\n * Updates every observer from observers list and notifies them of queued\r\n * entries.\r\n *\r\n * @private\r\n * @returns {boolean} Returns \"true\" if any observer has detected changes in\r\n * dimensions of it's elements.\r\n */\r\n ResizeObserverController.prototype.updateObservers_ = function () {\r\n // Collect observers that have active observations.\r\n var activeObservers = this.observers_.filter(function (observer) {\r\n return observer.gatherActive(), observer.hasActive();\r\n });\r\n // Deliver notifications in a separate cycle in order to avoid any\r\n // collisions between observers, e.g. when multiple instances of\r\n // ResizeObserver are tracking the same element and the callback of one\r\n // of them changes content dimensions of the observed target. Sometimes\r\n // this may result in notifications being blocked for the rest of observers.\r\n activeObservers.forEach(function (observer) { return observer.broadcastActive(); });\r\n return activeObservers.length > 0;\r\n };\r\n /**\r\n * Initializes DOM listeners.\r\n *\r\n * @private\r\n * @returns {void}\r\n */\r\n ResizeObserverController.prototype.connect_ = function () {\r\n // Do nothing if running in a non-browser environment or if listeners\r\n // have been already added.\r\n if (!isBrowser || this.connected_) {\r\n return;\r\n }\r\n // Subscription to the \"Transitionend\" event is used as a workaround for\r\n // delayed transitions. This way it's possible to capture at least the\r\n // final state of an element.\r\n document.addEventListener('transitionend', this.onTransitionEnd_);\r\n window.addEventListener('resize', this.refresh);\r\n if (mutationObserverSupported) {\r\n this.mutationsObserver_ = new MutationObserver(this.refresh);\r\n this.mutationsObserver_.observe(document, {\r\n attributes: true,\r\n childList: true,\r\n characterData: true,\r\n subtree: true\r\n });\r\n }\r\n else {\r\n document.addEventListener('DOMSubtreeModified', this.refresh);\r\n this.mutationEventsAdded_ = true;\r\n }\r\n this.connected_ = true;\r\n };\r\n /**\r\n * Removes DOM listeners.\r\n *\r\n * @private\r\n * @returns {void}\r\n */\r\n ResizeObserverController.prototype.disconnect_ = function () {\r\n // Do nothing if running in a non-browser environment or if listeners\r\n // have been already removed.\r\n if (!isBrowser || !this.connected_) {\r\n return;\r\n }\r\n document.removeEventListener('transitionend', this.onTransitionEnd_);\r\n window.removeEventListener('resize', this.refresh);\r\n if (this.mutationsObserver_) {\r\n this.mutationsObserver_.disconnect();\r\n }\r\n if (this.mutationEventsAdded_) {\r\n document.removeEventListener('DOMSubtreeModified', this.refresh);\r\n }\r\n this.mutationsObserver_ = null;\r\n this.mutationEventsAdded_ = false;\r\n this.connected_ = false;\r\n };\r\n /**\r\n * \"Transitionend\" event handler.\r\n *\r\n * @private\r\n * @param {TransitionEvent} event\r\n * @returns {void}\r\n */\r\n ResizeObserverController.prototype.onTransitionEnd_ = function (_a) {\r\n var _b = _a.propertyName, propertyName = _b === void 0 ? '' : _b;\r\n // Detect whether transition may affect dimensions of an element.\r\n var isReflowProperty = transitionKeys.some(function (key) {\r\n return !!~propertyName.indexOf(key);\r\n });\r\n if (isReflowProperty) {\r\n this.refresh();\r\n }\r\n };\r\n /**\r\n * Returns instance of the ResizeObserverController.\r\n *\r\n * @returns {ResizeObserverController}\r\n */\r\n ResizeObserverController.getInstance = function () {\r\n if (!this.instance_) {\r\n this.instance_ = new ResizeObserverController();\r\n }\r\n return this.instance_;\r\n };\r\n /**\r\n * Holds reference to the controller's instance.\r\n *\r\n * @private {ResizeObserverController}\r\n */\r\n ResizeObserverController.instance_ = null;\r\n return ResizeObserverController;\r\n}());\n\n/**\r\n * Defines non-writable/enumerable properties of the provided target object.\r\n *\r\n * @param {Object} target - Object for which to define properties.\r\n * @param {Object} props - Properties to be defined.\r\n * @returns {Object} Target object.\r\n */\r\nvar defineConfigurable = (function (target, props) {\r\n for (var _i = 0, _a = Object.keys(props); _i < _a.length; _i++) {\r\n var key = _a[_i];\r\n Object.defineProperty(target, key, {\r\n value: props[key],\r\n enumerable: false,\r\n writable: false,\r\n configurable: true\r\n });\r\n }\r\n return target;\r\n});\n\n/**\r\n * Returns the global object associated with provided element.\r\n *\r\n * @param {Object} target\r\n * @returns {Object}\r\n */\r\nvar getWindowOf = (function (target) {\r\n // Assume that the element is an instance of Node, which means that it\r\n // has the \"ownerDocument\" property from which we can retrieve a\r\n // corresponding global object.\r\n var ownerGlobal = target && target.ownerDocument && target.ownerDocument.defaultView;\r\n // Return the local global object if it's not possible extract one from\r\n // provided element.\r\n return ownerGlobal || global$1;\r\n});\n\n// Placeholder of an empty content rectangle.\r\nvar emptyRect = createRectInit(0, 0, 0, 0);\r\n/**\r\n * Converts provided string to a number.\r\n *\r\n * @param {number|string} value\r\n * @returns {number}\r\n */\r\nfunction toFloat(value) {\r\n return parseFloat(value) || 0;\r\n}\r\n/**\r\n * Extracts borders size from provided styles.\r\n *\r\n * @param {CSSStyleDeclaration} styles\r\n * @param {...string} positions - Borders positions (top, right, ...)\r\n * @returns {number}\r\n */\r\nfunction getBordersSize(styles) {\r\n var positions = [];\r\n for (var _i = 1; _i < arguments.length; _i++) {\r\n positions[_i - 1] = arguments[_i];\r\n }\r\n return positions.reduce(function (size, position) {\r\n var value = styles['border-' + position + '-width'];\r\n return size + toFloat(value);\r\n }, 0);\r\n}\r\n/**\r\n * Extracts paddings sizes from provided styles.\r\n *\r\n * @param {CSSStyleDeclaration} styles\r\n * @returns {Object} Paddings box.\r\n */\r\nfunction getPaddings(styles) {\r\n var positions = ['top', 'right', 'bottom', 'left'];\r\n var paddings = {};\r\n for (var _i = 0, positions_1 = positions; _i < positions_1.length; _i++) {\r\n var position = positions_1[_i];\r\n var value = styles['padding-' + position];\r\n paddings[position] = toFloat(value);\r\n }\r\n return paddings;\r\n}\r\n/**\r\n * Calculates content rectangle of provided SVG element.\r\n *\r\n * @param {SVGGraphicsElement} target - Element content rectangle of which needs\r\n * to be calculated.\r\n * @returns {DOMRectInit}\r\n */\r\nfunction getSVGContentRect(target) {\r\n var bbox = target.getBBox();\r\n return createRectInit(0, 0, bbox.width, bbox.height);\r\n}\r\n/**\r\n * Calculates content rectangle of provided HTMLElement.\r\n *\r\n * @param {HTMLElement} target - Element for which to calculate the content rectangle.\r\n * @returns {DOMRectInit}\r\n */\r\nfunction getHTMLElementContentRect(target) {\r\n // Client width & height properties can't be\r\n // used exclusively as they provide rounded values.\r\n var clientWidth = target.clientWidth, clientHeight = target.clientHeight;\r\n // By this condition we can catch all non-replaced inline, hidden and\r\n // detached elements. Though elements with width & height properties less\r\n // than 0.5 will be discarded as well.\r\n //\r\n // Without it we would need to implement separate methods for each of\r\n // those cases and it's not possible to perform a precise and performance\r\n // effective test for hidden elements. E.g. even jQuery's ':visible' filter\r\n // gives wrong results for elements with width & height less than 0.5.\r\n if (!clientWidth && !clientHeight) {\r\n return emptyRect;\r\n }\r\n var styles = getWindowOf(target).getComputedStyle(target);\r\n var paddings = getPaddings(styles);\r\n var horizPad = paddings.left + paddings.right;\r\n var vertPad = paddings.top + paddings.bottom;\r\n // Computed styles of width & height are being used because they are the\r\n // only dimensions available to JS that contain non-rounded values. It could\r\n // be possible to utilize the getBoundingClientRect if only it's data wasn't\r\n // affected by CSS transformations let alone paddings, borders and scroll bars.\r\n var width = toFloat(styles.width), height = toFloat(styles.height);\r\n // Width & height include paddings and borders when the 'border-box' box\r\n // model is applied (except for IE).\r\n if (styles.boxSizing === 'border-box') {\r\n // Following conditions are required to handle Internet Explorer which\r\n // doesn't include paddings and borders to computed CSS dimensions.\r\n //\r\n // We can say that if CSS dimensions + paddings are equal to the \"client\"\r\n // properties then it's either IE, and thus we don't need to subtract\r\n // anything, or an element merely doesn't have paddings/borders styles.\r\n if (Math.round(width + horizPad) !== clientWidth) {\r\n width -= getBordersSize(styles, 'left', 'right') + horizPad;\r\n }\r\n if (Math.round(height + vertPad) !== clientHeight) {\r\n height -= getBordersSize(styles, 'top', 'bottom') + vertPad;\r\n }\r\n }\r\n // Following steps can't be applied to the document's root element as its\r\n // client[Width/Height] properties represent viewport area of the window.\r\n // Besides, it's as well not necessary as the itself neither has\r\n // rendered scroll bars nor it can be clipped.\r\n if (!isDocumentElement(target)) {\r\n // In some browsers (only in Firefox, actually) CSS width & height\r\n // include scroll bars size which can be removed at this step as scroll\r\n // bars are the only difference between rounded dimensions + paddings\r\n // and \"client\" properties, though that is not always true in Chrome.\r\n var vertScrollbar = Math.round(width + horizPad) - clientWidth;\r\n var horizScrollbar = Math.round(height + vertPad) - clientHeight;\r\n // Chrome has a rather weird rounding of \"client\" properties.\r\n // E.g. for an element with content width of 314.2px it sometimes gives\r\n // the client width of 315px and for the width of 314.7px it may give\r\n // 314px. And it doesn't happen all the time. So just ignore this delta\r\n // as a non-relevant.\r\n if (Math.abs(vertScrollbar) !== 1) {\r\n width -= vertScrollbar;\r\n }\r\n if (Math.abs(horizScrollbar) !== 1) {\r\n height -= horizScrollbar;\r\n }\r\n }\r\n return createRectInit(paddings.left, paddings.top, width, height);\r\n}\r\n/**\r\n * Checks whether provided element is an instance of the SVGGraphicsElement.\r\n *\r\n * @param {Element} target - Element to be checked.\r\n * @returns {boolean}\r\n */\r\nvar isSVGGraphicsElement = (function () {\r\n // Some browsers, namely IE and Edge, don't have the SVGGraphicsElement\r\n // interface.\r\n if (typeof SVGGraphicsElement !== 'undefined') {\r\n return function (target) { return target instanceof getWindowOf(target).SVGGraphicsElement; };\r\n }\r\n // If it's so, then check that element is at least an instance of the\r\n // SVGElement and that it has the \"getBBox\" method.\r\n // eslint-disable-next-line no-extra-parens\r\n return function (target) { return (target instanceof getWindowOf(target).SVGElement &&\r\n typeof target.getBBox === 'function'); };\r\n})();\r\n/**\r\n * Checks whether provided element is a document element ().\r\n *\r\n * @param {Element} target - Element to be checked.\r\n * @returns {boolean}\r\n */\r\nfunction isDocumentElement(target) {\r\n return target === getWindowOf(target).document.documentElement;\r\n}\r\n/**\r\n * Calculates an appropriate content rectangle for provided html or svg element.\r\n *\r\n * @param {Element} target - Element content rectangle of which needs to be calculated.\r\n * @returns {DOMRectInit}\r\n */\r\nfunction getContentRect(target) {\r\n if (!isBrowser) {\r\n return emptyRect;\r\n }\r\n if (isSVGGraphicsElement(target)) {\r\n return getSVGContentRect(target);\r\n }\r\n return getHTMLElementContentRect(target);\r\n}\r\n/**\r\n * Creates rectangle with an interface of the DOMRectReadOnly.\r\n * Spec: https://drafts.fxtf.org/geometry/#domrectreadonly\r\n *\r\n * @param {DOMRectInit} rectInit - Object with rectangle's x/y coordinates and dimensions.\r\n * @returns {DOMRectReadOnly}\r\n */\r\nfunction createReadOnlyRect(_a) {\r\n var x = _a.x, y = _a.y, width = _a.width, height = _a.height;\r\n // If DOMRectReadOnly is available use it as a prototype for the rectangle.\r\n var Constr = typeof DOMRectReadOnly !== 'undefined' ? DOMRectReadOnly : Object;\r\n var rect = Object.create(Constr.prototype);\r\n // Rectangle's properties are not writable and non-enumerable.\r\n defineConfigurable(rect, {\r\n x: x, y: y, width: width, height: height,\r\n top: y,\r\n right: x + width,\r\n bottom: height + y,\r\n left: x\r\n });\r\n return rect;\r\n}\r\n/**\r\n * Creates DOMRectInit object based on the provided dimensions and the x/y coordinates.\r\n * Spec: https://drafts.fxtf.org/geometry/#dictdef-domrectinit\r\n *\r\n * @param {number} x - X coordinate.\r\n * @param {number} y - Y coordinate.\r\n * @param {number} width - Rectangle's width.\r\n * @param {number} height - Rectangle's height.\r\n * @returns {DOMRectInit}\r\n */\r\nfunction createRectInit(x, y, width, height) {\r\n return { x: x, y: y, width: width, height: height };\r\n}\n\n/**\r\n * Class that is responsible for computations of the content rectangle of\r\n * provided DOM element and for keeping track of it's changes.\r\n */\r\nvar ResizeObservation = /** @class */ (function () {\r\n /**\r\n * Creates an instance of ResizeObservation.\r\n *\r\n * @param {Element} target - Element to be observed.\r\n */\r\n function ResizeObservation(target) {\r\n /**\r\n * Broadcasted width of content rectangle.\r\n *\r\n * @type {number}\r\n */\r\n this.broadcastWidth = 0;\r\n /**\r\n * Broadcasted height of content rectangle.\r\n *\r\n * @type {number}\r\n */\r\n this.broadcastHeight = 0;\r\n /**\r\n * Reference to the last observed content rectangle.\r\n *\r\n * @private {DOMRectInit}\r\n */\r\n this.contentRect_ = createRectInit(0, 0, 0, 0);\r\n this.target = target;\r\n }\r\n /**\r\n * Updates content rectangle and tells whether it's width or height properties\r\n * have changed since the last broadcast.\r\n *\r\n * @returns {boolean}\r\n */\r\n ResizeObservation.prototype.isActive = function () {\r\n var rect = getContentRect(this.target);\r\n this.contentRect_ = rect;\r\n return (rect.width !== this.broadcastWidth ||\r\n rect.height !== this.broadcastHeight);\r\n };\r\n /**\r\n * Updates 'broadcastWidth' and 'broadcastHeight' properties with a data\r\n * from the corresponding properties of the last observed content rectangle.\r\n *\r\n * @returns {DOMRectInit} Last observed content rectangle.\r\n */\r\n ResizeObservation.prototype.broadcastRect = function () {\r\n var rect = this.contentRect_;\r\n this.broadcastWidth = rect.width;\r\n this.broadcastHeight = rect.height;\r\n return rect;\r\n };\r\n return ResizeObservation;\r\n}());\n\nvar ResizeObserverEntry = /** @class */ (function () {\r\n /**\r\n * Creates an instance of ResizeObserverEntry.\r\n *\r\n * @param {Element} target - Element that is being observed.\r\n * @param {DOMRectInit} rectInit - Data of the element's content rectangle.\r\n */\r\n function ResizeObserverEntry(target, rectInit) {\r\n var contentRect = createReadOnlyRect(rectInit);\r\n // According to the specification following properties are not writable\r\n // and are also not enumerable in the native implementation.\r\n //\r\n // Property accessors are not being used as they'd require to define a\r\n // private WeakMap storage which may cause memory leaks in browsers that\r\n // don't support this type of collections.\r\n defineConfigurable(this, { target: target, contentRect: contentRect });\r\n }\r\n return ResizeObserverEntry;\r\n}());\n\nvar ResizeObserverSPI = /** @class */ (function () {\r\n /**\r\n * Creates a new instance of ResizeObserver.\r\n *\r\n * @param {ResizeObserverCallback} callback - Callback function that is invoked\r\n * when one of the observed elements changes it's content dimensions.\r\n * @param {ResizeObserverController} controller - Controller instance which\r\n * is responsible for the updates of observer.\r\n * @param {ResizeObserver} callbackCtx - Reference to the public\r\n * ResizeObserver instance which will be passed to callback function.\r\n */\r\n function ResizeObserverSPI(callback, controller, callbackCtx) {\r\n /**\r\n * Collection of resize observations that have detected changes in dimensions\r\n * of elements.\r\n *\r\n * @private {Array}\r\n */\r\n this.activeObservations_ = [];\r\n /**\r\n * Registry of the ResizeObservation instances.\r\n *\r\n * @private {Map}\r\n */\r\n this.observations_ = new MapShim();\r\n if (typeof callback !== 'function') {\r\n throw new TypeError('The callback provided as parameter 1 is not a function.');\r\n }\r\n this.callback_ = callback;\r\n this.controller_ = controller;\r\n this.callbackCtx_ = callbackCtx;\r\n }\r\n /**\r\n * Starts observing provided element.\r\n *\r\n * @param {Element} target - Element to be observed.\r\n * @returns {void}\r\n */\r\n ResizeObserverSPI.prototype.observe = function (target) {\r\n if (!arguments.length) {\r\n throw new TypeError('1 argument required, but only 0 present.');\r\n }\r\n // Do nothing if current environment doesn't have the Element interface.\r\n if (typeof Element === 'undefined' || !(Element instanceof Object)) {\r\n return;\r\n }\r\n if (!(target instanceof getWindowOf(target).Element)) {\r\n throw new TypeError('parameter 1 is not of type \"Element\".');\r\n }\r\n var observations = this.observations_;\r\n // Do nothing if element is already being observed.\r\n if (observations.has(target)) {\r\n return;\r\n }\r\n observations.set(target, new ResizeObservation(target));\r\n this.controller_.addObserver(this);\r\n // Force the update of observations.\r\n this.controller_.refresh();\r\n };\r\n /**\r\n * Stops observing provided element.\r\n *\r\n * @param {Element} target - Element to stop observing.\r\n * @returns {void}\r\n */\r\n ResizeObserverSPI.prototype.unobserve = function (target) {\r\n if (!arguments.length) {\r\n throw new TypeError('1 argument required, but only 0 present.');\r\n }\r\n // Do nothing if current environment doesn't have the Element interface.\r\n if (typeof Element === 'undefined' || !(Element instanceof Object)) {\r\n return;\r\n }\r\n if (!(target instanceof getWindowOf(target).Element)) {\r\n throw new TypeError('parameter 1 is not of type \"Element\".');\r\n }\r\n var observations = this.observations_;\r\n // Do nothing if element is not being observed.\r\n if (!observations.has(target)) {\r\n return;\r\n }\r\n observations.delete(target);\r\n if (!observations.size) {\r\n this.controller_.removeObserver(this);\r\n }\r\n };\r\n /**\r\n * Stops observing all elements.\r\n *\r\n * @returns {void}\r\n */\r\n ResizeObserverSPI.prototype.disconnect = function () {\r\n this.clearActive();\r\n this.observations_.clear();\r\n this.controller_.removeObserver(this);\r\n };\r\n /**\r\n * Collects observation instances the associated element of which has changed\r\n * it's content rectangle.\r\n *\r\n * @returns {void}\r\n */\r\n ResizeObserverSPI.prototype.gatherActive = function () {\r\n var _this = this;\r\n this.clearActive();\r\n this.observations_.forEach(function (observation) {\r\n if (observation.isActive()) {\r\n _this.activeObservations_.push(observation);\r\n }\r\n });\r\n };\r\n /**\r\n * Invokes initial callback function with a list of ResizeObserverEntry\r\n * instances collected from active resize observations.\r\n *\r\n * @returns {void}\r\n */\r\n ResizeObserverSPI.prototype.broadcastActive = function () {\r\n // Do nothing if observer doesn't have active observations.\r\n if (!this.hasActive()) {\r\n return;\r\n }\r\n var ctx = this.callbackCtx_;\r\n // Create ResizeObserverEntry instance for every active observation.\r\n var entries = this.activeObservations_.map(function (observation) {\r\n return new ResizeObserverEntry(observation.target, observation.broadcastRect());\r\n });\r\n this.callback_.call(ctx, entries, ctx);\r\n this.clearActive();\r\n };\r\n /**\r\n * Clears the collection of active observations.\r\n *\r\n * @returns {void}\r\n */\r\n ResizeObserverSPI.prototype.clearActive = function () {\r\n this.activeObservations_.splice(0);\r\n };\r\n /**\r\n * Tells whether observer has active observations.\r\n *\r\n * @returns {boolean}\r\n */\r\n ResizeObserverSPI.prototype.hasActive = function () {\r\n return this.activeObservations_.length > 0;\r\n };\r\n return ResizeObserverSPI;\r\n}());\n\n// Registry of internal observers. If WeakMap is not available use current shim\r\n// for the Map collection as it has all required methods and because WeakMap\r\n// can't be fully polyfilled anyway.\r\nvar observers = typeof WeakMap !== 'undefined' ? new WeakMap() : new MapShim();\r\n/**\r\n * ResizeObserver API. Encapsulates the ResizeObserver SPI implementation\r\n * exposing only those methods and properties that are defined in the spec.\r\n */\r\nvar ResizeObserver = /** @class */ (function () {\r\n /**\r\n * Creates a new instance of ResizeObserver.\r\n *\r\n * @param {ResizeObserverCallback} callback - Callback that is invoked when\r\n * dimensions of the observed elements change.\r\n */\r\n function ResizeObserver(callback) {\r\n if (!(this instanceof ResizeObserver)) {\r\n throw new TypeError('Cannot call a class as a function.');\r\n }\r\n if (!arguments.length) {\r\n throw new TypeError('1 argument required, but only 0 present.');\r\n }\r\n var controller = ResizeObserverController.getInstance();\r\n var observer = new ResizeObserverSPI(callback, controller, this);\r\n observers.set(this, observer);\r\n }\r\n return ResizeObserver;\r\n}());\r\n// Expose public methods of ResizeObserver.\r\n[\r\n 'observe',\r\n 'unobserve',\r\n 'disconnect'\r\n].forEach(function (method) {\r\n ResizeObserver.prototype[method] = function () {\r\n var _a;\r\n return (_a = observers.get(this))[method].apply(_a, arguments);\r\n };\r\n});\n\nvar index = (function () {\r\n // Export existing implementation if available.\r\n if (typeof global$1.ResizeObserver !== 'undefined') {\r\n return global$1.ResizeObserver;\r\n }\r\n return ResizeObserver;\r\n})();\n\nexport default index;\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ResizeObserver from \"resize-observer-polyfill\"\nimport {\n NEVER,\n Observable,\n Subject,\n defer,\n filter,\n finalize,\n map,\n merge,\n of,\n shareReplay,\n startWith,\n switchMap,\n tap\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Element offset\n */\nexport interface ElementSize {\n width: number /* Element width */\n height: number /* Element height */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Resize observer entry subject\n */\nconst entry$ = new Subject()\n\n/**\n * Resize observer observable\n *\n * This observable will create a `ResizeObserver` on the first subscription\n * and will automatically terminate it when there are no more subscribers.\n * It's quite important to centralize observation in a single `ResizeObserver`,\n * as the performance difference can be quite dramatic, as the link shows.\n *\n * @see https://bit.ly/3iIYfEm - Google Groups on performance\n */\nconst observer$ = defer(() => of(\n new ResizeObserver(entries => {\n for (const entry of entries)\n entry$.next(entry)\n })\n))\n .pipe(\n switchMap(observer => merge(NEVER, of(observer))\n .pipe(\n finalize(() => observer.disconnect())\n )\n ),\n shareReplay(1)\n )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element size\n *\n * @param el - Element\n *\n * @returns Element size\n */\nexport function getElementSize(\n el: HTMLElement\n): ElementSize {\n return {\n width: el.offsetWidth,\n height: el.offsetHeight\n }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element size\n *\n * This function returns an observable that subscribes to a single internal\n * instance of `ResizeObserver` upon subscription, and emit resize events until\n * termination. Note that this function should not be called with the same\n * element twice, as the first unsubscription will terminate observation.\n *\n * Sadly, we can't use the `DOMRect` objects returned by the observer, because\n * we need the emitted values to be consistent with `getElementSize`, which will\n * return the used values (rounded) and not actual values (unrounded). Thus, we\n * use the `offset*` properties. See the linked GitHub issue.\n *\n * @see https://bit.ly/3m0k3he - GitHub issue\n *\n * @param el - Element\n *\n * @returns Element size observable\n */\nexport function watchElementSize(\n el: HTMLElement\n): Observable {\n return observer$\n .pipe(\n tap(observer => observer.observe(el)),\n switchMap(observer => entry$\n .pipe(\n filter(({ target }) => target === el),\n finalize(() => observer.unobserve(el)),\n map(() => getElementSize(el))\n )\n ),\n startWith(getElementSize(el))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ElementSize } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element content size (= scroll width and height)\n *\n * @param el - Element\n *\n * @returns Element content size\n */\nexport function getElementContentSize(\n el: HTMLElement\n): ElementSize {\n return {\n width: el.scrollWidth,\n height: el.scrollHeight\n }\n}\n\n/**\n * Retrieve the overflowing container of an element, if any\n *\n * @param el - Element\n *\n * @returns Overflowing container or nothing\n */\nexport function getElementContainer(\n el: HTMLElement\n): HTMLElement | undefined {\n let parent = el.parentElement\n while (parent)\n if (\n el.scrollWidth <= parent.scrollWidth &&\n el.scrollHeight <= parent.scrollHeight\n )\n parent = (el = parent).parentElement\n else\n break\n\n /* Return overflowing container */\n return parent ? el : undefined\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n NEVER,\n Observable,\n Subject,\n defer,\n distinctUntilChanged,\n filter,\n finalize,\n map,\n merge,\n of,\n shareReplay,\n switchMap,\n tap\n} from \"rxjs\"\n\nimport {\n getElementContentSize,\n getElementSize,\n watchElementContentOffset\n} from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Intersection observer entry subject\n */\nconst entry$ = new Subject()\n\n/**\n * Intersection observer observable\n *\n * This observable will create an `IntersectionObserver` on first subscription\n * and will automatically terminate it when there are no more subscribers.\n *\n * @see https://bit.ly/3iIYfEm - Google Groups on performance\n */\nconst observer$ = defer(() => of(\n new IntersectionObserver(entries => {\n for (const entry of entries)\n entry$.next(entry)\n }, {\n threshold: 0\n })\n))\n .pipe(\n switchMap(observer => merge(NEVER, of(observer))\n .pipe(\n finalize(() => observer.disconnect())\n )\n ),\n shareReplay(1)\n )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element visibility\n *\n * @param el - Element\n *\n * @returns Element visibility observable\n */\nexport function watchElementVisibility(\n el: HTMLElement\n): Observable {\n return observer$\n .pipe(\n tap(observer => observer.observe(el)),\n switchMap(observer => entry$\n .pipe(\n filter(({ target }) => target === el),\n finalize(() => observer.unobserve(el)),\n map(({ isIntersecting }) => isIntersecting)\n )\n )\n )\n}\n\n/**\n * Watch element boundary\n *\n * This function returns an observable which emits whether the bottom content\n * boundary (= scroll offset) of an element is within a certain threshold.\n *\n * @param el - Element\n * @param threshold - Threshold\n *\n * @returns Element boundary observable\n */\nexport function watchElementBoundary(\n el: HTMLElement, threshold = 16\n): Observable {\n return watchElementContentOffset(el)\n .pipe(\n map(({ y }) => {\n const visible = getElementSize(el)\n const content = getElementContentSize(el)\n return y >= (\n content.height - visible.height - threshold\n )\n }),\n distinctUntilChanged()\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n fromEvent,\n map,\n startWith\n} from \"rxjs\"\n\nimport { getElement } from \"../element\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Toggle\n */\nexport type Toggle =\n | \"drawer\" /* Toggle for drawer */\n | \"search\" /* Toggle for search */\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Toggle map\n */\nconst toggles: Record = {\n drawer: getElement(\"[data-md-toggle=drawer]\"),\n search: getElement(\"[data-md-toggle=search]\")\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve the value of a toggle\n *\n * @param name - Toggle\n *\n * @returns Toggle value\n */\nexport function getToggle(name: Toggle): boolean {\n return toggles[name].checked\n}\n\n/**\n * Set toggle\n *\n * Simulating a click event seems to be the most cross-browser compatible way\n * of changing the value while also emitting a `change` event. Before, Material\n * used `CustomEvent` to programmatically change the value of a toggle, but this\n * is a much simpler and cleaner solution which doesn't require a polyfill.\n *\n * @param name - Toggle\n * @param value - Toggle value\n */\nexport function setToggle(name: Toggle, value: boolean): void {\n if (toggles[name].checked !== value)\n toggles[name].click()\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch toggle\n *\n * @param name - Toggle\n *\n * @returns Toggle value observable\n */\nexport function watchToggle(name: Toggle): Observable {\n const el = toggles[name]\n return fromEvent(el, \"change\")\n .pipe(\n map(() => el.checked),\n startWith(el.checked)\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n filter,\n fromEvent,\n map,\n share\n} from \"rxjs\"\n\nimport { getActiveElement } from \"../element\"\nimport { getToggle } from \"../toggle\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Keyboard mode\n */\nexport type KeyboardMode =\n | \"global\" /* Global */\n | \"search\" /* Search is open */\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Keyboard\n */\nexport interface Keyboard {\n mode: KeyboardMode /* Keyboard mode */\n type: string /* Key type */\n claim(): void /* Key claim */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Check whether an element may receive keyboard input\n *\n * @param el - Element\n * @param type - Key type\n *\n * @returns Test result\n */\nfunction isSusceptibleToKeyboard(\n el: HTMLElement, type: string\n): boolean {\n switch (el.constructor) {\n\n /* Input elements */\n case HTMLInputElement:\n /* @ts-expect-error - omit unnecessary type cast */\n if (el.type === \"radio\")\n return /^Arrow/.test(type)\n else\n return true\n\n /* Select element and textarea */\n case HTMLSelectElement:\n case HTMLTextAreaElement:\n return true\n\n /* Everything else */\n default:\n return el.isContentEditable\n }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch keyboard\n *\n * @returns Keyboard observable\n */\nexport function watchKeyboard(): Observable {\n return fromEvent(window, \"keydown\")\n .pipe(\n filter(ev => !(ev.metaKey || ev.ctrlKey)),\n map(ev => ({\n mode: getToggle(\"search\") ? \"search\" : \"global\",\n type: ev.key,\n claim() {\n ev.preventDefault()\n ev.stopPropagation()\n }\n } as Keyboard)),\n filter(({ mode, type }) => {\n if (mode === \"global\") {\n const active = getActiveElement()\n if (typeof active !== \"undefined\")\n return !isSusceptibleToKeyboard(active, type)\n }\n return true\n }),\n share()\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Subject } from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve location\n *\n * This function returns a `URL` object (and not `Location`) to normalize the\n * typings across the application. Furthermore, locations need to be tracked\n * without setting them and `Location` is a singleton which represents the\n * current location.\n *\n * @returns URL\n */\nexport function getLocation(): URL {\n return new URL(location.href)\n}\n\n/**\n * Set location\n *\n * @param url - URL to change to\n */\nexport function setLocation(url: URL): void {\n location.href = url.href\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch location\n *\n * @returns Location subject\n */\nexport function watchLocation(): Subject {\n return new Subject()\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { JSX as JSXInternal } from \"preact\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * HTML attributes\n */\ntype Attributes =\n & JSXInternal.HTMLAttributes\n & JSXInternal.SVGAttributes\n & Record\n\n/**\n * Child element\n */\ntype Child =\n | HTMLElement\n | Text\n | string\n | number\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Append a child node to an element\n *\n * @param el - Element\n * @param child - Child node(s)\n */\nfunction appendChild(el: HTMLElement, child: Child | Child[]): void {\n\n /* Handle primitive types (including raw HTML) */\n if (typeof child === \"string\" || typeof child === \"number\") {\n el.innerHTML += child.toString()\n\n /* Handle nodes */\n } else if (child instanceof Node) {\n el.appendChild(child)\n\n /* Handle nested children */\n } else if (Array.isArray(child)) {\n for (const node of child)\n appendChild(el, node)\n }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * JSX factory\n *\n * @template T - Element type\n *\n * @param tag - HTML tag\n * @param attributes - HTML attributes\n * @param children - Child elements\n *\n * @returns Element\n */\nexport function h(\n tag: T, attributes?: Attributes | null, ...children: Child[]\n): HTMLElementTagNameMap[T]\n\nexport function h(\n tag: string, attributes?: Attributes | null, ...children: Child[]\n): T\n\nexport function h(\n tag: string, attributes?: Attributes | null, ...children: Child[]\n): T {\n const el = document.createElement(tag)\n\n /* Set attributes, if any */\n if (attributes)\n for (const attr of Object.keys(attributes)) {\n if (typeof attributes[attr] === \"undefined\")\n continue\n\n /* Set default attribute or boolean */\n if (typeof attributes[attr] !== \"boolean\")\n el.setAttribute(attr, attributes[attr])\n else\n el.setAttribute(attr, \"\")\n }\n\n /* Append child nodes */\n for (const child of children)\n appendChild(el, child)\n\n /* Return element */\n return el as T\n}\n\n/* ----------------------------------------------------------------------------\n * Namespace\n * ------------------------------------------------------------------------- */\n\nexport declare namespace h {\n namespace JSX {\n type Element = HTMLElement\n type IntrinsicElements = JSXInternal.IntrinsicElements\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Truncate a string after the given number of characters\n *\n * This is not a very reasonable approach, since the summaries kind of suck.\n * It would be better to create something more intelligent, highlighting the\n * search occurrences and making a better summary out of it, but this note was\n * written three years ago, so who knows if we'll ever fix it.\n *\n * @param value - Value to be truncated\n * @param n - Number of characters\n *\n * @returns Truncated value\n */\nexport function truncate(value: string, n: number): string {\n let i = n\n if (value.length > i) {\n while (value[i] !== \" \" && --i > 0) { /* keep eating */ }\n return `${value.substring(0, i)}...`\n }\n return value\n}\n\n/**\n * Round a number for display with repository facts\n *\n * This is a reverse-engineered version of GitHub's weird rounding algorithm\n * for stars, forks and all other numbers. While all numbers below `1,000` are\n * returned as-is, bigger numbers are converted to fixed numbers:\n *\n * - `1,049` => `1k`\n * - `1,050` => `1.1k`\n * - `1,949` => `1.9k`\n * - `1,950` => `2k`\n *\n * @param value - Original value\n *\n * @returns Rounded value\n */\nexport function round(value: number): string {\n if (value > 999) {\n const digits = +((value - 950) % 1000 > 99)\n return `${((value + 0.000001) / 1000).toFixed(digits)}k`\n } else {\n return value.toString()\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n filter,\n fromEvent,\n map,\n shareReplay,\n startWith\n} from \"rxjs\"\n\nimport { getOptionalElement } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve location hash\n *\n * @returns Location hash\n */\nexport function getLocationHash(): string {\n return location.hash.substring(1)\n}\n\n/**\n * Set location hash\n *\n * Setting a new fragment identifier via `location.hash` will have no effect\n * if the value doesn't change. When a new fragment identifier is set, we want\n * the browser to target the respective element at all times, which is why we\n * use this dirty little trick.\n *\n * @param hash - Location hash\n */\nexport function setLocationHash(hash: string): void {\n const el = h(\"a\", { href: hash })\n el.addEventListener(\"click\", ev => ev.stopPropagation())\n el.click()\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch location hash\n *\n * @returns Location hash observable\n */\nexport function watchLocationHash(): Observable {\n return fromEvent(window, \"hashchange\")\n .pipe(\n map(getLocationHash),\n startWith(getLocationHash()),\n filter(hash => hash.length > 0),\n shareReplay(1)\n )\n}\n\n/**\n * Watch location target\n *\n * @returns Location target observable\n */\nexport function watchLocationTarget(): Observable {\n return watchLocationHash()\n .pipe(\n map(id => getOptionalElement(`[id=\"${id}\"]`)!),\n filter(el => typeof el !== \"undefined\")\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n fromEvent,\n fromEventPattern,\n map,\n merge,\n startWith,\n switchMap\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch media query\n *\n * Note that although `MediaQueryList.addListener` is deprecated we have to\n * use it, because it's the only way to ensure proper downward compatibility.\n *\n * @see https://bit.ly/3dUBH2m - GitHub issue\n *\n * @param query - Media query\n *\n * @returns Media observable\n */\nexport function watchMedia(query: string): Observable {\n const media = matchMedia(query)\n return fromEventPattern(next => (\n media.addListener(() => next(media.matches))\n ))\n .pipe(\n startWith(media.matches)\n )\n}\n\n/**\n * Watch print mode\n *\n * @returns Print observable\n */\nexport function watchPrint(): Observable {\n const media = matchMedia(\"print\")\n return merge(\n fromEvent(window, \"beforeprint\").pipe(map(() => true)),\n fromEvent(window, \"afterprint\").pipe(map(() => false))\n )\n .pipe(\n startWith(media.matches)\n )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Toggle an observable with a media observable\n *\n * @template T - Data type\n *\n * @param query$ - Media observable\n * @param factory - Observable factory\n *\n * @returns Toggled observable\n */\nexport function at(\n query$: Observable, factory: () => Observable\n): Observable {\n return query$\n .pipe(\n switchMap(active => active ? factory() : EMPTY)\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n catchError,\n from,\n map,\n of,\n shareReplay,\n switchMap,\n throwError\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch the given URL\n *\n * If the request fails (e.g. when dispatched from `file://` locations), the\n * observable will complete without emitting a value.\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Response observable\n */\nexport function request(\n url: URL | string, options: RequestInit = { credentials: \"same-origin\" }\n): Observable {\n return from(fetch(`${url}`, options))\n .pipe(\n catchError(() => EMPTY),\n switchMap(res => res.status !== 200\n ? throwError(() => new Error(res.statusText))\n : of(res)\n )\n )\n}\n\n/**\n * Fetch JSON from the given URL\n *\n * @template T - Data type\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Data observable\n */\nexport function requestJSON(\n url: URL | string, options?: RequestInit\n): Observable {\n return request(url, options)\n .pipe(\n switchMap(res => res.json()),\n shareReplay(1)\n )\n}\n\n/**\n * Fetch XML from the given URL\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Data observable\n */\nexport function requestXML(\n url: URL | string, options?: RequestInit\n): Observable {\n const dom = new DOMParser()\n return request(url, options)\n .pipe(\n switchMap(res => res.text()),\n map(res => dom.parseFromString(res, \"text/xml\")),\n shareReplay(1)\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n defer,\n finalize,\n fromEvent,\n map,\n merge,\n switchMap,\n take,\n throwError\n} from \"rxjs\"\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create and load a `script` element\n *\n * This function returns an observable that will emit when the script was\n * successfully loaded, or throw an error if it didn't.\n *\n * @param src - Script URL\n *\n * @returns Script observable\n */\nexport function watchScript(src: string): Observable {\n const script = h(\"script\", { src })\n return defer(() => {\n document.head.appendChild(script)\n return merge(\n fromEvent(script, \"load\"),\n fromEvent(script, \"error\")\n .pipe(\n switchMap(() => (\n throwError(() => new ReferenceError(`Invalid script: ${src}`))\n ))\n )\n )\n .pipe(\n map(() => undefined),\n finalize(() => document.head.removeChild(script)),\n take(1)\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n fromEvent,\n map,\n merge,\n startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport offset\n */\nexport interface ViewportOffset {\n x: number /* Horizontal offset */\n y: number /* Vertical offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve viewport offset\n *\n * On iOS Safari, viewport offset can be negative due to overflow scrolling.\n * As this may induce strange behaviors downstream, we'll just limit it to 0.\n *\n * @returns Viewport offset\n */\nexport function getViewportOffset(): ViewportOffset {\n return {\n x: Math.max(0, scrollX),\n y: Math.max(0, scrollY)\n }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport offset\n *\n * @returns Viewport offset observable\n */\nexport function watchViewportOffset(): Observable {\n return merge(\n fromEvent(window, \"scroll\", { passive: true }),\n fromEvent(window, \"resize\", { passive: true })\n )\n .pipe(\n map(getViewportOffset),\n startWith(getViewportOffset())\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n fromEvent,\n map,\n startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport size\n */\nexport interface ViewportSize {\n width: number /* Viewport width */\n height: number /* Viewport height */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve viewport size\n *\n * @returns Viewport size\n */\nexport function getViewportSize(): ViewportSize {\n return {\n width: innerWidth,\n height: innerHeight\n }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport size\n *\n * @returns Viewport size observable\n */\nexport function watchViewportSize(): Observable {\n return fromEvent(window, \"resize\", { passive: true })\n .pipe(\n map(getViewportSize),\n startWith(getViewportSize())\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n combineLatest,\n map,\n shareReplay\n} from \"rxjs\"\n\nimport {\n ViewportOffset,\n watchViewportOffset\n} from \"../offset\"\nimport {\n ViewportSize,\n watchViewportSize\n} from \"../size\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport\n */\nexport interface Viewport {\n offset: ViewportOffset /* Viewport offset */\n size: ViewportSize /* Viewport size */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport\n *\n * @returns Viewport observable\n */\nexport function watchViewport(): Observable {\n return combineLatest([\n watchViewportOffset(),\n watchViewportSize()\n ])\n .pipe(\n map(([offset, size]) => ({ offset, size })),\n shareReplay(1)\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n combineLatest,\n distinctUntilKeyChanged,\n map\n} from \"rxjs\"\n\nimport { Header } from \"~/components\"\n\nimport { getElementOffset } from \"../../element\"\nimport { Viewport } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
/* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport relative to element\n *\n * @param el - Element\n * @param options - Options\n *\n * @returns Viewport observable\n */\nexport function watchViewportAt(\n el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable {\n const size$ = viewport$\n .pipe(\n distinctUntilKeyChanged(\"size\")\n )\n\n /* Compute element offset */\n const offset$ = combineLatest([size$, header$])\n .pipe(\n map(() => getElementOffset(el))\n )\n\n /* Compute relative viewport, return hot observable */\n return combineLatest([header$, viewport$, offset$])\n .pipe(\n map(([{ height }, { offset, size }, { x, y }]) => ({\n offset: {\n x: offset.x - x,\n y: offset.y - y + height\n },\n size\n }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n fromEvent,\n map,\n share,\n switchMap,\n tap,\n throttle\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Worker message\n */\nexport interface WorkerMessage {\n type: unknown /* Message type */\n data?: unknown /* Message data */\n}\n\n/**\n * Worker handler\n *\n * @template T - Message type\n */\nexport interface WorkerHandler<\n T extends WorkerMessage\n> {\n tx$: Subject /* Message transmission subject */\n rx$: Observable /* Message receive observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n *\n * @template T - Worker message type\n */\ninterface WatchOptions {\n tx$: Observable /* Message transmission observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch a web worker\n *\n * This function returns an observable that sends all values emitted by the\n * message observable to the web worker. Web worker communication is expected\n * to be bidirectional (request-response) and synchronous. Messages that are\n * emitted during a pending request are throttled, the last one is emitted.\n *\n * @param worker - Web worker\n * @param options - Options\n *\n * @returns Worker message observable\n */\nexport function watchWorker(\n worker: Worker, { tx$ }: WatchOptions\n): Observable {\n\n /* Intercept messages from worker-like objects */\n const rx$ = fromEvent(worker, \"message\")\n .pipe(\n map(({ data }) => data as T)\n )\n\n /* Send and receive messages, return hot observable */\n return tx$\n .pipe(\n throttle(() => rx$, { leading: true, trailing: true }),\n tap(message => worker.postMessage(message)),\n switchMap(() => rx$),\n share()\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { getElement, getLocation } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Feature flag\n */\nexport type Flag =\n | \"announce.dismiss\" /* Dismissable announcement bar */\n | \"content.code.annotate\" /* Code annotations */\n | \"content.lazy\" /* Lazy content elements */\n | \"content.tabs.link\" /* Link content tabs */\n | \"header.autohide\" /* Hide header */\n | \"navigation.expand\" /* Automatic expansion */\n | \"navigation.indexes\" /* Section pages */\n | \"navigation.instant\" /* Instant loading */\n | \"navigation.sections\" /* Section navigation */\n | \"navigation.tabs\" /* Tabs navigation */\n | \"navigation.tabs.sticky\" /* Tabs navigation (sticky) */\n | \"navigation.top\" /* Back-to-top button */\n | \"navigation.tracking\" /* Anchor tracking */\n | \"search.highlight\" /* Search highlighting */\n | \"search.share\" /* Search sharing */\n | \"search.suggest\" /* Search suggestions */\n | \"toc.follow\" /* Following table of contents */\n | \"toc.integrate\" /* Integrated table of contents */\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Translation\n */\nexport type Translation =\n | \"clipboard.copy\" /* Copy to clipboard */\n | \"clipboard.copied\" /* Copied to clipboard */\n | \"search.config.lang\" /* Search language */\n | \"search.config.pipeline\" /* Search pipeline */\n | \"search.config.separator\" /* Search separator */\n | \"search.placeholder\" /* Search */\n | \"search.result.placeholder\" /* Type to start searching */\n | \"search.result.none\" /* No matching documents */\n | \"search.result.one\" /* 1 matching document */\n | \"search.result.other\" /* # matching documents */\n | \"search.result.more.one\" /* 1 more on this page */\n | \"search.result.more.other\" /* # more on this page */\n | \"search.result.term.missing\" /* Missing */\n | \"select.version.title\" /* Version selector */\n\n/**\n * Translations\n */\nexport type Translations = Record\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Versioning\n */\nexport interface Versioning {\n provider: \"mike\" /* Version provider */\n default?: string /* Default version */\n}\n\n/**\n * Configuration\n */\nexport interface Config {\n base: string /* Base URL */\n features: Flag[] /* Feature flags */\n translations: Translations /* Translations */\n search: string /* Search worker URL */\n tags?: Record /* Tags mapping */\n version?: Versioning /* Versioning */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve global configuration and make base URL absolute\n */\nconst script = getElement(\"#__config\")\nconst config: Config = JSON.parse(script.textContent!)\nconfig.base = `${new URL(config.base, getLocation())}`\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve global configuration\n *\n * @returns Global configuration\n */\nexport function configuration(): Config {\n return config\n}\n\n/**\n * Check whether a feature flag is enabled\n *\n * @param flag - Feature flag\n *\n * @returns Test result\n */\nexport function feature(flag: Flag): boolean {\n return config.features.includes(flag)\n}\n\n/**\n * Retrieve the translation for the given key\n *\n * @param key - Key to be translated\n * @param value - Positional value, if any\n *\n * @returns Translation\n */\nexport function translation(\n key: Translation, value?: string | number\n): string {\n return typeof value !== \"undefined\"\n ? config.translations[key].replace(\"#\", value.toString())\n : config.translations[key]\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { getElement, getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Component type\n */\nexport type ComponentType =\n | \"announce\" /* Announcement bar */\n | \"container\" /* Container */\n | \"consent\" /* Consent */\n | \"content\" /* Content */\n | \"dialog\" /* Dialog */\n | \"header\" /* Header */\n | \"header-title\" /* Header title */\n | \"header-topic\" /* Header topic */\n | \"main\" /* Main area */\n | \"outdated\" /* Version warning */\n | \"palette\" /* Color palette */\n | \"search\" /* Search */\n | \"search-query\" /* Search input */\n | \"search-result\" /* Search results */\n | \"search-share\" /* Search sharing */\n | \"search-suggest\" /* Search suggestions */\n | \"sidebar\" /* Sidebar */\n | \"skip\" /* Skip link */\n | \"source\" /* Repository information */\n | \"tabs\" /* Navigation tabs */\n | \"toc\" /* Table of contents */\n | \"top\" /* Back-to-top button */\n\n/**\n * Component\n *\n * @template T - Component type\n * @template U - Reference type\n */\nexport type Component<\n T extends {} = {},\n U extends HTMLElement = HTMLElement\n> =\n T & {\n ref: U /* Component reference */\n }\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Component type map\n */\ninterface ComponentTypeMap {\n \"announce\": HTMLElement /* Announcement bar */\n \"container\": HTMLElement /* Container */\n \"consent\": HTMLElement /* Consent */\n \"content\": HTMLElement /* Content */\n \"dialog\": HTMLElement /* Dialog */\n \"header\": HTMLElement /* Header */\n \"header-title\": HTMLElement /* Header title */\n \"header-topic\": HTMLElement /* Header topic */\n \"main\": HTMLElement /* Main area */\n \"outdated\": HTMLElement /* Version warning */\n \"palette\": HTMLElement /* Color palette */\n \"search\": HTMLElement /* Search */\n \"search-query\": HTMLInputElement /* Search input */\n \"search-result\": HTMLElement /* Search results */\n \"search-share\": HTMLAnchorElement /* Search sharing */\n \"search-suggest\": HTMLElement /* Search suggestions */\n \"sidebar\": HTMLElement /* Sidebar */\n \"skip\": HTMLAnchorElement /* Skip link */\n \"source\": HTMLAnchorElement /* Repository information */\n \"tabs\": HTMLElement /* Navigation tabs */\n \"toc\": HTMLElement /* Table of contents */\n \"top\": HTMLAnchorElement /* Back-to-top button */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve the element for a given component or throw a reference error\n *\n * @template T - Component type\n *\n * @param type - Component type\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getComponentElement(\n type: T, node: ParentNode = document\n): ComponentTypeMap[T] {\n return getElement(`[data-md-component=${type}]`, node)\n}\n\n/**\n * Retrieve all elements for a given component\n *\n * @template T - Component type\n *\n * @param type - Component type\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getComponentElements(\n type: T, node: ParentNode = document\n): ComponentTypeMap[T][] {\n return getElements(`[data-md-component=${type}]`, node)\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n Subject,\n defer,\n finalize,\n fromEvent,\n map,\n startWith,\n tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { getElement } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Announcement bar\n */\nexport interface Announce {\n hash: number /* Content hash */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch announcement bar\n *\n * @param el - Announcement bar element\n *\n * @returns Announcement bar observable\n */\nexport function watchAnnounce(\n el: HTMLElement\n): Observable {\n const button = getElement(\".md-typeset > :first-child\", el)\n return fromEvent(button, \"click\", { once: true })\n .pipe(\n map(() => getElement(\".md-typeset\", el)),\n map(content => ({ hash: __md_hash(content.innerHTML) }))\n )\n}\n\n/**\n * Mount announcement bar\n *\n * @param el - Announcement bar element\n *\n * @returns Announcement bar component observable\n */\nexport function mountAnnounce(\n el: HTMLElement\n): Observable> {\n if (!feature(\"announce.dismiss\") || !el.childElementCount)\n return EMPTY\n\n /* Mount component on subscription */\n return defer(() => {\n const push$ = new Subject()\n push$\n .pipe(\n startWith({ hash: __md_get(\"__announce\") })\n )\n .subscribe(({ hash }) => {\n if (hash && hash === (__md_get(\"__announce\") ?? hash)) {\n el.hidden = true\n\n /* Persist preference in local storage */\n __md_set(\"__announce\", hash)\n }\n })\n\n /* Create and return component */\n return watchAnnounce(el)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n finalize,\n map,\n tap\n} from \"rxjs\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Consent\n */\nexport interface Consent {\n hidden: boolean /* Consent is hidden */\n}\n\n/**\n * Consent defaults\n */\nexport interface ConsentDefaults {\n analytics?: boolean /* Consent for Analytics */\n github?: boolean /* Consent for GitHub */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n target$: Observable /* Target observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n target$: Observable /* Target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch consent\n *\n * @param el - Consent element\n * @param options - Options\n *\n * @returns Consent observable\n */\nexport function watchConsent(\n el: HTMLElement, { target$ }: WatchOptions\n): Observable {\n return target$\n .pipe(\n map(target => ({ hidden: target !== el }))\n )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount consent\n *\n * @param el - Consent element\n * @param options - Options\n *\n * @returns Consent component observable\n */\nexport function mountConsent(\n el: HTMLElement, options: MountOptions\n): Observable> {\n const internal$ = new Subject()\n internal$.subscribe(({ hidden }) => {\n el.hidden = hidden\n })\n\n /* Create and return component */\n return watchConsent(el, options)\n .pipe(\n tap(state => internal$.next(state)),\n finalize(() => internal$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ClipboardJS from \"clipboard\"\nimport {\n EMPTY,\n Observable,\n Subject,\n defer,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n filter,\n finalize,\n map,\n mergeWith,\n switchMap,\n take,\n tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n getElementContentSize,\n watchElementSize,\n watchElementVisibility\n} from \"~/browser\"\nimport { renderClipboardButton } from \"~/templates\"\n\nimport { Component } from \"../../../_\"\nimport {\n Annotation,\n mountAnnotationList\n} from \"../../annotation\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Code block\n */\nexport interface CodeBlock {\n scrollable: boolean /* Code block overflows */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n target$: Observable /* Location target observable */\n print$: Observable /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Global sequence number for code blocks\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find candidate list element directly following a code block\n *\n * @param el - Code block element\n *\n * @returns List element or nothing\n */\nfunction findCandidateList(el: HTMLElement): HTMLElement | undefined {\n if (el.nextElementSibling) {\n const sibling = el.nextElementSibling as HTMLElement\n if (sibling.tagName === \"OL\")\n return sibling\n\n /* Skip empty paragraphs - see https://bit.ly/3r4ZJ2O */\n else if (sibling.tagName === \"P\" && !sibling.children.length)\n return findCandidateList(sibling)\n }\n\n /* Everything else */\n return undefined\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch code block\n *\n * This function monitors size changes of the viewport, as well as switches of\n * content tabs with embedded code blocks, as both may trigger overflow.\n *\n * @param el - Code block element\n *\n * @returns Code block observable\n */\nexport function watchCodeBlock(\n el: HTMLElement\n): Observable {\n return watchElementSize(el)\n .pipe(\n map(({ width }) => {\n const content = getElementContentSize(el)\n return {\n scrollable: content.width > width\n }\n }),\n distinctUntilKeyChanged(\"scrollable\")\n )\n}\n\n/**\n * Mount code block\n *\n * This function ensures that an overflowing code block is focusable through\n * keyboard, so it can be scrolled without a mouse to improve on accessibility.\n * Furthermore, if code annotations are enabled, they are mounted if and only\n * if the code block is currently visible, e.g., not in a hidden content tab.\n *\n * Note that code blocks may be mounted eagerly or lazily. If they're mounted\n * lazily (on first visibility), code annotation anchor links will not work,\n * as they are evaluated on initial page load, and code annotations in general\n * might feel a little bumpier.\n *\n * @param el - Code block element\n * @param options - Options\n *\n * @returns Code block and annotation component observable\n */\nexport function mountCodeBlock(\n el: HTMLElement, options: MountOptions\n): Observable> {\n const { matches: hover } = matchMedia(\"(hover)\")\n\n /* Defer mounting of code block - see https://bit.ly/3vHVoVD */\n const factory$ = defer(() => {\n const push$ = new Subject()\n push$.subscribe(({ scrollable }) => {\n if (scrollable && hover)\n el.setAttribute(\"tabindex\", \"0\")\n else\n el.removeAttribute(\"tabindex\")\n })\n\n /* Render button for Clipboard.js integration */\n if (ClipboardJS.isSupported()) {\n const parent = el.closest(\"pre\")!\n parent.id = `__code_${++sequence}`\n parent.insertBefore(\n renderClipboardButton(parent.id),\n el\n )\n }\n\n /* Handle code annotations */\n const container = el.closest(\".highlight\")\n if (container instanceof HTMLElement) {\n const list = findCandidateList(container)\n\n /* Mount code annotations, if enabled */\n if (typeof list !== \"undefined\" && (\n container.classList.contains(\"annotate\") ||\n feature(\"content.code.annotate\")\n )) {\n const annotations$ = mountAnnotationList(list, el, options)\n\n /* Create and return component */\n return watchCodeBlock(el)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state })),\n mergeWith(\n watchElementSize(container)\n .pipe(\n map(({ width, height }) => width && height),\n distinctUntilChanged(),\n switchMap(active => active ? annotations$ : EMPTY)\n )\n )\n )\n }\n }\n\n /* Create and return component */\n return watchCodeBlock(el)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n\n /* Mount code block lazily */\n if (feature(\"content.lazy\"))\n return watchElementVisibility(el)\n .pipe(\n filter(visible => visible),\n take(1),\n switchMap(() => factory$)\n )\n\n /* Mount code block */\n return factory$\n}\n", "/*\n * Copyright (c) 2016-2021 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a tooltip\n *\n * @param id - Tooltip identifier\n *\n * @returns Element\n */\nexport function renderTooltip(id?: string): HTMLElement {\n return (\n
\n
\n
\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\nimport { renderTooltip } from \"../tooltip\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render an annotation\n *\n * @param id - Annotation identifier\n * @param prefix - Tooltip identifier prefix\n *\n * @returns Element\n */\nexport function renderAnnotation(\n id: string | number, prefix?: string\n): HTMLElement {\n prefix = prefix ? `${prefix}_annotation_${id}` : undefined\n\n /* Render tooltip with anchor, if given */\n if (prefix) {\n const anchor = prefix ? `#${prefix}` : undefined\n return (\n \n )\n } else {\n return (\n \n )\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { translation } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a 'copy-to-clipboard' button\n *\n * @param id - Unique identifier\n *\n * @returns Element\n */\nexport function renderClipboardButton(id: string): HTMLElement {\n return (\n code`}\n >\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ComponentChild } from \"preact\"\n\nimport { configuration, feature, translation } from \"~/_\"\nimport {\n SearchDocument,\n SearchMetadata,\n SearchResultItem\n} from \"~/integrations/search\"\nimport { h, truncate } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Render flag\n */\nconst enum Flag {\n TEASER = 1, /* Render teaser */\n PARENT = 2 /* Render as parent */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper function\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a search document\n *\n * @param document - Search document\n * @param flag - Render flags\n *\n * @returns Element\n */\nfunction renderSearchDocument(\n document: SearchDocument & SearchMetadata, flag: Flag\n): HTMLElement {\n const parent = flag & Flag.PARENT\n const teaser = flag & Flag.TEASER\n\n /* Render missing query terms */\n const missing = Object.keys(document.terms)\n .filter(key => !document.terms[key])\n .reduce((list, key) => [\n ...list, {key}, \" \"\n ], [])\n .slice(0, -1)\n\n /* Assemble query string for highlighting */\n const url = new URL(document.location)\n if (feature(\"search.highlight\"))\n url.searchParams.set(\"h\", Object.entries(document.terms)\n .filter(([, match]) => match)\n .reduce((highlight, [value]) => `${highlight} ${value}`.trim(), \"\")\n )\n\n /* Render article or section, depending on flags */\n const { tags } = configuration()\n return (\n \n \n {parent > 0 &&
}\n

{document.title}

\n {teaser > 0 && document.text.length > 0 &&\n

\n {truncate(document.text, 320)}\n

\n }\n {document.tags && (\n
\n {document.tags.map(tag => {\n const id = tag.replace(/<[^>]+>/g, \"\")\n const type = tags\n ? id in tags\n ? `md-tag-icon md-tag-icon--${tags[id]}`\n : \"md-tag-icon\"\n : \"\"\n return (\n {tag}\n )\n })}\n
\n )}\n {teaser > 0 && missing.length > 0 &&\n

\n {translation(\"search.result.term.missing\")}: {...missing}\n

\n }\n \n \n )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a search result\n *\n * @param result - Search result\n *\n * @returns Element\n */\nexport function renderSearchResultItem(\n result: SearchResultItem\n): HTMLElement {\n const threshold = result[0].score\n const docs = [...result]\n\n /* Find and extract parent article */\n const parent = docs.findIndex(doc => !doc.location.includes(\"#\"))\n const [article] = docs.splice(parent, 1)\n\n /* Determine last index above threshold */\n let index = docs.findIndex(doc => doc.score < threshold)\n if (index === -1)\n index = docs.length\n\n /* Partition sections */\n const best = docs.slice(0, index)\n const more = docs.slice(index)\n\n /* Render children */\n const children = [\n renderSearchDocument(article, Flag.PARENT | +(!parent && index === 0)),\n ...best.map(section => renderSearchDocument(section, Flag.TEASER)),\n ...more.length ? [\n
\n \n {more.length > 0 && more.length === 1\n ? translation(\"search.result.more.one\")\n : translation(\"search.result.more.other\", more.length)\n }\n \n {...more.map(section => renderSearchDocument(section, Flag.TEASER))}\n
\n ] : []\n ]\n\n /* Render search result */\n return (\n
  • \n {children}\n
    • \n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { SourceFacts } from \"~/components\"\nimport { h, round } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render repository facts\n *\n * @param facts - Repository facts\n *\n * @returns Element\n */\nexport function renderSourceFacts(facts: SourceFacts): HTMLElement {\n return (\n
      \n {Object.entries(facts).map(([key, value]) => (\n
    • \n {typeof value === \"number\" ? round(value) : value}\n
      • \n ))}\n
    \n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Tabbed control type\n */\ntype TabbedControlType =\n | \"prev\"\n | \"next\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render control for content tabs\n *\n * @param type - Control type\n *\n * @returns Element\n */\nexport function renderTabbedControl(\n type: TabbedControlType\n): HTMLElement {\n const classes = `tabbed-control tabbed-control--${type}`\n return (\n \n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a table inside a wrapper to improve scrolling on mobile\n *\n * @param table - Table element\n *\n * @returns Element\n */\nexport function renderTable(table: HTMLElement): HTMLElement {\n return (\n
    \n
    \n {table}\n
    \n
    \n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { configuration, translation } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Version\n */\nexport interface Version {\n version: string /* Version identifier */\n title: string /* Version title */\n aliases: string[] /* Version aliases */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a version\n *\n * @param version - Version\n *\n * @returns Element\n */\nfunction renderVersion(version: Version): HTMLElement {\n const config = configuration()\n\n /* Ensure trailing slash - see https://bit.ly/3rL5u3f */\n const url = new URL(`../${version.version}/`, config.base)\n return (\n
  • \n \n {version.title}\n \n
    • \n )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a version selector\n *\n * @param versions - Versions\n * @param active - Active version\n *\n * @returns Element\n */\nexport function renderVersionSelector(\n versions: Version[], active: Version\n): HTMLElement {\n return (\n
    \n \n {active.title}\n \n
      \n {versions.map(renderVersion)}\n
    \n
    \n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n animationFrameScheduler,\n auditTime,\n combineLatest,\n debounceTime,\n defer,\n delay,\n filter,\n finalize,\n fromEvent,\n map,\n merge,\n switchMap,\n take,\n takeLast,\n takeUntil,\n tap,\n throttleTime,\n withLatestFrom\n} from \"rxjs\"\n\nimport {\n ElementOffset,\n getActiveElement,\n getElementSize,\n watchElementContentOffset,\n watchElementFocus,\n watchElementOffset,\n watchElementVisibility\n} from \"~/browser\"\n\nimport { Component } from \"../../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Annotation\n */\nexport interface Annotation {\n active: boolean /* Annotation is active */\n offset: ElementOffset /* Annotation offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n target$: Observable /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch annotation\n *\n * @param el - Annotation element\n * @param container - Containing element\n *\n * @returns Annotation observable\n */\nexport function watchAnnotation(\n el: HTMLElement, container: HTMLElement\n): Observable {\n const offset$ = defer(() => combineLatest([\n watchElementOffset(el),\n watchElementContentOffset(container)\n ]))\n .pipe(\n map(([{ x, y }, scroll]): ElementOffset => {\n const { width, height } = getElementSize(el)\n return ({\n x: x - scroll.x + width / 2,\n y: y - scroll.y + height / 2\n })\n })\n )\n\n /* Actively watch annotation on focus */\n return watchElementFocus(el)\n .pipe(\n switchMap(active => offset$\n .pipe(\n map(offset => ({ active, offset })),\n take(+!active || Infinity)\n )\n )\n )\n}\n\n/**\n * Mount annotation\n *\n * @param el - Annotation element\n * @param container - Containing element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotation(\n el: HTMLElement, container: HTMLElement, { target$ }: MountOptions\n): Observable> {\n const [tooltip, index] = Array.from(el.children)\n\n /* Mount component on subscription */\n return defer(() => {\n const push$ = new Subject()\n const done$ = push$.pipe(takeLast(1))\n push$.subscribe({\n\n /* Handle emission */\n next({ offset }) {\n el.style.setProperty(\"--md-tooltip-x\", `${offset.x}px`)\n el.style.setProperty(\"--md-tooltip-y\", `${offset.y}px`)\n },\n\n /* Handle complete */\n complete() {\n el.style.removeProperty(\"--md-tooltip-x\")\n el.style.removeProperty(\"--md-tooltip-y\")\n }\n })\n\n /* Start animation only when annotation is visible */\n watchElementVisibility(el)\n .pipe(\n takeUntil(done$)\n )\n .subscribe(visible => {\n el.toggleAttribute(\"data-md-visible\", visible)\n })\n\n /* Toggle tooltip presence to mitigate empty lines when copying */\n merge(\n push$.pipe(filter(({ active }) => active)),\n push$.pipe(debounceTime(250), filter(({ active }) => !active))\n )\n .subscribe({\n\n /* Handle emission */\n next({ active }) {\n if (active)\n el.prepend(tooltip)\n else\n tooltip.remove()\n },\n\n /* Handle complete */\n complete() {\n el.prepend(tooltip)\n }\n })\n\n /* Toggle tooltip visibility */\n push$\n .pipe(\n auditTime(16, animationFrameScheduler)\n )\n .subscribe(({ active }) => {\n tooltip.classList.toggle(\"md-tooltip--active\", active)\n })\n\n /* Track relative origin of tooltip */\n push$\n .pipe(\n throttleTime(125, animationFrameScheduler),\n filter(() => !!el.offsetParent),\n map(() => el.offsetParent!.getBoundingClientRect()),\n map(({ x }) => x)\n )\n .subscribe({\n\n /* Handle emission */\n next(origin) {\n if (origin)\n el.style.setProperty(\"--md-tooltip-0\", `${-origin}px`)\n else\n el.style.removeProperty(\"--md-tooltip-0\")\n },\n\n /* Handle complete */\n complete() {\n el.style.removeProperty(\"--md-tooltip-0\")\n }\n })\n\n /* Allow to copy link without scrolling to anchor */\n fromEvent(index, \"click\")\n .pipe(\n takeUntil(done$),\n filter(ev => !(ev.metaKey || ev.ctrlKey))\n )\n .subscribe(ev => ev.preventDefault())\n\n /* Allow to open link in new tab or blur on close */\n fromEvent(index, \"mousedown\")\n .pipe(\n takeUntil(done$),\n withLatestFrom(push$)\n )\n .subscribe(([ev, { active }]) => {\n\n /* Open in new tab */\n if (ev.button !== 0 || ev.metaKey || ev.ctrlKey) {\n ev.preventDefault()\n\n /* Close annotation */\n } else if (active) {\n ev.preventDefault()\n\n /* Focus parent annotation, if any */\n const parent = el.parentElement!.closest(\".md-annotation\")\n if (parent instanceof HTMLElement)\n parent.focus()\n else\n getActiveElement()?.blur()\n }\n })\n\n /* Open and focus annotation on location target */\n target$\n .pipe(\n takeUntil(done$),\n filter(target => target === tooltip),\n delay(125)\n )\n .subscribe(() => el.focus())\n\n /* Create and return component */\n return watchAnnotation(el, container)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n Subject,\n defer,\n finalize,\n merge,\n share,\n takeLast,\n takeUntil\n} from \"rxjs\"\n\nimport {\n getElement,\n getElements,\n getOptionalElement\n} from \"~/browser\"\nimport { renderAnnotation } from \"~/templates\"\n\nimport { Component } from \"../../../_\"\nimport {\n Annotation,\n mountAnnotation\n} from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n target$: Observable /* Location target observable */\n print$: Observable /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find all annotation markers in the given code block\n *\n * @param container - Containing element\n *\n * @returns Annotation markers\n */\nfunction findAnnotationMarkers(container: HTMLElement): Text[] {\n const markers: Text[] = []\n for (const el of getElements(\".c, .c1, .cm\", container)) {\n const nodes: Text[] = []\n\n /* Find all text nodes in current element */\n const it = document.createNodeIterator(el, NodeFilter.SHOW_TEXT)\n for (let node = it.nextNode(); node; node = it.nextNode())\n nodes.push(node as Text)\n\n /* Find all markers in each text node */\n for (let text of nodes) {\n let match: RegExpExecArray | null\n\n /* Split text at marker and add to list */\n while ((match = /(\\(\\d+\\))(!)?/.exec(text.textContent!))) {\n const [, id, force] = match\n if (typeof force === \"undefined\") {\n const marker = text.splitText(match.index)\n text = marker.splitText(id.length)\n markers.push(marker)\n\n /* Replace entire text with marker */\n } else {\n text.textContent = id\n markers.push(text)\n break\n }\n }\n }\n }\n return markers\n}\n\n/**\n * Swap the child nodes of two elements\n *\n * @param source - Source element\n * @param target - Target element\n */\nfunction swap(source: HTMLElement, target: HTMLElement): void {\n target.append(...Array.from(source.childNodes))\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount annotation list\n *\n * This function analyzes the containing code block and checks for markers\n * referring to elements in the given annotation list. If no markers are found,\n * the list is left untouched. Otherwise, list elements are rendered as\n * annotations inside the code block.\n *\n * @param el - Annotation list element\n * @param container - Containing element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotationList(\n el: HTMLElement, container: HTMLElement, { target$, print$ }: MountOptions\n): Observable> {\n\n /* Compute prefix for tooltip anchors */\n const parent = container.closest(\"[id]\")\n const prefix = parent?.id\n\n /* Find and replace all markers with empty annotations */\n const annotations = new Map()\n for (const marker of findAnnotationMarkers(container)) {\n const [, id] = marker.textContent!.match(/\\((\\d+)\\)/)!\n if (getOptionalElement(`li:nth-child(${id})`, el)) {\n annotations.set(id, renderAnnotation(id, prefix))\n marker.replaceWith(annotations.get(id)!)\n }\n }\n\n /* Keep list if there are no annotations to render */\n if (annotations.size === 0)\n return EMPTY\n\n /* Mount component on subscription */\n return defer(() => {\n const done$ = new Subject()\n\n /* Retrieve container pairs for swapping */\n const pairs: [HTMLElement, HTMLElement][] = []\n for (const [id, annotation] of annotations)\n pairs.push([\n getElement(\".md-typeset\", annotation),\n getElement(`li:nth-child(${id})`, el)\n ])\n\n /* Handle print mode - see https://bit.ly/3rgPdpt */\n print$\n .pipe(\n takeUntil(done$.pipe(takeLast(1)))\n )\n .subscribe(active => {\n el.hidden = !active\n\n /* Show annotations in code block or list (print) */\n for (const [inner, child] of pairs)\n if (!active)\n swap(child, inner)\n else\n swap(inner, child)\n })\n\n /* Create and return component */\n return merge(...[...annotations]\n .map(([, annotation]) => (\n mountAnnotation(annotation, container, { target$ })\n ))\n )\n .pipe(\n finalize(() => done$.complete()),\n share()\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n map,\n of,\n shareReplay,\n tap\n} from \"rxjs\"\n\nimport { watchScript } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../../_\"\n\nimport themeCSS from \"./index.css\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mermaid diagram\n */\nexport interface Mermaid {}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Mermaid instance observable\n */\nlet mermaid$: Observable\n\n/**\n * Global sequence number for diagrams\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch Mermaid script\n *\n * @returns Mermaid scripts observable\n */\nfunction fetchScripts(): Observable {\n return typeof mermaid === \"undefined\" || mermaid instanceof Element\n ? watchScript(\"https://unpkg.com/mermaid@9.1.7/dist/mermaid.min.js\")\n : of(undefined)\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount Mermaid diagram\n *\n * @param el - Code block element\n *\n * @returns Mermaid diagram component observable\n */\nexport function mountMermaid(\n el: HTMLElement\n): Observable> {\n el.classList.remove(\"mermaid\") // Hack: mitigate https://bit.ly/3CiN6Du\n mermaid$ ||= fetchScripts()\n .pipe(\n tap(() => mermaid.initialize({\n startOnLoad: false,\n themeCSS,\n sequence: {\n actorFontSize: \"16px\", // Hack: mitigate https://bit.ly/3y0NEi3\n messageFontSize: \"16px\",\n noteFontSize: \"16px\"\n }\n })),\n map(() => undefined),\n shareReplay(1)\n )\n\n /* Render diagram */\n mermaid$.subscribe(() => {\n el.classList.add(\"mermaid\") // Hack: mitigate https://bit.ly/3CiN6Du\n const id = `__mermaid_${sequence++}`\n const host = h(\"div\", { class: \"mermaid\" })\n mermaid.mermaidAPI.render(id, el.textContent, (svg: string) => {\n\n /* Create a shadow root and inject diagram */\n const shadow = host.attachShadow({ mode: \"closed\" })\n shadow.innerHTML = svg\n\n /* Replace code block with diagram */\n el.replaceWith(host)\n })\n })\n\n /* Create and return component */\n return mermaid$\n .pipe(\n map(() => ({ ref: el }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n defer,\n filter,\n finalize,\n map,\n merge,\n tap\n} from \"rxjs\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Details\n */\nexport interface Details {\n action: \"open\" | \"close\" /* Details state */\n reveal?: boolean /* Details is revealed */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n target$: Observable /* Location target observable */\n print$: Observable /* Media print observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n target$: Observable /* Location target observable */\n print$: Observable /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch details\n *\n * @param el - Details element\n * @param options - Options\n *\n * @returns Details observable\n */\nexport function watchDetails(\n el: HTMLDetailsElement, { target$, print$ }: WatchOptions\n): Observable
    {\n let open = true\n return merge(\n\n /* Open and focus details on location target */\n target$\n .pipe(\n map(target => target.closest(\"details:not([open])\")!),\n filter(details => el === details),\n map(() => ({\n action: \"open\", reveal: true\n }) as Details)\n ),\n\n /* Open details on print and close afterwards */\n print$\n .pipe(\n filter(active => active || !open),\n tap(() => open = el.open),\n map(active => ({\n action: active ? \"open\" : \"close\"\n }) as Details)\n )\n )\n}\n\n/**\n * Mount details\n *\n * This function ensures that `details` tags are opened on anchor jumps and\n * prior to printing, so the whole content of the page is visible.\n *\n * @param el - Details element\n * @param options - Options\n *\n * @returns Details component observable\n */\nexport function mountDetails(\n el: HTMLDetailsElement, options: MountOptions\n): Observable> {\n return defer(() => {\n const push$ = new Subject
    ()\n push$.subscribe(({ action, reveal }) => {\n el.toggleAttribute(\"open\", action === \"open\")\n if (reveal)\n el.scrollIntoView()\n })\n\n /* Create and return component */\n return watchDetails(el, options)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Observable, of } from \"rxjs\"\n\nimport { renderTable } from \"~/templates\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Data table\n */\nexport interface DataTable {}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Sentinel for replacement\n */\nconst sentinel = h(\"table\")\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount data table\n *\n * This function wraps a data table in another scrollable container, so it can\n * be smoothly scrolled on smaller screen sizes and won't break the layout.\n *\n * @param el - Data table element\n *\n * @returns Data table component observable\n */\nexport function mountDataTable(\n el: HTMLElement\n): Observable> {\n el.replaceWith(sentinel)\n sentinel.replaceWith(renderTable(el))\n\n /* Create and return component */\n return of({ ref: el })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n animationFrameScheduler,\n asyncScheduler,\n auditTime,\n combineLatest,\n defer,\n finalize,\n fromEvent,\n map,\n merge,\n skip,\n startWith,\n subscribeOn,\n takeLast,\n takeUntil,\n tap,\n withLatestFrom\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n Viewport,\n getElement,\n getElementContentOffset,\n getElementContentSize,\n getElementOffset,\n getElementSize,\n getElements,\n watchElementContentOffset,\n watchElementSize\n} from \"~/browser\"\nimport { renderTabbedControl } from \"~/templates\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Content tabs\n */\nexport interface ContentTabs {\n active: HTMLLabelElement /* Active tab label */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch content tabs\n *\n * @param el - Content tabs element\n *\n * @returns Content tabs observable\n */\nexport function watchContentTabs(\n el: HTMLElement\n): Observable {\n const inputs = getElements(\":scope > input\", el)\n const initial = inputs.find(input => input.checked) || inputs[0]\n return merge(...inputs.map(input => fromEvent(input, \"change\")\n .pipe(\n map(() => getElement(`label[for=\"${input.id}\"]`))\n )\n ))\n .pipe(\n startWith(getElement(`label[for=\"${initial.id}\"]`)),\n map(active => ({ active }))\n )\n}\n\n/**\n * Mount content tabs\n *\n * This function scrolls the active tab into view. While this functionality is\n * provided by browsers as part of `scrollInfoView`, browsers will always also\n * scroll the vertical axis, which we do not want. Thus, we decided to provide\n * this functionality ourselves.\n *\n * @param el - Content tabs element\n * @param options - Options\n *\n * @returns Content tabs component observable\n */\nexport function mountContentTabs(\n el: HTMLElement, { viewport$ }: MountOptions\n): Observable> {\n\n /* Render content tab previous button for pagination */\n const prev = renderTabbedControl(\"prev\")\n el.append(prev)\n\n /* Render content tab next button for pagination */\n const next = renderTabbedControl(\"next\")\n el.append(next)\n\n /* Mount component on subscription */\n const container = getElement(\".tabbed-labels\", el)\n return defer(() => {\n const push$ = new Subject()\n const done$ = push$.pipe(takeLast(1))\n combineLatest([push$, watchElementSize(el)])\n .pipe(\n auditTime(1, animationFrameScheduler),\n takeUntil(done$)\n )\n .subscribe({\n\n /* Handle emission */\n next([{ active }, size]) {\n const offset = getElementOffset(active)\n const { width } = getElementSize(active)\n\n /* Set tab indicator offset and width */\n el.style.setProperty(\"--md-indicator-x\", `${offset.x}px`)\n el.style.setProperty(\"--md-indicator-width\", `${width}px`)\n\n /* Scroll container to active content tab */\n const content = getElementContentOffset(container)\n if (\n offset.x < content.x ||\n offset.x + width > content.x + size.width\n )\n container.scrollTo({\n left: Math.max(0, offset.x - 16),\n behavior: \"smooth\"\n })\n },\n\n /* Handle complete */\n complete() {\n el.style.removeProperty(\"--md-indicator-x\")\n el.style.removeProperty(\"--md-indicator-width\")\n }\n })\n\n /* Hide content tab buttons on borders */\n combineLatest([\n watchElementContentOffset(container),\n watchElementSize(container)\n ])\n .pipe(\n takeUntil(done$)\n )\n .subscribe(([offset, size]) => {\n const content = getElementContentSize(container)\n prev.hidden = offset.x < 16\n next.hidden = offset.x > content.width - size.width - 16\n })\n\n /* Paginate content tab container on click */\n merge(\n fromEvent(prev, \"click\").pipe(map(() => -1)),\n fromEvent(next, \"click\").pipe(map(() => +1))\n )\n .pipe(\n takeUntil(done$)\n )\n .subscribe(direction => {\n const { width } = getElementSize(container)\n container.scrollBy({\n left: width * direction,\n behavior: \"smooth\"\n })\n })\n\n /* Set up linking of content tabs, if enabled */\n if (feature(\"content.tabs.link\"))\n push$.pipe(\n skip(1),\n withLatestFrom(viewport$)\n )\n .subscribe(([{ active }, { offset }]) => {\n const tab = active.innerText.trim()\n if (active.hasAttribute(\"data-md-switching\")) {\n active.removeAttribute(\"data-md-switching\")\n\n /* Determine viewport offset of active tab */\n } else {\n const y = el.offsetTop - offset.y\n\n /* Passively activate other tabs */\n for (const set of getElements(\"[data-tabs]\"))\n for (const input of getElements(\n \":scope > input\", set\n )) {\n const label = getElement(`label[for=\"${input.id}\"]`)\n if (\n label !== active &&\n label.innerText.trim() === tab\n ) {\n label.setAttribute(\"data-md-switching\", \"\")\n input.click()\n break\n }\n }\n\n /* Bring active tab into view */\n window.scrollTo({\n top: el.offsetTop - y\n })\n\n /* Persist active tabs in local storage */\n const tabs = __md_get(\"__tabs\") || []\n __md_set(\"__tabs\", [...new Set([tab, ...tabs])])\n }\n })\n\n /* Create and return component */\n return watchContentTabs(el)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n .pipe(\n subscribeOn(asyncScheduler)\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Observable, merge } from \"rxjs\"\n\nimport { Viewport, getElements } from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Annotation } from \"../annotation\"\nimport {\n CodeBlock,\n Mermaid,\n mountCodeBlock,\n mountMermaid\n} from \"../code\"\nimport {\n Details,\n mountDetails\n} from \"../details\"\nimport {\n DataTable,\n mountDataTable\n} from \"../table\"\nimport {\n ContentTabs,\n mountContentTabs\n} from \"../tabs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Content\n */\nexport type Content =\n | Annotation\n | ContentTabs\n | CodeBlock\n | Mermaid\n | DataTable\n | Details\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n target$: Observable /* Location target observable */\n print$: Observable /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount content\n *\n * This function mounts all components that are found in the content of the\n * actual article, including code blocks, data tables and details.\n *\n * @param el - Content element\n * @param options - Options\n *\n * @returns Content component observable\n */\nexport function mountContent(\n el: HTMLElement, { viewport$, target$, print$ }: MountOptions\n): Observable> {\n return merge(\n\n /* Code blocks */\n ...getElements(\"pre:not(.mermaid) > code\", el)\n .map(child => mountCodeBlock(child, { target$, print$ })),\n\n /* Mermaid diagrams */\n ...getElements(\"pre.mermaid\", el)\n .map(child => mountMermaid(child)),\n\n /* Data tables */\n ...getElements(\"table:not([class])\", el)\n .map(child => mountDataTable(child)),\n\n /* Details */\n ...getElements(\"details\", el)\n .map(child => mountDetails(child, { target$, print$ })),\n\n /* Content tabs */\n ...getElements(\"[data-tabs]\", el)\n .map(child => mountContentTabs(child, { viewport$ }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n defer,\n delay,\n finalize,\n map,\n merge,\n of,\n switchMap,\n tap\n} from \"rxjs\"\n\nimport { getElement } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Dialog\n */\nexport interface Dialog {\n message: string /* Dialog message */\n active: boolean /* Dialog is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n alert$: Subject /* Alert subject */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n alert$: Subject /* Alert subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch dialog\n *\n * @param _el - Dialog element\n * @param options - Options\n *\n * @returns Dialog observable\n */\nexport function watchDialog(\n _el: HTMLElement, { alert$ }: WatchOptions\n): Observable {\n return alert$\n .pipe(\n switchMap(message => merge(\n of(true),\n of(false).pipe(delay(2000))\n )\n .pipe(\n map(active => ({ message, active }))\n )\n )\n )\n}\n\n/**\n * Mount dialog\n *\n * This function reveals the dialog in the right corner when a new alert is\n * emitted through the subject that is passed as part of the options.\n *\n * @param el - Dialog element\n * @param options - Options\n *\n * @returns Dialog component observable\n */\nexport function mountDialog(\n el: HTMLElement, options: MountOptions\n): Observable> {\n const inner = getElement(\".md-typeset\", el)\n return defer(() => {\n const push$ = new Subject()\n push$.subscribe(({ message, active }) => {\n el.classList.toggle(\"md-dialog--active\", active)\n inner.textContent = message\n })\n\n /* Create and return component */\n return watchDialog(el, options)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n bufferCount,\n combineLatest,\n combineLatestWith,\n defer,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n filter,\n map,\n of,\n shareReplay,\n startWith,\n switchMap,\n takeLast,\n takeUntil\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n Viewport,\n watchElementSize,\n watchToggle\n} from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Main } from \"../../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Header\n */\nexport interface Header {\n height: number /* Header visible height */\n hidden: boolean /* Header is hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n main$: Observable
    /* Main area observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Compute whether the header is hidden\n *\n * If the user scrolls past a certain threshold, the header can be hidden when\n * scrolling down, and shown when scrolling up.\n *\n * @param options - Options\n *\n * @returns Toggle observable\n */\nfunction isHidden({ viewport$ }: WatchOptions): Observable {\n if (!feature(\"header.autohide\"))\n return of(false)\n\n /* Compute direction and turning point */\n const direction$ = viewport$\n .pipe(\n map(({ offset: { y } }) => y),\n bufferCount(2, 1),\n map(([a, b]) => [a < b, b] as const),\n distinctUntilKeyChanged(0)\n )\n\n /* Compute whether header should be hidden */\n const hidden$ = combineLatest([viewport$, direction$])\n .pipe(\n filter(([{ offset }, [, y]]) => Math.abs(y - offset.y) > 100),\n map(([, [direction]]) => direction),\n distinctUntilChanged()\n )\n\n /* Compute threshold for hiding */\n const search$ = watchToggle(\"search\")\n return combineLatest([viewport$, search$])\n .pipe(\n map(([{ offset }, search]) => offset.y > 400 && !search),\n distinctUntilChanged(),\n switchMap(active => active ? hidden$ : of(false)),\n startWith(false)\n )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch header\n *\n * @param el - Header element\n * @param options - Options\n *\n * @returns Header observable\n */\nexport function watchHeader(\n el: HTMLElement, options: WatchOptions\n): Observable
    {\n return defer(() => combineLatest([\n watchElementSize(el),\n isHidden(options)\n ]))\n .pipe(\n map(([{ height }, hidden]) => ({\n height,\n hidden\n })),\n distinctUntilChanged((a, b) => (\n a.height === b.height &&\n a.hidden === b.hidden\n )),\n shareReplay(1)\n )\n}\n\n/**\n * Mount header\n *\n * This function manages the different states of the header, i.e. whether it's\n * hidden or rendered with a shadow. This depends heavily on the main area.\n *\n * @param el - Header element\n * @param options - Options\n *\n * @returns Header component observable\n */\nexport function mountHeader(\n el: HTMLElement, { header$, main$ }: MountOptions\n): Observable> {\n return defer(() => {\n const push$ = new Subject
    ()\n const done$ = push$.pipe(takeLast(1))\n push$\n .pipe(\n distinctUntilKeyChanged(\"active\"),\n combineLatestWith(header$)\n )\n .subscribe(([{ active }, { hidden }]) => {\n el.classList.toggle(\"md-header--shadow\", active && !hidden)\n el.hidden = hidden\n })\n\n /* Link to main area */\n main$.subscribe(push$)\n\n /* Create and return component */\n return header$\n .pipe(\n takeUntil(done$),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n Subject,\n defer,\n distinctUntilKeyChanged,\n finalize,\n map,\n tap\n} from \"rxjs\"\n\nimport {\n Viewport,\n getElementSize,\n getOptionalElement,\n watchViewportAt\n} from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Header } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Header\n */\nexport interface HeaderTitle {\n active: boolean /* Header title is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch header title\n *\n * @param el - Heading element\n * @param options - Options\n *\n * @returns Header title observable\n */\nexport function watchHeaderTitle(\n el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable {\n return watchViewportAt(el, { viewport$, header$ })\n .pipe(\n map(({ offset: { y } }) => {\n const { height } = getElementSize(el)\n return {\n active: y >= height\n }\n }),\n distinctUntilKeyChanged(\"active\")\n )\n}\n\n/**\n * Mount header title\n *\n * This function swaps the header title from the site title to the title of the\n * current page when the user scrolls past the first headline.\n *\n * @param el - Header title element\n * @param options - Options\n *\n * @returns Header title component observable\n */\nexport function mountHeaderTitle(\n el: HTMLElement, options: MountOptions\n): Observable> {\n return defer(() => {\n const push$ = new Subject()\n push$.subscribe(({ active }) => {\n el.classList.toggle(\"md-header__title--active\", active)\n })\n\n /* Obtain headline, if any */\n const heading = getOptionalElement(\"article h1\")\n if (typeof heading === \"undefined\")\n return EMPTY\n\n /* Create and return component */\n return watchHeaderTitle(heading, options)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n combineLatest,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n map,\n switchMap\n} from \"rxjs\"\n\nimport {\n Viewport,\n watchElementSize\n} from \"~/browser\"\n\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Main area\n */\nexport interface Main {\n offset: number /* Main area top offset */\n height: number /* Main area visible height */\n active: boolean /* Main area is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch main area\n *\n * This function returns an observable that computes the visual parameters of\n * the main area which depends on the viewport vertical offset and height, as\n * well as the height of the header element, if the header is fixed.\n *\n * @param el - Main area element\n * @param options - Options\n *\n * @returns Main area observable\n */\nexport function watchMain(\n el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable
    {\n\n /* Compute necessary adjustment for header */\n const adjust$ = header$\n .pipe(\n map(({ height }) => height),\n distinctUntilChanged()\n )\n\n /* Compute the main area's top and bottom borders */\n const border$ = adjust$\n .pipe(\n switchMap(() => watchElementSize(el)\n .pipe(\n map(({ height }) => ({\n top: el.offsetTop,\n bottom: el.offsetTop + height\n })),\n distinctUntilKeyChanged(\"bottom\")\n )\n )\n )\n\n /* Compute the main area's offset, visible height and if we scrolled past */\n return combineLatest([adjust$, border$, viewport$])\n .pipe(\n map(([header, { top, bottom }, { offset: { y }, size: { height } }]) => {\n height = Math.max(0, height\n - Math.max(0, top - y, header)\n - Math.max(0, height + y - bottom)\n )\n return {\n offset: top - header,\n height,\n active: top - header <= y\n }\n }),\n distinctUntilChanged((a, b) => (\n a.offset === b.offset &&\n a.height === b.height &&\n a.active === b.active\n ))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n asyncScheduler,\n defer,\n finalize,\n fromEvent,\n map,\n mergeMap,\n observeOn,\n of,\n shareReplay,\n startWith,\n tap\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Palette colors\n */\nexport interface PaletteColor {\n scheme?: string /* Color scheme */\n primary?: string /* Primary color */\n accent?: string /* Accent color */\n}\n\n/**\n * Palette\n */\nexport interface Palette {\n index: number /* Palette index */\n color: PaletteColor /* Palette colors */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch color palette\n *\n * @param inputs - Color palette element\n *\n * @returns Color palette observable\n */\nexport function watchPalette(\n inputs: HTMLInputElement[]\n): Observable {\n const current = __md_get(\"__palette\") || {\n index: inputs.findIndex(input => matchMedia(\n input.getAttribute(\"data-md-color-media\")!\n ).matches)\n }\n\n /* Emit changes in color palette */\n return of(...inputs)\n .pipe(\n mergeMap(input => fromEvent(input, \"change\")\n .pipe(\n map(() => input)\n )\n ),\n startWith(inputs[Math.max(0, current.index)]),\n map(input => ({\n index: inputs.indexOf(input),\n color: {\n scheme: input.getAttribute(\"data-md-color-scheme\"),\n primary: input.getAttribute(\"data-md-color-primary\"),\n accent: input.getAttribute(\"data-md-color-accent\")\n }\n } as Palette)),\n shareReplay(1)\n )\n}\n\n/**\n * Mount color palette\n *\n * @param el - Color palette element\n *\n * @returns Color palette component observable\n */\nexport function mountPalette(\n el: HTMLElement\n): Observable> {\n return defer(() => {\n const push$ = new Subject()\n push$.subscribe(palette => {\n document.body.setAttribute(\"data-md-color-switching\", \"\")\n\n /* Set color palette */\n for (const [key, value] of Object.entries(palette.color))\n document.body.setAttribute(`data-md-color-${key}`, value)\n\n /* Toggle visibility */\n for (let index = 0; index < inputs.length; index++) {\n const label = inputs[index].nextElementSibling\n if (label instanceof HTMLElement)\n label.hidden = palette.index !== index\n }\n\n /* Persist preference in local storage */\n __md_set(\"__palette\", palette)\n })\n\n /* Revert transition durations after color switch */\n push$.pipe(observeOn(asyncScheduler))\n .subscribe(() => {\n document.body.removeAttribute(\"data-md-color-switching\")\n })\n\n /* Create and return component */\n const inputs = getElements(\"input\", el)\n return watchPalette(inputs)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ClipboardJS from \"clipboard\"\nimport {\n Observable,\n Subject,\n map,\n tap\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport { getElement } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n alert$: Subject /* Alert subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Extract text to copy\n *\n * @param el - HTML element\n *\n * @returns Extracted text\n */\nfunction extract(el: HTMLElement): string {\n el.setAttribute(\"data-md-copying\", \"\")\n const text = el.innerText\n el.removeAttribute(\"data-md-copying\")\n return text\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up Clipboard.js integration\n *\n * @param options - Options\n */\nexport function setupClipboardJS(\n { alert$ }: SetupOptions\n): void {\n if (ClipboardJS.isSupported()) {\n new Observable(subscriber => {\n new ClipboardJS(\"[data-clipboard-target], [data-clipboard-text]\", {\n text: el => (\n el.getAttribute(\"data-clipboard-text\")! ||\n extract(getElement(\n el.getAttribute(\"data-clipboard-target\")!\n ))\n )\n })\n .on(\"success\", ev => subscriber.next(ev))\n })\n .pipe(\n tap(ev => {\n const trigger = ev.trigger as HTMLElement\n trigger.focus()\n }),\n map(() => translation(\"clipboard.copied\"))\n )\n .subscribe(alert$)\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n catchError,\n defaultIfEmpty,\n map,\n of,\n tap\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport { getElements, requestXML } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Sitemap, i.e. a list of URLs\n */\nexport type Sitemap = string[]\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Preprocess a list of URLs\n *\n * This function replaces the `site_url` in the sitemap with the actual base\n * URL, to allow instant loading to work in occasions like Netlify previews.\n *\n * @param urls - URLs\n *\n * @returns URL path parts\n */\nfunction preprocess(urls: Sitemap): Sitemap {\n if (urls.length < 2)\n return [\"\"]\n\n /* Take the first two URLs and remove everything after the last slash */\n const [root, next] = [...urls]\n .sort((a, b) => a.length - b.length)\n .map(url => url.replace(/[^/]+$/, \"\"))\n\n /* Compute common prefix */\n let index = 0\n if (root === next)\n index = root.length\n else\n while (root.charCodeAt(index) === next.charCodeAt(index))\n index++\n\n /* Remove common prefix and return in original order */\n return urls.map(url => url.replace(root.slice(0, index), \"\"))\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch the sitemap for the given base URL\n *\n * @param base - Base URL\n *\n * @returns Sitemap observable\n */\nexport function fetchSitemap(base?: URL): Observable {\n const cached = __md_get(\"__sitemap\", sessionStorage, base)\n if (cached) {\n return of(cached)\n } else {\n const config = configuration()\n return requestXML(new URL(\"sitemap.xml\", base || config.base))\n .pipe(\n map(sitemap => preprocess(getElements(\"loc\", sitemap)\n .map(node => node.textContent!)\n )),\n catchError(() => EMPTY), // @todo refactor instant loading\n defaultIfEmpty([]),\n tap(sitemap => __md_set(\"__sitemap\", sitemap, sessionStorage, base))\n )\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n NEVER,\n Observable,\n Subject,\n bufferCount,\n catchError,\n concatMap,\n debounceTime,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n filter,\n fromEvent,\n map,\n merge,\n of,\n sample,\n share,\n skip,\n skipUntil,\n switchMap\n} from \"rxjs\"\n\nimport { configuration, feature } from \"~/_\"\nimport {\n Viewport,\n ViewportOffset,\n getElements,\n getOptionalElement,\n request,\n setLocation,\n setLocationHash\n} from \"~/browser\"\nimport { getComponentElement } from \"~/components\"\nimport { h } from \"~/utilities\"\n\nimport { fetchSitemap } from \"../sitemap\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * History state\n */\nexport interface HistoryState {\n url: URL /* State URL */\n offset?: ViewportOffset /* State viewport offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n document$: Subject /* Document subject */\n location$: Subject /* Location subject */\n viewport$: Observable /* Viewport observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up instant loading\n *\n * When fetching, theoretically, we could use `responseType: \"document\"`, but\n * since all MkDocs links are relative, we need to make sure that the current\n * location matches the document we just loaded. Otherwise any relative links\n * in the document could use the old location.\n *\n * This is the reason why we need to synchronize history events and the process\n * of fetching the document for navigation changes (except `popstate` events):\n *\n * 1. Fetch document via `XMLHTTPRequest`\n * 2. Set new location via `history.pushState`\n * 3. Parse and emit fetched document\n *\n * For `popstate` events, we must not use `history.pushState`, or the forward\n * history will be irreversibly overwritten. In case the request fails, the\n * location change is dispatched regularly.\n *\n * @param options - Options\n */\nexport function setupInstantLoading(\n { document$, location$, viewport$ }: SetupOptions\n): void {\n const config = configuration()\n if (location.protocol === \"file:\")\n return\n\n /* Disable automatic scroll restoration */\n if (\"scrollRestoration\" in history) {\n history.scrollRestoration = \"manual\"\n\n /* Hack: ensure that reloads restore viewport offset */\n fromEvent(window, \"beforeunload\")\n .subscribe(() => {\n history.scrollRestoration = \"auto\"\n })\n }\n\n /* Hack: ensure absolute favicon link to omit 404s when switching */\n const favicon = getOptionalElement(\"link[rel=icon]\")\n if (typeof favicon !== \"undefined\")\n favicon.href = favicon.href\n\n /* Intercept internal navigation */\n const push$ = fetchSitemap()\n .pipe(\n map(paths => paths.map(path => `${new URL(path, config.base)}`)),\n switchMap(urls => fromEvent(document.body, \"click\")\n .pipe(\n filter(ev => !ev.metaKey && !ev.ctrlKey),\n switchMap(ev => {\n if (ev.target instanceof Element) {\n const el = ev.target.closest(\"a\")\n if (el && !el.target) {\n const url = new URL(el.href)\n\n /* Canonicalize URL */\n url.search = \"\"\n url.hash = \"\"\n\n /* Check if URL should be intercepted */\n if (\n url.pathname !== location.pathname &&\n urls.includes(url.toString())\n ) {\n ev.preventDefault()\n return of({\n url: new URL(el.href)\n })\n }\n }\n }\n return NEVER\n })\n )\n ),\n share()\n )\n\n /* Intercept history back and forward */\n const pop$ = fromEvent(window, \"popstate\")\n .pipe(\n filter(ev => ev.state !== null),\n map(ev => ({\n url: new URL(location.href),\n offset: ev.state\n })),\n share()\n )\n\n /* Emit location change */\n merge(push$, pop$)\n .pipe(\n distinctUntilChanged((a, b) => a.url.href === b.url.href),\n map(({ url }) => url)\n )\n .subscribe(location$)\n\n /* Fetch document via `XMLHTTPRequest` */\n const response$ = location$\n .pipe(\n distinctUntilKeyChanged(\"pathname\"),\n switchMap(url => request(url.href)\n .pipe(\n catchError(() => {\n setLocation(url)\n return NEVER\n })\n )\n ),\n share()\n )\n\n /* Set new location via `history.pushState` */\n push$\n .pipe(\n sample(response$)\n )\n .subscribe(({ url }) => {\n history.pushState({}, \"\", `${url}`)\n })\n\n /* Parse and emit fetched document */\n const dom = new DOMParser()\n response$\n .pipe(\n switchMap(res => res.text()),\n map(res => dom.parseFromString(res, \"text/html\"))\n )\n .subscribe(document$)\n\n /* Replace meta tags and components */\n document$\n .pipe(\n skip(1)\n )\n .subscribe(replacement => {\n for (const selector of [\n\n /* Meta tags */\n \"title\",\n \"link[rel=canonical]\",\n \"meta[name=author]\",\n \"meta[name=description]\",\n\n /* Components */\n \"[data-md-component=announce]\",\n \"[data-md-component=container]\",\n \"[data-md-component=header-topic]\",\n \"[data-md-component=outdated]\",\n \"[data-md-component=logo]\",\n \"[data-md-component=skip]\",\n ...feature(\"navigation.tabs.sticky\")\n ? [\"[data-md-component=tabs]\"]\n : []\n ]) {\n const source = getOptionalElement(selector)\n const target = getOptionalElement(selector, replacement)\n if (\n typeof source !== \"undefined\" &&\n typeof target !== \"undefined\"\n ) {\n source.replaceWith(target)\n }\n }\n })\n\n /* Re-evaluate scripts */\n document$\n .pipe(\n skip(1),\n map(() => getComponentElement(\"container\")),\n switchMap(el => getElements(\"script\", el)),\n concatMap(el => {\n const script = h(\"script\")\n if (el.src) {\n for (const name of el.getAttributeNames())\n script.setAttribute(name, el.getAttribute(name)!)\n el.replaceWith(script)\n\n /* Complete when script is loaded */\n return new Observable(observer => {\n script.onload = () => observer.complete()\n })\n\n /* Complete immediately */\n } else {\n script.textContent = el.textContent\n el.replaceWith(script)\n return EMPTY\n }\n })\n )\n .subscribe()\n\n /* Emit history state change */\n merge(push$, pop$)\n .pipe(\n sample(document$)\n )\n .subscribe(({ url, offset }) => {\n if (url.hash && !offset) {\n setLocationHash(url.hash)\n } else {\n window.scrollTo(0, offset?.y || 0)\n }\n })\n\n /* Debounce update of viewport offset */\n viewport$\n .pipe(\n skipUntil(push$),\n debounceTime(250),\n distinctUntilKeyChanged(\"offset\")\n )\n .subscribe(({ offset }) => {\n history.replaceState(offset, \"\")\n })\n\n /* Set viewport offset from history */\n merge(push$, pop$)\n .pipe(\n bufferCount(2, 1),\n filter(([a, b]) => a.url.pathname === b.url.pathname),\n map(([, state]) => state)\n )\n .subscribe(({ offset }) => {\n window.scrollTo(0, offset?.y || 0)\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport escapeHTML from \"escape-html\"\n\nimport { SearchIndexDocument } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search document\n */\nexport interface SearchDocument extends SearchIndexDocument {\n parent?: SearchIndexDocument /* Parent article */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search document mapping\n */\nexport type SearchDocumentMap = Map\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search document mapping\n *\n * @param docs - Search index documents\n *\n * @returns Search document map\n */\nexport function setupSearchDocumentMap(\n docs: SearchIndexDocument[]\n): SearchDocumentMap {\n const documents = new Map()\n const parents = new Set()\n for (const doc of docs) {\n const [path, hash] = doc.location.split(\"#\")\n\n /* Extract location, title and tags */\n const location = doc.location\n const title = doc.title\n const tags = doc.tags\n\n /* Escape and cleanup text */\n const text = escapeHTML(doc.text)\n .replace(/\\s+(?=[,.:;!?])/g, \"\")\n .replace(/\\s+/g, \" \")\n\n /* Handle section */\n if (hash) {\n const parent = documents.get(path)!\n\n /* Ignore first section, override article */\n if (!parents.has(parent)) {\n parent.title = doc.title\n parent.text = text\n\n /* Remember that we processed the article */\n parents.add(parent)\n\n /* Add subsequent section */\n } else {\n documents.set(location, {\n location,\n title,\n text,\n parent\n })\n }\n\n /* Add article */\n } else {\n documents.set(location, {\n location,\n title,\n text,\n ...tags && { tags }\n })\n }\n }\n return documents\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport escapeHTML from \"escape-html\"\n\nimport { SearchIndexConfig } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlight function\n *\n * @param value - Value\n *\n * @returns Highlighted value\n */\nexport type SearchHighlightFn = (value: string) => string\n\n/**\n * Search highlight factory function\n *\n * @param query - Query value\n *\n * @returns Search highlight function\n */\nexport type SearchHighlightFactoryFn = (query: string) => SearchHighlightFn\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search highlighter\n *\n * @param config - Search index configuration\n * @param escape - Whether to escape HTML\n *\n * @returns Search highlight factory function\n */\nexport function setupSearchHighlighter(\n config: SearchIndexConfig, escape: boolean\n): SearchHighlightFactoryFn {\n const separator = new RegExp(config.separator, \"img\")\n const highlight = (_: unknown, data: string, term: string) => {\n return `${data}${term}`\n }\n\n /* Return factory function */\n return (query: string) => {\n query = query\n .replace(/[\\s*+\\-:~^]+/g, \" \")\n .trim()\n\n /* Create search term match expression */\n const match = new RegExp(`(^|${config.separator})(${\n query\n .replace(/[|\\\\{}()[\\]^$+*?.-]/g, \"\\\\$&\")\n .replace(separator, \"|\")\n })`, \"img\")\n\n /* Highlight string value */\n return value => (\n escape\n ? escapeHTML(value)\n : value\n )\n .replace(match, highlight)\n .replace(/<\\/mark>(\\s+)]*>/img, \"$1\")\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search transformation function\n *\n * @param value - Query value\n *\n * @returns Transformed query value\n */\nexport type SearchTransformFn = (value: string) => string\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Default transformation function\n *\n * 1. Search for terms in quotation marks and prepend a `+` modifier to denote\n * that the resulting document must contain all terms, converting the query\n * to an `AND` query (as opposed to the default `OR` behavior). While users\n * may expect terms enclosed in quotation marks to map to span queries, i.e.\n * for which order is important, Lunr.js doesn't support them, so the best\n * we can do is to convert the terms to an `AND` query.\n *\n * 2. Replace control characters which are not located at the beginning of the\n * query or preceded by white space, or are not followed by a non-whitespace\n * character or are at the end of the query string. Furthermore, filter\n * unmatched quotation marks.\n *\n * 3. Trim excess whitespace from left and right.\n *\n * @param query - Query value\n *\n * @returns Transformed query value\n */\nexport function defaultTransform(query: string): string {\n return query\n .split(/\"([^\"]+)\"/g) /* => 1 */\n .map((terms, index) => index & 1\n ? terms.replace(/^\\b|^(?![^\\x00-\\x7F]|$)|\\s+/g, \" +\")\n : terms\n )\n .join(\"\")\n .replace(/\"|(?:^|\\s+)[*+\\-:^~]+(?=\\s+|$)/g, \"\") /* => 2 */\n .trim() /* => 3 */\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { SearchIndex, SearchResult } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search message type\n */\nexport const enum SearchMessageType {\n SETUP, /* Search index setup */\n READY, /* Search index ready */\n QUERY, /* Search query */\n RESULT /* Search results */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Message containing the data necessary to setup the search index\n */\nexport interface SearchSetupMessage {\n type: SearchMessageType.SETUP /* Message type */\n data: SearchIndex /* Message data */\n}\n\n/**\n * Message indicating the search index is ready\n */\nexport interface SearchReadyMessage {\n type: SearchMessageType.READY /* Message type */\n}\n\n/**\n * Message containing a search query\n */\nexport interface SearchQueryMessage {\n type: SearchMessageType.QUERY /* Message type */\n data: string /* Message data */\n}\n\n/**\n * Message containing results for a search query\n */\nexport interface SearchResultMessage {\n type: SearchMessageType.RESULT /* Message type */\n data: SearchResult /* Message data */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Message exchanged with the search worker\n */\nexport type SearchMessage =\n | SearchSetupMessage\n | SearchReadyMessage\n | SearchQueryMessage\n | SearchResultMessage\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Type guard for search setup messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchSetupMessage(\n message: SearchMessage\n): message is SearchSetupMessage {\n return message.type === SearchMessageType.SETUP\n}\n\n/**\n * Type guard for search ready messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchReadyMessage(\n message: SearchMessage\n): message is SearchReadyMessage {\n return message.type === SearchMessageType.READY\n}\n\n/**\n * Type guard for search query messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchQueryMessage(\n message: SearchMessage\n): message is SearchQueryMessage {\n return message.type === SearchMessageType.QUERY\n}\n\n/**\n * Type guard for search result messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchResultMessage(\n message: SearchMessage\n): message is SearchResultMessage {\n return message.type === SearchMessageType.RESULT\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n ObservableInput,\n Subject,\n from,\n map,\n share\n} from \"rxjs\"\n\nimport { configuration, feature, translation } from \"~/_\"\nimport { WorkerHandler, watchWorker } from \"~/browser\"\n\nimport { SearchIndex } from \"../../_\"\nimport {\n SearchOptions,\n SearchPipeline\n} from \"../../options\"\nimport {\n SearchMessage,\n SearchMessageType,\n SearchSetupMessage,\n isSearchResultMessage\n} from \"../message\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search worker\n */\nexport type SearchWorker = WorkerHandler\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up search index\n *\n * @param data - Search index\n *\n * @returns Search index\n */\nfunction setupSearchIndex({ config, docs }: SearchIndex): SearchIndex {\n\n /* Override default language with value from translation */\n if (config.lang.length === 1 && config.lang[0] === \"en\")\n config.lang = [\n translation(\"search.config.lang\")\n ]\n\n /* Override default separator with value from translation */\n if (config.separator === \"[\\\\s\\\\-]+\")\n config.separator = translation(\"search.config.separator\")\n\n /* Set pipeline from translation */\n const pipeline = translation(\"search.config.pipeline\")\n .split(/\\s*,\\s*/)\n .filter(Boolean) as SearchPipeline\n\n /* Determine search options */\n const options: SearchOptions = {\n pipeline,\n suggestions: feature(\"search.suggest\")\n }\n\n /* Return search index after defaulting */\n return { config, docs, options }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up search worker\n *\n * This function creates a web worker to set up and query the search index,\n * which is done using Lunr.js. The index must be passed as an observable to\n * enable hacks like _localsearch_ via search index embedding as JSON.\n *\n * @param url - Worker URL\n * @param index - Search index observable input\n *\n * @returns Search worker\n */\nexport function setupSearchWorker(\n url: string, index: ObservableInput\n): SearchWorker {\n const config = configuration()\n const worker = new Worker(url)\n\n /* Create communication channels and resolve relative links */\n const tx$ = new Subject()\n const rx$ = watchWorker(worker, { tx$ })\n .pipe(\n map(message => {\n if (isSearchResultMessage(message)) {\n for (const result of message.data.items)\n for (const document of result)\n document.location = `${new URL(document.location, config.base)}`\n }\n return message\n }),\n share()\n )\n\n /* Set up search index */\n from(index)\n .pipe(\n map(data => ({\n type: SearchMessageType.SETUP,\n data: setupSearchIndex(data)\n } as SearchSetupMessage))\n )\n .subscribe(tx$.next.bind(tx$))\n\n /* Return search worker */\n return { tx$, rx$ }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Subject,\n catchError,\n combineLatest,\n filter,\n fromEvent,\n map,\n of,\n switchMap,\n withLatestFrom\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport {\n getElement,\n getLocation,\n requestJSON,\n setLocation\n} from \"~/browser\"\nimport { getComponentElements } from \"~/components\"\nimport {\n Version,\n renderVersionSelector\n} from \"~/templates\"\n\nimport { fetchSitemap } from \"../sitemap\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n document$: Subject /* Document subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up version selector\n *\n * @param options - Options\n */\nexport function setupVersionSelector(\n { document$ }: SetupOptions\n): void {\n const config = configuration()\n const versions$ = requestJSON(\n new URL(\"../versions.json\", config.base)\n )\n .pipe(\n catchError(() => EMPTY) // @todo refactor instant loading\n )\n\n /* Determine current version */\n const current$ = versions$\n .pipe(\n map(versions => {\n const [, current] = config.base.match(/([^/]+)\\/?$/)!\n return versions.find(({ version, aliases }) => (\n version === current || aliases.includes(current)\n )) || versions[0]\n })\n )\n\n /* Intercept inter-version navigation */\n versions$\n .pipe(\n map(versions => new Map(versions.map(version => [\n `${new URL(`../${version.version}/`, config.base)}`,\n version\n ]))),\n switchMap(urls => fromEvent(document.body, \"click\")\n .pipe(\n filter(ev => !ev.metaKey && !ev.ctrlKey),\n withLatestFrom(current$),\n switchMap(([ev, current]) => {\n if (ev.target instanceof Element) {\n const el = ev.target.closest(\"a\")\n if (el && !el.target && urls.has(el.href)) {\n const url = el.href\n // This is a temporary hack to detect if a version inside the\n // version selector or on another part of the site was clicked.\n // If we're inside the version selector, we definitely want to\n // find the same page, as we might have different deployments\n // due to aliases. However, if we're outside the version\n // selector, we must abort here, because we might otherwise\n // interfere with instant loading. We need to refactor this\n // at some point together with instant loading.\n //\n // See https://github.com/squidfunk/mkdocs-material/issues/4012\n if (!ev.target.closest(\".md-version\")) {\n const version = urls.get(url)!\n if (version === current)\n return EMPTY\n }\n ev.preventDefault()\n return of(url)\n }\n }\n return EMPTY\n }),\n switchMap(url => {\n const { version } = urls.get(url)!\n return fetchSitemap(new URL(url))\n .pipe(\n map(sitemap => {\n const location = getLocation()\n const path = location.href.replace(config.base, \"\")\n return sitemap.includes(path.split(\"#\")[0])\n ? new URL(`../${version}/${path}`, config.base)\n : new URL(url)\n })\n )\n })\n )\n )\n )\n .subscribe(url => setLocation(url))\n\n /* Render version selector and warning */\n combineLatest([versions$, current$])\n .subscribe(([versions, current]) => {\n const topic = getElement(\".md-header__topic\")\n topic.appendChild(renderVersionSelector(versions, current))\n })\n\n /* Integrate outdated version banner with instant loading */\n document$.pipe(switchMap(() => current$))\n .subscribe(current => {\n\n /* Check if version state was already determined */\n let outdated = __md_get(\"__outdated\", sessionStorage)\n if (outdated === null) {\n const latest = config.version?.default || \"latest\"\n outdated = !current.aliases.includes(latest)\n\n /* Persist version state in session storage */\n __md_set(\"__outdated\", outdated, sessionStorage)\n }\n\n /* Unhide outdated version banner */\n if (outdated)\n for (const warning of getComponentElements(\"outdated\"))\n warning.hidden = false\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n combineLatest,\n delay,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n filter,\n finalize,\n fromEvent,\n map,\n merge,\n share,\n shareReplay,\n startWith,\n take,\n takeLast,\n takeUntil,\n tap\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport {\n getLocation,\n setToggle,\n watchElementFocus,\n watchToggle\n} from \"~/browser\"\nimport {\n SearchMessageType,\n SearchQueryMessage,\n SearchWorker,\n defaultTransform,\n isSearchReadyMessage\n} from \"~/integrations\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search query\n */\nexport interface SearchQuery {\n value: string /* Query value */\n focus: boolean /* Query focus */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch search query\n *\n * Note that the focus event which triggers re-reading the current query value\n * is delayed by `1ms` so the input's empty state is allowed to propagate.\n *\n * @param el - Search query element\n * @param worker - Search worker\n *\n * @returns Search query observable\n */\nexport function watchSearchQuery(\n el: HTMLInputElement, { rx$ }: SearchWorker\n): Observable {\n const fn = __search?.transform || defaultTransform\n\n /* Immediately show search dialog */\n const { searchParams } = getLocation()\n if (searchParams.has(\"q\"))\n setToggle(\"search\", true)\n\n /* Intercept query parameter (deep link) */\n const param$ = rx$\n .pipe(\n filter(isSearchReadyMessage),\n take(1),\n map(() => searchParams.get(\"q\") || \"\")\n )\n\n /* Remove query parameter when search is closed */\n watchToggle(\"search\")\n .pipe(\n filter(active => !active),\n take(1)\n )\n .subscribe(() => {\n const url = new URL(location.href)\n url.searchParams.delete(\"q\")\n history.replaceState({}, \"\", `${url}`)\n })\n\n /* Set query from parameter */\n param$.subscribe(value => { // TODO: not ideal - find a better way\n if (value) {\n el.value = value\n el.focus()\n }\n })\n\n /* Intercept focus and input events */\n const focus$ = watchElementFocus(el)\n const value$ = merge(\n fromEvent(el, \"keyup\"),\n fromEvent(el, \"focus\").pipe(delay(1)),\n param$\n )\n .pipe(\n map(() => fn(el.value)),\n startWith(\"\"),\n distinctUntilChanged(),\n )\n\n /* Combine into single observable */\n return combineLatest([value$, focus$])\n .pipe(\n map(([value, focus]) => ({ value, focus })),\n shareReplay(1)\n )\n}\n\n/**\n * Mount search query\n *\n * @param el - Search query element\n * @param worker - Search worker\n *\n * @returns Search query component observable\n */\nexport function mountSearchQuery(\n el: HTMLInputElement, { tx$, rx$ }: SearchWorker\n): Observable> {\n const push$ = new Subject()\n const done$ = push$.pipe(takeLast(1))\n\n /* Handle value changes */\n push$\n .pipe(\n distinctUntilKeyChanged(\"value\"),\n map(({ value }): SearchQueryMessage => ({\n type: SearchMessageType.QUERY,\n data: value\n }))\n )\n .subscribe(tx$.next.bind(tx$))\n\n /* Handle focus changes */\n push$\n .pipe(\n distinctUntilKeyChanged(\"focus\")\n )\n .subscribe(({ focus }) => {\n if (focus) {\n setToggle(\"search\", focus)\n el.placeholder = \"\"\n } else {\n el.placeholder = translation(\"search.placeholder\")\n }\n })\n\n /* Handle reset */\n fromEvent(el.form!, \"reset\")\n .pipe(\n takeUntil(done$)\n )\n .subscribe(() => el.focus())\n\n /* Create and return component */\n return watchSearchQuery(el, { tx$, rx$ })\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state })),\n share()\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n bufferCount,\n filter,\n finalize,\n map,\n merge,\n of,\n skipUntil,\n switchMap,\n take,\n tap,\n withLatestFrom,\n zipWith\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport {\n getElement,\n watchElementBoundary\n} from \"~/browser\"\nimport {\n SearchResult,\n SearchWorker,\n isSearchReadyMessage,\n isSearchResultMessage\n} from \"~/integrations\"\nimport { renderSearchResultItem } from \"~/templates\"\nimport { round } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\nimport { SearchQuery } from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n query$: Observable /* Search query observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search result list\n *\n * This function performs a lazy rendering of the search results, depending on\n * the vertical offset of the search result container.\n *\n * @param el - Search result list element\n * @param worker - Search worker\n * @param options - Options\n *\n * @returns Search result list component observable\n */\nexport function mountSearchResult(\n el: HTMLElement, { rx$ }: SearchWorker, { query$ }: MountOptions\n): Observable> {\n const push$ = new Subject()\n const boundary$ = watchElementBoundary(el.parentElement!)\n .pipe(\n filter(Boolean)\n )\n\n /* Retrieve nested components */\n const meta = getElement(\":scope > :first-child\", el)\n const list = getElement(\":scope > :last-child\", el)\n\n /* Wait until search is ready */\n const ready$ = rx$\n .pipe(\n filter(isSearchReadyMessage),\n take(1)\n )\n\n /* Update search result metadata */\n push$\n .pipe(\n withLatestFrom(query$),\n skipUntil(ready$)\n )\n .subscribe(([{ items }, { value }]) => {\n if (value) {\n switch (items.length) {\n\n /* No results */\n case 0:\n meta.textContent = translation(\"search.result.none\")\n break\n\n /* One result */\n case 1:\n meta.textContent = translation(\"search.result.one\")\n break\n\n /* Multiple result */\n default:\n meta.textContent = translation(\n \"search.result.other\",\n round(items.length)\n )\n }\n } else {\n meta.textContent = translation(\"search.result.placeholder\")\n }\n })\n\n /* Update search result list */\n push$\n .pipe(\n tap(() => list.innerHTML = \"\"),\n switchMap(({ items }) => merge(\n of(...items.slice(0, 10)),\n of(...items.slice(10))\n .pipe(\n bufferCount(4),\n zipWith(boundary$),\n switchMap(([chunk]) => chunk)\n )\n ))\n )\n .subscribe(result => list.appendChild(\n renderSearchResultItem(result)\n ))\n\n /* Filter search result message */\n const result$ = rx$\n .pipe(\n filter(isSearchResultMessage),\n map(({ data }) => data)\n )\n\n /* Create and return component */\n return result$\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n finalize,\n fromEvent,\n map,\n tap\n} from \"rxjs\"\n\nimport { getLocation } from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { SearchQuery } from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search sharing\n */\nexport interface SearchShare {\n url: URL /* Deep link for sharing */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n query$: Observable /* Search query observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n query$: Observable /* Search query observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search sharing\n *\n * @param _el - Search sharing element\n * @param options - Options\n *\n * @returns Search sharing observable\n */\nexport function watchSearchShare(\n _el: HTMLElement, { query$ }: WatchOptions\n): Observable {\n return query$\n .pipe(\n map(({ value }) => {\n const url = getLocation()\n url.hash = \"\"\n url.searchParams.delete(\"h\")\n url.searchParams.set(\"q\", value)\n return { url }\n })\n )\n}\n\n/**\n * Mount search sharing\n *\n * @param el - Search sharing element\n * @param options - Options\n *\n * @returns Search sharing component observable\n */\nexport function mountSearchShare(\n el: HTMLAnchorElement, options: MountOptions\n): Observable> {\n const push$ = new Subject()\n push$.subscribe(({ url }) => {\n el.setAttribute(\"data-clipboard-text\", el.href)\n el.href = `${url}`\n })\n\n /* Prevent following of link */\n fromEvent(el, \"click\")\n .subscribe(ev => ev.preventDefault())\n\n /* Create and return component */\n return watchSearchShare(el, options)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n asyncScheduler,\n combineLatestWith,\n distinctUntilChanged,\n filter,\n finalize,\n fromEvent,\n map,\n merge,\n observeOn,\n tap\n} from \"rxjs\"\n\nimport { Keyboard } from \"~/browser\"\nimport {\n SearchResult,\n SearchWorker,\n isSearchResultMessage\n} from \"~/integrations\"\n\nimport { Component, getComponentElement } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search suggestions\n */\nexport interface SearchSuggest {}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n keyboard$: Observable /* Keyboard observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search suggestions\n *\n * This function will perform a lazy rendering of the search results, depending\n * on the vertical offset of the search result container.\n *\n * @param el - Search result list element\n * @param worker - Search worker\n * @param options - Options\n *\n * @returns Search result list component observable\n */\nexport function mountSearchSuggest(\n el: HTMLElement, { rx$ }: SearchWorker, { keyboard$ }: MountOptions\n): Observable> {\n const push$ = new Subject()\n\n /* Retrieve query component and track all changes */\n const query = getComponentElement(\"search-query\")\n const query$ = merge(\n fromEvent(query, \"keydown\"),\n fromEvent(query, \"focus\")\n )\n .pipe(\n observeOn(asyncScheduler),\n map(() => query.value),\n distinctUntilChanged(),\n )\n\n /* Update search suggestions */\n push$\n .pipe(\n combineLatestWith(query$),\n map(([{ suggestions }, value]) => {\n const words = value.split(/([\\s-]+)/)\n if (suggestions?.length && words[words.length - 1]) {\n const last = suggestions[suggestions.length - 1]\n if (last.startsWith(words[words.length - 1]))\n words[words.length - 1] = last\n } else {\n words.length = 0\n }\n return words\n })\n )\n .subscribe(words => el.innerHTML = words\n .join(\"\")\n .replace(/\\s/g, \" \")\n )\n\n /* Set up search keyboard handlers */\n keyboard$\n .pipe(\n filter(({ mode }) => mode === \"search\")\n )\n .subscribe(key => {\n switch (key.type) {\n\n /* Right arrow: accept current suggestion */\n case \"ArrowRight\":\n if (\n el.innerText.length &&\n query.selectionStart === query.value.length\n )\n query.value = el.innerText\n break\n }\n })\n\n /* Filter search result message */\n const result$ = rx$\n .pipe(\n filter(isSearchResultMessage),\n map(({ data }) => data)\n )\n\n /* Create and return component */\n return result$\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(() => ({ ref: el }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n NEVER,\n Observable,\n ObservableInput,\n filter,\n merge,\n mergeWith,\n sample,\n take\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport {\n Keyboard,\n getActiveElement,\n getElements,\n setToggle\n} from \"~/browser\"\nimport {\n SearchIndex,\n SearchResult,\n isSearchQueryMessage,\n isSearchReadyMessage,\n setupSearchWorker\n} from \"~/integrations\"\n\nimport {\n Component,\n getComponentElement,\n getComponentElements\n} from \"../../_\"\nimport {\n SearchQuery,\n mountSearchQuery\n} from \"../query\"\nimport { mountSearchResult } from \"../result\"\nimport {\n SearchShare,\n mountSearchShare\n} from \"../share\"\nimport {\n SearchSuggest,\n mountSearchSuggest\n} from \"../suggest\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search\n */\nexport type Search =\n | SearchQuery\n | SearchResult\n | SearchShare\n | SearchSuggest\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n index$: ObservableInput /* Search index observable */\n keyboard$: Observable /* Keyboard observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search\n *\n * This function sets up the search functionality, including the underlying\n * web worker and all keyboard bindings.\n *\n * @param el - Search element\n * @param options - Options\n *\n * @returns Search component observable\n */\nexport function mountSearch(\n el: HTMLElement, { index$, keyboard$ }: MountOptions\n): Observable> {\n const config = configuration()\n try {\n const url = __search?.worker || config.search\n const worker = setupSearchWorker(url, index$)\n\n /* Retrieve query and result components */\n const query = getComponentElement(\"search-query\", el)\n const result = getComponentElement(\"search-result\", el)\n\n /* Re-emit query when search is ready */\n const { tx$, rx$ } = worker\n tx$\n .pipe(\n filter(isSearchQueryMessage),\n sample(rx$.pipe(filter(isSearchReadyMessage))),\n take(1)\n )\n .subscribe(tx$.next.bind(tx$))\n\n /* Set up search keyboard handlers */\n keyboard$\n .pipe(\n filter(({ mode }) => mode === \"search\")\n )\n .subscribe(key => {\n const active = getActiveElement()\n switch (key.type) {\n\n /* Enter: go to first (best) result */\n case \"Enter\":\n if (active === query) {\n const anchors = new Map()\n for (const anchor of getElements(\n \":first-child [href]\", result\n )) {\n const article = anchor.firstElementChild!\n anchors.set(anchor, parseFloat(\n article.getAttribute(\"data-md-score\")!\n ))\n }\n\n /* Go to result with highest score, if any */\n if (anchors.size) {\n const [[best]] = [...anchors].sort(([, a], [, b]) => b - a)\n best.click()\n }\n\n /* Otherwise omit form submission */\n key.claim()\n }\n break\n\n /* Escape or Tab: close search */\n case \"Escape\":\n case \"Tab\":\n setToggle(\"search\", false)\n query.blur()\n break\n\n /* Vertical arrows: select previous or next search result */\n case \"ArrowUp\":\n case \"ArrowDown\":\n if (typeof active === \"undefined\") {\n query.focus()\n } else {\n const els = [query, ...getElements(\n \":not(details) > [href], summary, details[open] [href]\",\n result\n )]\n const i = Math.max(0, (\n Math.max(0, els.indexOf(active)) + els.length + (\n key.type === \"ArrowUp\" ? -1 : +1\n )\n ) % els.length)\n els[i].focus()\n }\n\n /* Prevent scrolling of page */\n key.claim()\n break\n\n /* All other keys: hand to search query */\n default:\n if (query !== getActiveElement())\n query.focus()\n }\n })\n\n /* Set up global keyboard handlers */\n keyboard$\n .pipe(\n filter(({ mode }) => mode === \"global\"),\n )\n .subscribe(key => {\n switch (key.type) {\n\n /* Open search and select query */\n case \"f\":\n case \"s\":\n case \"/\":\n query.focus()\n query.select()\n\n /* Prevent scrolling of page */\n key.claim()\n break\n }\n })\n\n /* Create and return component */\n const query$ = mountSearchQuery(query, worker)\n const result$ = mountSearchResult(result, worker, { query$ })\n return merge(query$, result$)\n .pipe(\n mergeWith(\n\n /* Search sharing */\n ...getComponentElements(\"search-share\", el)\n .map(child => mountSearchShare(child, { query$ })),\n\n /* Search suggestions */\n ...getComponentElements(\"search-suggest\", el)\n .map(child => mountSearchSuggest(child, worker, { keyboard$ }))\n )\n )\n\n /* Gracefully handle broken search */\n } catch (err) {\n el.hidden = true\n return NEVER\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n ObservableInput,\n combineLatest,\n filter,\n map,\n startWith\n} from \"rxjs\"\n\nimport { getLocation } from \"~/browser\"\nimport {\n SearchIndex,\n setupSearchHighlighter\n} from \"~/integrations\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlighting\n */\nexport interface SearchHighlight {\n nodes: Map /* Map of replacements */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n index$: ObservableInput /* Search index observable */\n location$: Observable /* Location observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search highlighting\n *\n * @param el - Content element\n * @param options - Options\n *\n * @returns Search highlighting component observable\n */\nexport function mountSearchHiglight(\n el: HTMLElement, { index$, location$ }: MountOptions\n): Observable> {\n return combineLatest([\n index$,\n location$\n .pipe(\n startWith(getLocation()),\n filter(url => !!url.searchParams.get(\"h\"))\n )\n ])\n .pipe(\n map(([index, url]) => setupSearchHighlighter(index.config, true)(\n url.searchParams.get(\"h\")!\n )),\n map(fn => {\n const nodes = new Map()\n\n /* Traverse text nodes and collect matches */\n const it = document.createNodeIterator(el, NodeFilter.SHOW_TEXT)\n for (let node = it.nextNode(); node; node = it.nextNode()) {\n if (node.parentElement?.offsetHeight) {\n const original = node.textContent!\n const replaced = fn(original)\n if (replaced.length > original.length)\n nodes.set(node as ChildNode, replaced)\n }\n }\n\n /* Replace original nodes with matches */\n for (const [node, text] of nodes) {\n const { childNodes } = h(\"span\", null, text)\n node.replaceWith(...Array.from(childNodes))\n }\n\n /* Return component */\n return { ref: el, nodes }\n })\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n animationFrameScheduler,\n auditTime,\n combineLatest,\n defer,\n distinctUntilChanged,\n finalize,\n map,\n observeOn,\n take,\n tap,\n withLatestFrom\n} from \"rxjs\"\n\nimport {\n Viewport,\n getElement,\n getElementContainer,\n getElementOffset,\n getElementSize,\n getElements\n} from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Sidebar\n */\nexport interface Sidebar {\n height: number /* Sidebar height */\n locked: boolean /* Sidebar is locked */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n main$: Observable
    /* Main area observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n main$: Observable
    /* Main area observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch sidebar\n *\n * This function returns an observable that computes the visual parameters of\n * the sidebar which depends on the vertical viewport offset, as well as the\n * height of the main area. When the page is scrolled beyond the header, the\n * sidebar is locked and fills the remaining space.\n *\n * @param el - Sidebar element\n * @param options - Options\n *\n * @returns Sidebar observable\n */\nexport function watchSidebar(\n el: HTMLElement, { viewport$, main$ }: WatchOptions\n): Observable {\n const parent = el.parentElement!\n const adjust =\n parent.offsetTop -\n parent.parentElement!.offsetTop\n\n /* Compute the sidebar's available height and if it should be locked */\n return combineLatest([main$, viewport$])\n .pipe(\n map(([{ offset, height }, { offset: { y } }]) => {\n height = height\n + Math.min(adjust, Math.max(0, y - offset))\n - adjust\n return {\n height,\n locked: y >= offset + adjust\n }\n }),\n distinctUntilChanged((a, b) => (\n a.height === b.height &&\n a.locked === b.locked\n ))\n )\n}\n\n/**\n * Mount sidebar\n *\n * This function doesn't set the height of the actual sidebar, but of its first\n * child \u2013 the `.md-sidebar__scrollwrap` element in order to mitigiate jittery\n * sidebars when the footer is scrolled into view. At some point we switched\n * from `absolute` / `fixed` positioning to `sticky` positioning, significantly\n * reducing jitter in some browsers (respectively Firefox and Safari) when\n * scrolling from the top. However, top-aligned sticky positioning means that\n * the sidebar snaps to the bottom when the end of the container is reached.\n * This is what leads to the mentioned jitter, as the sidebar's height may be\n * updated too slowly.\n *\n * This behaviour can be mitigiated by setting the height of the sidebar to `0`\n * while preserving the padding, and the height on its first element.\n *\n * @param el - Sidebar element\n * @param options - Options\n *\n * @returns Sidebar component observable\n */\nexport function mountSidebar(\n el: HTMLElement, { header$, ...options }: MountOptions\n): Observable> {\n const inner = getElement(\".md-sidebar__scrollwrap\", el)\n const { y } = getElementOffset(inner)\n return defer(() => {\n const push$ = new Subject()\n push$\n .pipe(\n auditTime(0, animationFrameScheduler),\n withLatestFrom(header$)\n )\n .subscribe({\n\n /* Handle emission */\n next([{ height }, { height: offset }]) {\n inner.style.height = `${height - 2 * y}px`\n el.style.top = `${offset}px`\n },\n\n /* Handle complete */\n complete() {\n inner.style.height = \"\"\n el.style.top = \"\"\n }\n })\n\n /* Bring active item into view on initial load */\n push$\n .pipe(\n observeOn(animationFrameScheduler),\n take(1)\n )\n .subscribe(() => {\n for (const item of getElements(\".md-nav__link--active[href]\", el)) {\n const container = getElementContainer(item)\n if (typeof container !== \"undefined\") {\n const offset = item.offsetTop - container.offsetTop\n const { height } = getElementSize(container)\n container.scrollTo({\n top: offset - height / 2\n })\n }\n }\n })\n\n /* Create and return component */\n return watchSidebar(el, options)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Repo, User } from \"github-types\"\nimport {\n EMPTY,\n Observable,\n catchError,\n defaultIfEmpty,\n map,\n zip\n} from \"rxjs\"\n\nimport { requestJSON } from \"~/browser\"\n\nimport { SourceFacts } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * GitHub release (partial)\n */\ninterface Release {\n tag_name: string /* Tag name */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch GitHub repository facts\n *\n * @param user - GitHub user or organization\n * @param repo - GitHub repository\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFactsFromGitHub(\n user: string, repo?: string\n): Observable {\n if (typeof repo !== \"undefined\") {\n const url = `https://api.github.com/repos/${user}/${repo}`\n return zip(\n\n /* Fetch version */\n requestJSON(`${url}/releases/latest`)\n .pipe(\n catchError(() => EMPTY), // @todo refactor instant loading\n map(release => ({\n version: release.tag_name\n })),\n defaultIfEmpty({})\n ),\n\n /* Fetch stars and forks */\n requestJSON(url)\n .pipe(\n catchError(() => EMPTY), // @todo refactor instant loading\n map(info => ({\n stars: info.stargazers_count,\n forks: info.forks_count\n })),\n defaultIfEmpty({})\n )\n )\n .pipe(\n map(([release, info]) => ({ ...release, ...info }))\n )\n\n /* User or organization */\n } else {\n const url = `https://api.github.com/users/${user}`\n return requestJSON(url)\n .pipe(\n map(info => ({\n repositories: info.public_repos\n })),\n defaultIfEmpty({})\n )\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ProjectSchema } from \"gitlab\"\nimport {\n EMPTY,\n Observable,\n catchError,\n defaultIfEmpty,\n map\n} from \"rxjs\"\n\nimport { requestJSON } from \"~/browser\"\n\nimport { SourceFacts } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch GitLab repository facts\n *\n * @param base - GitLab base\n * @param project - GitLab project\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFactsFromGitLab(\n base: string, project: string\n): Observable {\n const url = `https://${base}/api/v4/projects/${encodeURIComponent(project)}`\n return requestJSON(url)\n .pipe(\n catchError(() => EMPTY), // @todo refactor instant loading\n map(({ star_count, forks_count }) => ({\n stars: star_count,\n forks: forks_count\n })),\n defaultIfEmpty({})\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { EMPTY, Observable } from \"rxjs\"\n\nimport { fetchSourceFactsFromGitHub } from \"../github\"\nimport { fetchSourceFactsFromGitLab } from \"../gitlab\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository facts for repositories\n */\nexport interface RepositoryFacts {\n stars?: number /* Number of stars */\n forks?: number /* Number of forks */\n version?: string /* Latest version */\n}\n\n/**\n * Repository facts for organizations\n */\nexport interface OrganizationFacts {\n repositories?: number /* Number of repositories */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Repository facts\n */\nexport type SourceFacts =\n | RepositoryFacts\n | OrganizationFacts\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch repository facts\n *\n * @param url - Repository URL\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFacts(\n url: string\n): Observable {\n const [type] = url.match(/(git(?:hub|lab))/i) || []\n switch (type.toLowerCase()) {\n\n /* GitHub repository */\n case \"github\":\n const [, user, repo] = url.match(/^.+github\\.com\\/([^/]+)\\/?([^/]+)?/i)!\n return fetchSourceFactsFromGitHub(user, repo)\n\n /* GitLab repository */\n case \"gitlab\":\n const [, base, slug] = url.match(/^.+?([^/]*gitlab[^/]+)\\/(.+?)\\/?$/i)!\n return fetchSourceFactsFromGitLab(base, slug)\n\n /* Everything else */\n default:\n return EMPTY\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n EMPTY,\n Observable,\n Subject,\n catchError,\n defer,\n filter,\n finalize,\n map,\n of,\n shareReplay,\n tap\n} from \"rxjs\"\n\nimport { getElement } from \"~/browser\"\nimport { ConsentDefaults } from \"~/components/consent\"\nimport { renderSourceFacts } from \"~/templates\"\n\nimport {\n Component,\n getComponentElements\n} from \"../../_\"\nimport {\n SourceFacts,\n fetchSourceFacts\n} from \"../facts\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository information\n */\nexport interface Source {\n facts: SourceFacts /* Repository facts */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository information observable\n */\nlet fetch$: Observable\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch repository information\n *\n * This function tries to read the repository facts from session storage, and\n * if unsuccessful, fetches them from the underlying provider.\n *\n * @param el - Repository information element\n *\n * @returns Repository information observable\n */\nexport function watchSource(\n el: HTMLAnchorElement\n): Observable {\n return fetch$ ||= defer(() => {\n const cached = __md_get(\"__source\", sessionStorage)\n if (cached) {\n return of(cached)\n } else {\n\n /* Check if consent is configured and was given */\n const els = getComponentElements(\"consent\")\n if (els.length) {\n const consent = __md_get(\"__consent\")\n if (!(consent && consent.github))\n return EMPTY\n }\n\n /* Fetch repository facts */\n return fetchSourceFacts(el.href)\n .pipe(\n tap(facts => __md_set(\"__source\", facts, sessionStorage))\n )\n }\n })\n .pipe(\n catchError(() => EMPTY),\n filter(facts => Object.keys(facts).length > 0),\n map(facts => ({ facts })),\n shareReplay(1)\n )\n}\n\n/**\n * Mount repository information\n *\n * @param el - Repository information element\n *\n * @returns Repository information component observable\n */\nexport function mountSource(\n el: HTMLAnchorElement\n): Observable> {\n const inner = getElement(\":scope > :last-child\", el)\n return defer(() => {\n const push$ = new Subject()\n push$.subscribe(({ facts }) => {\n inner.appendChild(renderSourceFacts(facts))\n inner.classList.add(\"md-source__repository--active\")\n })\n\n /* Create and return component */\n return watchSource(el)\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n defer,\n distinctUntilKeyChanged,\n finalize,\n map,\n of,\n switchMap,\n tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n Viewport,\n watchElementSize,\n watchViewportAt\n} from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Navigation tabs\n */\nexport interface Tabs {\n hidden: boolean /* Navigation tabs are hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch navigation tabs\n *\n * @param el - Navigation tabs element\n * @param options - Options\n *\n * @returns Navigation tabs observable\n */\nexport function watchTabs(\n el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable {\n return watchElementSize(document.body)\n .pipe(\n switchMap(() => watchViewportAt(el, { header$, viewport$ })),\n map(({ offset: { y } }) => {\n return {\n hidden: y >= 10\n }\n }),\n distinctUntilKeyChanged(\"hidden\")\n )\n}\n\n/**\n * Mount navigation tabs\n *\n * This function hides the navigation tabs when scrolling past the threshold\n * and makes them reappear in a nice CSS animation when scrolling back up.\n *\n * @param el - Navigation tabs element\n * @param options - Options\n *\n * @returns Navigation tabs component observable\n */\nexport function mountTabs(\n el: HTMLElement, options: MountOptions\n): Observable> {\n return defer(() => {\n const push$ = new Subject()\n push$.subscribe({\n\n /* Handle emission */\n next({ hidden }) {\n el.hidden = hidden\n },\n\n /* Handle complete */\n complete() {\n el.hidden = false\n }\n })\n\n /* Create and return component */\n return (\n feature(\"navigation.tabs.sticky\")\n ? of({ hidden: false })\n : watchTabs(el, options)\n )\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n bufferCount,\n combineLatestWith,\n debounceTime,\n defer,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n filter,\n finalize,\n map,\n merge,\n of,\n repeat,\n scan,\n share,\n skip,\n startWith,\n switchMap,\n takeLast,\n takeUntil,\n tap,\n withLatestFrom\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n Viewport,\n getElement,\n getElementContainer,\n getElementSize,\n getElements,\n getLocation,\n getOptionalElement,\n watchElementSize\n} from \"~/browser\"\n\nimport {\n Component,\n getComponentElement\n} from \"../_\"\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Table of contents\n */\nexport interface TableOfContents {\n prev: HTMLAnchorElement[][] /* Anchors (previous) */\n next: HTMLAnchorElement[][] /* Anchors (next) */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n target$: Observable /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch table of contents\n *\n * This is effectively a scroll spy implementation which will account for the\n * fixed header and automatically re-calculate anchor offsets when the viewport\n * is resized. The returned observable will only emit if the table of contents\n * needs to be repainted.\n *\n * This implementation tracks an anchor element's entire path starting from its\n * level up to the top-most anchor element, e.g. `[h3, h2, h1]`. Although the\n * Material theme currently doesn't make use of this information, it enables\n * the styling of the entire hierarchy through customization.\n *\n * Note that the current anchor is the last item of the `prev` anchor list.\n *\n * @param el - Table of contents element\n * @param options - Options\n *\n * @returns Table of contents observable\n */\nexport function watchTableOfContents(\n el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable {\n const table = new Map()\n\n /* Compute anchor-to-target mapping */\n const anchors = getElements(\"[href^=\\\\#]\", el)\n for (const anchor of anchors) {\n const id = decodeURIComponent(anchor.hash.substring(1))\n const target = getOptionalElement(`[id=\"${id}\"]`)\n if (typeof target !== \"undefined\")\n table.set(anchor, target)\n }\n\n /* Compute necessary adjustment for header */\n const adjust$ = header$\n .pipe(\n distinctUntilKeyChanged(\"height\"),\n map(({ height }) => {\n const main = getComponentElement(\"main\")\n const grid = getElement(\":scope > :first-child\", main)\n return height + 0.8 * (\n grid.offsetTop -\n main.offsetTop\n )\n }),\n share()\n )\n\n /* Compute partition of previous and next anchors */\n const partition$ = watchElementSize(document.body)\n .pipe(\n distinctUntilKeyChanged(\"height\"),\n\n /* Build index to map anchor paths to vertical offsets */\n switchMap(body => defer(() => {\n let path: HTMLAnchorElement[] = []\n return of([...table].reduce((index, [anchor, target]) => {\n while (path.length) {\n const last = table.get(path[path.length - 1])!\n if (last.tagName >= target.tagName) {\n path.pop()\n } else {\n break\n }\n }\n\n /* If the current anchor is hidden, continue with its parent */\n let offset = target.offsetTop\n while (!offset && target.parentElement) {\n target = target.parentElement\n offset = target.offsetTop\n }\n\n /* Map reversed anchor path to vertical offset */\n return index.set(\n [...path = [...path, anchor]].reverse(),\n offset\n )\n }, new Map()))\n })\n .pipe(\n\n /* Sort index by vertical offset (see https://bit.ly/30z6QSO) */\n map(index => new Map([...index].sort(([, a], [, b]) => a - b))),\n combineLatestWith(adjust$),\n\n /* Re-compute partition when viewport offset changes */\n switchMap(([index, adjust]) => viewport$\n .pipe(\n scan(([prev, next], { offset: { y }, size }) => {\n const last = y + size.height >= Math.floor(body.height)\n\n /* Look forward */\n while (next.length) {\n const [, offset] = next[0]\n if (offset - adjust < y || last) {\n prev = [...prev, next.shift()!]\n } else {\n break\n }\n }\n\n /* Look backward */\n while (prev.length) {\n const [, offset] = prev[prev.length - 1]\n if (offset - adjust >= y && !last) {\n next = [prev.pop()!, ...next]\n } else {\n break\n }\n }\n\n /* Return partition */\n return [prev, next]\n }, [[], [...index]]),\n distinctUntilChanged((a, b) => (\n a[0] === b[0] &&\n a[1] === b[1]\n ))\n )\n )\n )\n )\n )\n\n /* Compute and return anchor list migrations */\n return partition$\n .pipe(\n map(([prev, next]) => ({\n prev: prev.map(([path]) => path),\n next: next.map(([path]) => path)\n })),\n\n /* Extract anchor list migrations */\n startWith({ prev: [], next: [] }),\n bufferCount(2, 1),\n map(([a, b]) => {\n\n /* Moving down */\n if (a.prev.length < b.prev.length) {\n return {\n prev: b.prev.slice(Math.max(0, a.prev.length - 1), b.prev.length),\n next: []\n }\n\n /* Moving up */\n } else {\n return {\n prev: b.prev.slice(-1),\n next: b.next.slice(0, b.next.length - a.next.length)\n }\n }\n })\n )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount table of contents\n *\n * @param el - Table of contents element\n * @param options - Options\n *\n * @returns Table of contents component observable\n */\nexport function mountTableOfContents(\n el: HTMLElement, { viewport$, header$, target$ }: MountOptions\n): Observable> {\n return defer(() => {\n const push$ = new Subject()\n const done$ = push$.pipe(takeLast(1))\n push$.subscribe(({ prev, next }) => {\n\n /* Look forward */\n for (const [anchor] of next) {\n anchor.classList.remove(\"md-nav__link--passed\")\n anchor.classList.remove(\"md-nav__link--active\")\n }\n\n /* Look backward */\n for (const [index, [anchor]] of prev.entries()) {\n anchor.classList.add(\"md-nav__link--passed\")\n anchor.classList.toggle(\n \"md-nav__link--active\",\n index === prev.length - 1\n )\n }\n })\n\n /* Set up following, if enabled */\n if (feature(\"toc.follow\")) {\n\n /* Toggle smooth scrolling only for anchor clicks */\n const smooth$ = merge(\n viewport$.pipe(debounceTime(1), map(() => undefined)),\n viewport$.pipe(debounceTime(250), map(() => \"smooth\" as const))\n )\n\n /* Bring active anchor into view */\n push$\n .pipe(\n filter(({ prev }) => prev.length > 0),\n withLatestFrom(smooth$)\n )\n .subscribe(([{ prev }, behavior]) => {\n const [anchor] = prev[prev.length - 1]\n if (anchor.offsetHeight) {\n\n /* Retrieve overflowing container and scroll */\n const container = getElementContainer(anchor)\n if (typeof container !== \"undefined\") {\n const offset = anchor.offsetTop - container.offsetTop\n const { height } = getElementSize(container)\n container.scrollTo({\n top: offset - height / 2,\n behavior\n })\n }\n }\n })\n }\n\n /* Set up anchor tracking, if enabled */\n if (feature(\"navigation.tracking\"))\n viewport$\n .pipe(\n takeUntil(done$),\n distinctUntilKeyChanged(\"offset\"),\n debounceTime(250),\n skip(1),\n takeUntil(target$.pipe(skip(1))),\n repeat({ delay: 250 }),\n withLatestFrom(push$)\n )\n .subscribe(([, { prev }]) => {\n const url = getLocation()\n\n /* Set hash fragment to active anchor */\n const anchor = prev[prev.length - 1]\n if (anchor && anchor.length) {\n const [active] = anchor\n const { hash } = new URL(active.href)\n if (url.hash !== hash) {\n url.hash = hash\n history.replaceState({}, \"\", `${url}`)\n }\n\n /* Reset anchor when at the top */\n } else {\n url.hash = \"\"\n history.replaceState({}, \"\", `${url}`)\n }\n })\n\n /* Create and return component */\n return watchTableOfContents(el, { viewport$, header$ })\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n Subject,\n bufferCount,\n combineLatest,\n distinctUntilChanged,\n distinctUntilKeyChanged,\n endWith,\n finalize,\n map,\n repeat,\n skip,\n takeLast,\n takeUntil,\n tap\n} from \"rxjs\"\n\nimport { Viewport } from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Back-to-top button\n */\nexport interface BackToTop {\n hidden: boolean /* Back-to-top button is hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n viewport$: Observable /* Viewport observable */\n main$: Observable
    /* Main area observable */\n target$: Observable /* Location target observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n viewport$: Observable /* Viewport observable */\n header$: Observable
    /* Header observable */\n main$: Observable
    /* Main area observable */\n target$: Observable /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch back-to-top\n *\n * @param _el - Back-to-top element\n * @param options - Options\n *\n * @returns Back-to-top observable\n */\nexport function watchBackToTop(\n _el: HTMLElement, { viewport$, main$, target$ }: WatchOptions\n): Observable {\n\n /* Compute direction */\n const direction$ = viewport$\n .pipe(\n map(({ offset: { y } }) => y),\n bufferCount(2, 1),\n map(([a, b]) => a > b && b > 0),\n distinctUntilChanged()\n )\n\n /* Compute whether main area is active */\n const active$ = main$\n .pipe(\n map(({ active }) => active)\n )\n\n /* Compute threshold for hiding */\n return combineLatest([active$, direction$])\n .pipe(\n map(([active, direction]) => !(active && direction)),\n distinctUntilChanged(),\n takeUntil(target$.pipe(skip(1))),\n endWith(true),\n repeat({ delay: 250 }),\n map(hidden => ({ hidden }))\n )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount back-to-top\n *\n * @param el - Back-to-top element\n * @param options - Options\n *\n * @returns Back-to-top component observable\n */\nexport function mountBackToTop(\n el: HTMLElement, { viewport$, header$, main$, target$ }: MountOptions\n): Observable> {\n const push$ = new Subject()\n const done$ = push$.pipe(takeLast(1))\n push$.subscribe({\n\n /* Handle emission */\n next({ hidden }) {\n el.hidden = hidden\n if (hidden) {\n el.setAttribute(\"tabindex\", \"-1\")\n el.blur()\n } else {\n el.removeAttribute(\"tabindex\")\n }\n },\n\n /* Handle complete */\n complete() {\n el.style.top = \"\"\n el.hidden = true\n el.removeAttribute(\"tabindex\")\n }\n })\n\n /* Watch header height */\n header$\n .pipe(\n takeUntil(done$),\n distinctUntilKeyChanged(\"height\")\n )\n .subscribe(({ height }) => {\n el.style.top = `${height + 16}px`\n })\n\n /* Create and return component */\n return watchBackToTop(el, { viewport$, main$, target$ })\n .pipe(\n tap(state => push$.next(state)),\n finalize(() => push$.complete()),\n map(state => ({ ref: el, ...state }))\n )\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n fromEvent,\n map,\n mergeMap,\n switchMap,\n takeWhile,\n tap,\n withLatestFrom\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n document$: Observable /* Document observable */\n tablet$: Observable /* Media tablet observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch indeterminate checkboxes\n *\n * This function replaces the indeterminate \"pseudo state\" with the actual\n * indeterminate state, which is used to keep navigation always expanded.\n *\n * @param options - Options\n */\nexport function patchIndeterminate(\n { document$, tablet$ }: PatchOptions\n): void {\n document$\n .pipe(\n switchMap(() => getElements(\n // @todo `data-md-state` is deprecated and removed in v9\n \".md-toggle--indeterminate, [data-md-state=indeterminate]\"\n )),\n tap(el => {\n el.indeterminate = true\n el.checked = false\n }),\n mergeMap(el => fromEvent(el, \"change\")\n .pipe(\n takeWhile(() => el.classList.contains(\"md-toggle--indeterminate\")),\n map(() => el)\n )\n ),\n withLatestFrom(tablet$)\n )\n .subscribe(([el, tablet]) => {\n el.classList.remove(\"md-toggle--indeterminate\")\n if (tablet)\n el.checked = false\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n filter,\n fromEvent,\n map,\n mergeMap,\n switchMap,\n tap\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n document$: Observable /* Document observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Check whether the given device is an Apple device\n *\n * @returns Test result\n */\nfunction isAppleDevice(): boolean {\n return /(iPad|iPhone|iPod)/.test(navigator.userAgent)\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch all elements with `data-md-scrollfix` attributes\n *\n * This is a year-old patch which ensures that overflow scrolling works at the\n * top and bottom of containers on iOS by ensuring a `1px` scroll offset upon\n * the start of a touch event.\n *\n * @see https://bit.ly/2SCtAOO - Original source\n *\n * @param options - Options\n */\nexport function patchScrollfix(\n { document$ }: PatchOptions\n): void {\n document$\n .pipe(\n switchMap(() => getElements(\"[data-md-scrollfix]\")),\n tap(el => el.removeAttribute(\"data-md-scrollfix\")),\n filter(isAppleDevice),\n mergeMap(el => fromEvent(el, \"touchstart\")\n .pipe(\n map(() => el)\n )\n )\n )\n .subscribe(el => {\n const top = el.scrollTop\n\n /* We're at the top of the container */\n if (top === 0) {\n el.scrollTop = 1\n\n /* We're at the bottom of the container */\n } else if (top + el.offsetHeight === el.scrollHeight) {\n el.scrollTop = top - 1\n }\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n Observable,\n combineLatest,\n delay,\n map,\n of,\n switchMap,\n withLatestFrom\n} from \"rxjs\"\n\nimport {\n Viewport,\n watchToggle\n} from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n viewport$: Observable /* Viewport observable */\n tablet$: Observable /* Media tablet observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch the document body to lock when search is open\n *\n * For mobile and tablet viewports, the search is rendered full screen, which\n * leads to scroll leaking when at the top or bottom of the search result. This\n * function locks the body when the search is in full screen mode, and restores\n * the scroll position when leaving.\n *\n * @param options - Options\n */\nexport function patchScrolllock(\n { viewport$, tablet$ }: PatchOptions\n): void {\n combineLatest([watchToggle(\"search\"), tablet$])\n .pipe(\n map(([active, tablet]) => active && !tablet),\n switchMap(active => of(active)\n .pipe(\n delay(active ? 400 : 100)\n )\n ),\n withLatestFrom(viewport$)\n )\n .subscribe(([active, { offset: { y }}]) => {\n if (active) {\n document.body.setAttribute(\"data-md-scrolllock\", \"\")\n document.body.style.top = `-${y}px`\n } else {\n const value = -1 * parseInt(document.body.style.top, 10)\n document.body.removeAttribute(\"data-md-scrolllock\")\n document.body.style.top = \"\"\n if (value)\n window.scrollTo(0, value)\n }\n })\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Polyfills\n * ------------------------------------------------------------------------- */\n\n/* Polyfill `Object.entries` */\nif (!Object.entries)\n Object.entries = function (obj: object) {\n const data: [string, string][] = []\n for (const key of Object.keys(obj))\n // @ts-expect-error - ignore property access warning\n data.push([key, obj[key]])\n\n /* Return entries */\n return data\n }\n\n/* Polyfill `Object.values` */\nif (!Object.values)\n Object.values = function (obj: object) {\n const data: string[] = []\n for (const key of Object.keys(obj))\n // @ts-expect-error - ignore property access warning\n 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nu når og også om op os over på selv sig sin sine sit skal skulle som sådan thi til ud under var vi vil ville vor være været".split(" ")),e.Pipeline.registerFunction(e.da.stopWordFilter,"stopWordFilter-da")}}); \ No newline at end of file diff --git a/assets/javascripts/lunr/min/lunr.de.min.js b/assets/javascripts/lunr/min/lunr.de.min.js new file mode 100644 index 00000000..f3b5c108 --- /dev/null +++ b/assets/javascripts/lunr/min/lunr.de.min.js @@ -0,0 +1,18 @@ +/*! + * Lunr languages, `German` language + * https://github.com/MihaiValentin/lunr-languages + * + * Copyright 2014, Mihai Valentin + * http://www.mozilla.org/MPL/ + */ +/*! + * based on + * Snowball JavaScript Library v0.3 + * http://code.google.com/p/urim/ + * http://snowball.tartarus.org/ + * + * Copyright 2010, Oleg Mazko + * http://www.mozilla.org/MPL/ + */ + +!function(e,r){"function"==typeof define&&define.amd?define(r):"object"==typeof exports?module.exports=r():r()(e.lunr)}(this,function(){return function(e){if(void 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andere anderem anderen anderer anderes anderm andern anderr anders auch auf aus bei bin bis bist da damit dann das dasselbe dazu daß dein deine deinem deinen deiner deines dem demselben den denn denselben der derer derselbe derselben des desselben dessen dich die dies diese dieselbe dieselben diesem diesen dieser dieses dir doch dort du durch ein eine einem einen einer eines einig einige einigem einigen einiger einiges einmal er es etwas euch euer eure eurem euren eurer eures für gegen gewesen hab habe haben hat hatte hatten hier hin hinter ich ihm ihn ihnen ihr ihre ihrem ihren ihrer ihres im in indem ins ist jede jedem jeden jeder jedes jene jenem jenen jener jenes jetzt kann kein keine keinem keinen keiner keines können könnte machen man manche manchem manchen mancher manches mein meine meinem meinen meiner meines mich mir mit muss musste nach nicht nichts noch nun nur ob oder ohne sehr sein seine seinem seinen seiner seines selbst sich sie sind so solche solchem solchen solcher solches soll sollte sondern sonst um und uns unse unsem unsen unser unses unter viel vom von vor war waren warst was weg weil weiter welche welchem welchen welcher welches wenn werde werden wie wieder will wir wird wirst wo wollen wollte während würde würden zu zum zur zwar zwischen über".split(" ")),e.Pipeline.registerFunction(e.de.stopWordFilter,"stopWordFilter-de")}}); \ No newline at end of file diff --git a/assets/javascripts/lunr/min/lunr.du.min.js b/assets/javascripts/lunr/min/lunr.du.min.js new file mode 100644 index 00000000..49a0f3f0 --- /dev/null +++ b/assets/javascripts/lunr/min/lunr.du.min.js @@ -0,0 +1,18 @@ +/*! + * Lunr languages, `Dutch` language + * https://github.com/MihaiValentin/lunr-languages + * + * Copyright 2014, Mihai Valentin + * http://www.mozilla.org/MPL/ + */ +/*! + * based on + * Snowball JavaScript Library v0.3 + * http://code.google.com/p/urim/ + * http://snowball.tartarus.org/ + * + * Copyright 2010, Oleg Mazko + * http://www.mozilla.org/MPL/ + */ + +!function(e,r){"function"==typeof define&&define.amd?define(r):"object"==typeof exports?module.exports=r():r()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. 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Lunr languages, `Italian` language + * https://github.com/MihaiValentin/lunr-languages + * + * Copyright 2014, Mihai Valentin + * http://www.mozilla.org/MPL/ + */ +/*! + * based on + * Snowball JavaScript Library v0.3 + * http://code.google.com/p/urim/ + * http://snowball.tartarus.org/ + * + * Copyright 2010, Oleg Mazko + * http://www.mozilla.org/MPL/ + */ + +!function(e,r){"function"==typeof define&&define.amd?define(r):"object"==typeof exports?module.exports=r():r()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. Please include / require Lunr before this script.");if(void 0===e.stemmerSupport)throw new Error("Lunr stemmer support is not present. Please include / require Lunr stemmer support before this script.");e.it=function(){this.pipeline.reset(),this.pipeline.add(e.it.trimmer,e.it.stopWordFilter,e.it.stemmer),this.searchPipeline&&(this.searchPipeline.reset(),this.searchPipeline.add(e.it.stemmer))},e.it.wordCharacters="A-Za-zªºÀ-ÖØ-öø-ʸˠ-ˤᴀ-ᴥᴬ-ᵜᵢ-ᵥᵫ-ᵷᵹ-ᶾḀ-ỿⁱⁿₐ-ₜKÅℲⅎⅠ-ↈⱠ-ⱿꜢ-ꞇꞋ-ꞭꞰ-ꞷꟷ-ꟿꬰ-ꭚꭜ-ꭤff-stＡ-Ｚａ-ｚ",e.it.trimmer=e.trimmerSupport.generateTrimmer(e.it.wordCharacters),e.Pipeline.registerFunction(e.it.trimmer,"trimmer-it"),e.it.stemmer=function(){var r=e.stemmerSupport.Among,n=e.stemmerSupport.SnowballProgram,i=new function(){function e(e,r,n){return!(!x.eq_s(1,e)||(x.ket=x.cursor,!x.in_grouping(L,97,249)))&&(x.slice_from(r),x.cursor=n,!0)}function i(){for(var r,n,i,o,t=x.cursor;;){if(x.bra=x.cursor,r=x.find_among(h,7))switch(x.ket=x.cursor,r){case 1:x.slice_from("à");continue;case 2:x.slice_from("è");continue;case 3:x.slice_from("ì");continue;case 4:x.slice_from("ò");continue;case 5:x.slice_from("ù");continue;case 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a(){for(;!x.in_grouping(L,97,249);){if(x.cursor>=x.limit)return!1;x.cursor++}for(;!x.out_grouping(L,97,249);){if(x.cursor>=x.limit)return!1;x.cursor++}return!0}function u(){var e=x.cursor;k=x.limit,p=k,g=k,s(),x.cursor=e,a()&&(p=x.cursor,a()&&(g=x.cursor))}function c(){for(var e;;){if(x.bra=x.cursor,!(e=x.find_among(q,3)))break;switch(x.ket=x.cursor,e){case 1:x.slice_from("i");break;case 2:x.slice_from("u");break;case 3:if(x.cursor>=x.limit)return;x.cursor++}}}function w(){return k<=x.cursor}function l(){return p<=x.cursor}function m(){return g<=x.cursor}function f(){var e;if(x.ket=x.cursor,x.find_among_b(C,37)&&(x.bra=x.cursor,(e=x.find_among_b(z,5))&&w()))switch(e){case 1:x.slice_del();break;case 2:x.slice_from("e")}}function v(){var e;if(x.ket=x.cursor,!(e=x.find_among_b(S,51)))return!1;switch(x.bra=x.cursor,e){case 1:if(!m())return!1;x.slice_del();break;case 2:if(!m())return!1;x.slice_del(),x.ket=x.cursor,x.eq_s_b(2,"ic")&&(x.bra=x.cursor,m()&&x.slice_del());break;case 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r("irai",-1,1),new r("isci",-1,1),new r("endi",-1,1),new r("erei",-1,1),new r("irei",-1,1),new r("assi",-1,1),new r("ati",-1,1),new r("iti",-1,1),new r("eresti",-1,1),new r("iresti",-1,1),new r("uti",-1,1),new r("avi",-1,1),new r("evi",-1,1),new r("ivi",-1,1),new r("isco",-1,1),new r("ando",-1,1),new r("endo",-1,1),new r("Yamo",-1,1),new r("iamo",-1,1),new r("avamo",-1,1),new r("evamo",-1,1),new r("ivamo",-1,1),new r("eremo",-1,1),new r("iremo",-1,1),new r("assimo",-1,1),new r("ammo",-1,1),new r("emmo",-1,1),new r("eremmo",54,1),new r("iremmo",54,1),new r("immo",-1,1),new r("ano",-1,1),new r("iscano",58,1),new r("avano",58,1),new r("evano",58,1),new r("ivano",58,1),new r("eranno",-1,1),new r("iranno",-1,1),new r("ono",-1,1),new r("iscono",65,1),new r("arono",65,1),new r("erono",65,1),new r("irono",65,1),new r("erebbero",-1,1),new r("irebbero",-1,1),new r("assero",-1,1),new r("essero",-1,1),new r("issero",-1,1),new r("ato",-1,1),new r("ito",-1,1),new r("uto",-1,1),new r("avo",-1,1),new r("evo",-1,1),new r("ivo",-1,1),new r("ar",-1,1),new r("ir",-1,1),new r("erà",-1,1),new r("irà",-1,1),new r("erò",-1,1),new r("irò",-1,1)],L=[17,65,16,0,0,0,0,0,0,0,0,0,0,0,0,128,128,8,2,1],y=[17,65,0,0,0,0,0,0,0,0,0,0,0,0,0,128,128,8,2],U=[17],x=new n;this.setCurrent=function(e){x.setCurrent(e)},this.getCurrent=function(){return x.getCurrent()},this.stem=function(){var e=x.cursor;return i(),x.cursor=e,u(),x.limit_backward=e,x.cursor=x.limit,f(),x.cursor=x.limit,v()||(x.cursor=x.limit,b()),x.cursor=x.limit,_(),x.cursor=x.limit_backward,c(),!0}};return function(e){return"function"==typeof e.update?e.update(function(e){return i.setCurrent(e),i.stem(),i.getCurrent()}):(i.setCurrent(e),i.stem(),i.getCurrent())}}(),e.Pipeline.registerFunction(e.it.stemmer,"stemmer-it"),e.it.stopWordFilter=e.generateStopWordFilter("a abbia abbiamo abbiano abbiate ad agl agli ai al all alla alle allo anche avemmo avendo avesse avessero avessi avessimo aveste avesti avete aveva avevamo avevano avevate avevi avevo avrai avranno avrebbe avrebbero avrei avremmo avremo avreste avresti avrete avrà avrò avuta avute avuti avuto c che chi ci coi col come con contro cui da dagl dagli dai dal dall dalla dalle dallo degl degli dei del dell della delle dello di dov dove e ebbe ebbero ebbi ed era erano eravamo eravate eri ero essendo faccia facciamo facciano facciate faccio facemmo facendo facesse facessero facessi facessimo faceste facesti faceva facevamo facevano facevate facevi facevo fai fanno farai faranno farebbe farebbero farei faremmo faremo fareste faresti farete farà farò fece fecero feci fosse fossero fossi fossimo foste fosti fu fui fummo furono gli ha hai hanno ho i il in io l la le lei li lo loro lui ma mi mia mie miei mio ne negl negli nei nel nell nella nelle nello noi non nostra nostre nostri nostro o per perché più quale quanta quante quanti quanto quella quelle quelli quello questa queste questi questo sarai saranno sarebbe sarebbero sarei saremmo saremo sareste saresti sarete sarà sarò se sei si sia siamo siano siate siete sono sta stai stando stanno starai staranno starebbe starebbero starei staremmo staremo stareste staresti starete starà starò stava stavamo stavano stavate stavi stavo stemmo stesse stessero stessi stessimo steste stesti stette stettero stetti stia stiamo stiano stiate sto su sua sue sugl sugli sui sul sull sulla sulle sullo suo suoi ti tra tu tua tue tuo tuoi tutti tutto un una uno vi voi vostra vostre vostri vostro è".split(" ")),e.Pipeline.registerFunction(e.it.stopWordFilter,"stopWordFilter-it")}}); \ No newline at end of file diff --git a/assets/javascripts/lunr/min/lunr.ja.min.js b/assets/javascripts/lunr/min/lunr.ja.min.js new file mode 100644 index 00000000..5f254ebe --- /dev/null +++ b/assets/javascripts/lunr/min/lunr.ja.min.js @@ -0,0 +1 @@ +!function(e,r){"function"==typeof define&&define.amd?define(r):"object"==typeof exports?module.exports=r():r()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. Please include / require Lunr before this script.");if(void 0===e.stemmerSupport)throw new Error("Lunr stemmer support is not present. Please include / require Lunr stemmer support before this script.");var r="2"==e.version[0];e.ja=function(){this.pipeline.reset(),this.pipeline.add(e.ja.trimmer,e.ja.stopWordFilter,e.ja.stemmer),r?this.tokenizer=e.ja.tokenizer:(e.tokenizer&&(e.tokenizer=e.ja.tokenizer),this.tokenizerFn&&(this.tokenizerFn=e.ja.tokenizer))};var t=new e.TinySegmenter;e.ja.tokenizer=function(i){var n,o,s,p,a,u,m,l,c,f;if(!arguments.length||null==i||void 0==i)return[];if(Array.isArray(i))return i.map(function(t){return r?new e.Token(t.toLowerCase()):t.toLowerCase()});for(o=i.toString().toLowerCase().replace(/^\s+/,""),n=o.length-1;n>=0;n--)if(/\S/.test(o.charAt(n))){o=o.substring(0,n+1);break}for(a=[],s=o.length,c=0,l=0;c<=s;c++)if(u=o.charAt(c),m=c-l,u.match(/\s/)||c==s){if(m>0)for(p=t.segment(o.slice(l,c)).filter(function(e){return!!e}),f=l,n=0;n=C.limit)break;C.cursor++;continue}break}for(C.cursor=o,C.bra=o,C.eq_s(1,"y")?(C.ket=C.cursor,C.slice_from("Y")):C.cursor=o;;)if(e=C.cursor,C.in_grouping(q,97,232)){if(i=C.cursor,C.bra=i,C.eq_s(1,"i"))C.ket=C.cursor,C.in_grouping(q,97,232)&&(C.slice_from("I"),C.cursor=e);else if(C.cursor=i,C.eq_s(1,"y"))C.ket=C.cursor,C.slice_from("Y"),C.cursor=e;else if(n(e))break}else if(n(e))break}function n(r){return C.cursor=r,r>=C.limit||(C.cursor++,!1)}function o(){_=C.limit,d=_,t()||(_=C.cursor,_<3&&(_=3),t()||(d=C.cursor))}function t(){for(;!C.in_grouping(q,97,232);){if(C.cursor>=C.limit)return!0;C.cursor++}for(;!C.out_grouping(q,97,232);){if(C.cursor>=C.limit)return!0;C.cursor++}return!1}function s(){for(var r;;)if(C.bra=C.cursor,r=C.find_among(p,3))switch(C.ket=C.cursor,r){case 1:C.slice_from("y");break;case 2:C.slice_from("i");break;case 3:if(C.cursor>=C.limit)return;C.cursor++}}function u(){return _<=C.cursor}function c(){return d<=C.cursor}function a(){var r=C.limit-C.cursor;C.find_among_b(g,3)&&(C.cursor=C.limit-r,C.ket=C.cursor,C.cursor>C.limit_backward&&(C.cursor--,C.bra=C.cursor,C.slice_del()))}function l(){var r;w=!1,C.ket=C.cursor,C.eq_s_b(1,"e")&&(C.bra=C.cursor,u()&&(r=C.limit-C.cursor,C.out_grouping_b(q,97,232)&&(C.cursor=C.limit-r,C.slice_del(),w=!0,a())))}function m(){var r;u()&&(r=C.limit-C.cursor,C.out_grouping_b(q,97,232)&&(C.cursor=C.limit-r,C.eq_s_b(3,"gem")||(C.cursor=C.limit-r,C.slice_del(),a())))}function f(){var r,e,i,n,o,t,s=C.limit-C.cursor;if(C.ket=C.cursor,r=C.find_among_b(h,5))switch(C.bra=C.cursor,r){case 1:u()&&C.slice_from("heid");break;case 2:m();break;case 3:u()&&C.out_grouping_b(j,97,232)&&C.slice_del()}if(C.cursor=C.limit-s,l(),C.cursor=C.limit-s,C.ket=C.cursor,C.eq_s_b(4,"heid")&&(C.bra=C.cursor,c()&&(e=C.limit-C.cursor,C.eq_s_b(1,"c")||(C.cursor=C.limit-e,C.slice_del(),C.ket=C.cursor,C.eq_s_b(2,"en")&&(C.bra=C.cursor,m())))),C.cursor=C.limit-s,C.ket=C.cursor,r=C.find_among_b(k,6))switch(C.bra=C.cursor,r){case 1:if(c()){if(C.slice_del(),i=C.limit-C.cursor,C.ket=C.cursor,C.eq_s_b(2,"ig")&&(C.bra=C.cursor,c()&&(n=C.limit-C.cursor,!C.eq_s_b(1,"e")))){C.cursor=C.limit-n,C.slice_del();break}C.cursor=C.limit-i,a()}break;case 2:c()&&(o=C.limit-C.cursor,C.eq_s_b(1,"e")||(C.cursor=C.limit-o,C.slice_del()));break;case 3:c()&&(C.slice_del(),l());break;case 4:c()&&C.slice_del();break;case 5:c()&&w&&C.slice_del()}C.cursor=C.limit-s,C.out_grouping_b(z,73,232)&&(t=C.limit-C.cursor,C.find_among_b(v,4)&&C.out_grouping_b(q,97,232)&&(C.cursor=C.limit-t,C.ket=C.cursor,C.cursor>C.limit_backward&&(C.cursor--,C.bra=C.cursor,C.slice_del())))}var d,_,w,b=[new e("",-1,6),new e("á",0,1),new e("ä",0,1),new e("é",0,2),new e("ë",0,2),new e("í",0,3),new e("ï",0,3),new e("ó",0,4),new e("ö",0,4),new e("ú",0,5),new e("ü",0,5)],p=[new e("",-1,3),new e("I",0,2),new e("Y",0,1)],g=[new e("dd",-1,-1),new e("kk",-1,-1),new e("tt",-1,-1)],h=[new e("ene",-1,2),new e("se",-1,3),new e("en",-1,2),new e("heden",2,1),new e("s",-1,3)],k=[new e("end",-1,1),new e("ig",-1,2),new e("ing",-1,1),new e("lijk",-1,3),new e("baar",-1,4),new e("bar",-1,5)],v=[new e("aa",-1,-1),new e("ee",-1,-1),new e("oo",-1,-1),new e("uu",-1,-1)],q=[17,65,16,1,0,0,0,0,0,0,0,0,0,0,0,0,128],z=[1,0,0,17,65,16,1,0,0,0,0,0,0,0,0,0,0,0,0,128],j=[17,67,16,1,0,0,0,0,0,0,0,0,0,0,0,0,128],C=new i;this.setCurrent=function(r){C.setCurrent(r)},this.getCurrent=function(){return C.getCurrent()},this.stem=function(){var e=C.cursor;return r(),C.cursor=e,o(),C.limit_backward=e,C.cursor=C.limit,f(),C.cursor=C.limit_backward,s(),!0}};return function(r){return"function"==typeof r.update?r.update(function(r){return n.setCurrent(r),n.stem(),n.getCurrent()}):(n.setCurrent(r),n.stem(),n.getCurrent())}}(),r.Pipeline.registerFunction(r.nl.stemmer,"stemmer-nl"),r.nl.stopWordFilter=r.generateStopWordFilter(" aan al alles als altijd andere ben bij daar dan dat de der deze die dit doch doen door dus een eens en er ge geen geweest haar had heb hebben heeft hem het hier hij hoe hun iemand iets ik in is ja je kan kon kunnen maar me meer men met mij mijn moet na naar niet niets nog nu of om omdat onder ons ook op over reeds te tegen toch toen tot u uit uw van veel voor want waren was wat werd wezen wie wil worden wordt zal ze zelf zich zij zijn zo zonder zou".split(" 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dă ea ei el ele eram este eu eşti face fata fi fie fiecare fii fim fiu fiţi frumos fără graţie halbă iar ieri la le li lor lui lângă lîngă mai mea mei mele mereu meu mi mie mine mult multă mulţi mulţumesc mâine mîine mă ne nevoie nici nicăieri nimeni nimeri nimic nişte noastre noastră noi noroc nostru nouă noştri nu opt ori oricare orice oricine oricum oricând oricât oricînd oricît oriunde patra patru patrulea pe pentru peste pic poate pot prea prima primul prin puţin puţina puţină până pînă rog sa sale sau se spate spre sub sunt suntem sunteţi sută sînt sîntem sînteţi să săi său ta tale te timp tine toate toată tot totuşi toţi trei treia treilea tu tăi tău un una unde undeva unei uneia unele uneori unii unor unora unu unui unuia unul vi voastre voastră voi vostru vouă voştri vreme vreo vreun vă zece zero zi zice îi îl îmi împotriva în înainte înaintea încotro încât încît între întrucât întrucît îţi ăla ălea ăsta ăstea ăştia şapte şase şi ştiu ţi ţie".split(" ")),e.Pipeline.registerFunction(e.ro.stopWordFilter,"stopWordFilter-ro")}}); \ No newline at end of file diff --git a/assets/javascripts/lunr/min/lunr.ru.min.js b/assets/javascripts/lunr/min/lunr.ru.min.js new file mode 100644 index 00000000..186cc485 --- /dev/null +++ b/assets/javascripts/lunr/min/lunr.ru.min.js @@ -0,0 +1,18 @@ +/*! + * Lunr languages, `Russian` language + * https://github.com/MihaiValentin/lunr-languages + * + * Copyright 2014, Mihai Valentin + * http://www.mozilla.org/MPL/ + */ +/*! + * based on + * Snowball JavaScript Library v0.3 + * http://code.google.com/p/urim/ + * http://snowball.tartarus.org/ + * + * Copyright 2010, Oleg Mazko + * http://www.mozilla.org/MPL/ + */ + +!function(e,n){"function"==typeof define&&define.amd?define(n):"object"==typeof exports?module.exports=n():n()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. 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http://snowball.tartarus.org/ + * + * Copyright 2010, Oleg Mazko + * http://www.mozilla.org/MPL/ + */ + +!function(e,r){"function"==typeof define&&define.amd?define(r):"object"==typeof exports?module.exports=r():r()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. 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e;this.setCurrent=function(r){Dr.setCurrent(r)},this.getCurrent=function(){return Dr.getCurrent()},this.stem=function(){return!!($()&&(Dr.limit_backward=Dr.cursor,Dr.cursor=Dr.limit,J(),Dr.cursor=Dr.limit,nr&&(R(),Dr.cursor=Dr.limit_backward,er())))}};return function(r){return"function"==typeof r.update?r.update(function(r){return n.setCurrent(r),n.stem(),n.getCurrent()}):(n.setCurrent(r),n.stem(),n.getCurrent())}}(),r.Pipeline.registerFunction(r.tr.stemmer,"stemmer-tr"),r.tr.stopWordFilter=r.generateStopWordFilter("acaba altmış altı ama ancak arada aslında ayrıca bana bazı belki ben benden beni benim beri beş bile bin bir biri birkaç birkez birçok birşey birşeyi biz bizden bize bizi bizim bu buna bunda bundan bunlar bunları bunların bunu bunun burada böyle böylece da daha dahi de defa değil diye diğer doksan dokuz dolayı dolayısıyla dört edecek eden ederek edilecek ediliyor edilmesi ediyor elli en etmesi etti ettiği ettiğini eğer gibi göre halen hangi hatta hem henüz hep hepsi her 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b/assets/javascripts/lunr/min/lunr.vi.min.js new file mode 100644 index 00000000..22aed28c --- /dev/null +++ b/assets/javascripts/lunr/min/lunr.vi.min.js @@ -0,0 +1 @@ +!function(e,r){"function"==typeof define&&define.amd?define(r):"object"==typeof exports?module.exports=r():r()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. 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Register as an anonymous module. + define(factory) + } else if (typeof exports === 'object') { + /** + * Node. Does not work with strict CommonJS, but + * only CommonJS-like environments that support module.exports, + * like Node. + */ + module.exports = factory() + } else { + // Browser globals (root is window) + factory()(root.lunr); + } +}(this, function () { + /** + * Just return a value to define the module export. + * This example returns an object, but the module + * can return a function as the exported value. + */ + + return function(lunr) { + // TinySegmenter 0.1 -- Super compact Japanese tokenizer in Javascript + // (c) 2008 Taku Kudo + // TinySegmenter is freely distributable under the terms of a new BSD licence. + // For details, see http://chasen.org/~taku/software/TinySegmenter/LICENCE.txt + + function TinySegmenter() { + var patterns = { + "[一二三四五六七八九十百千万億兆]":"M", + "[一-龠々〆ヵヶ]":"H", + "[ぁ-ん]":"I", + "[ァ-ヴーｱ-ﾝﾞｰ]":"K", + "[a-zA-Zａ-ｚＡ-Ｚ]":"A", + "[0-9０-９]":"N" + } + this.chartype_ = []; + for (var i in patterns) { + var regexp = new RegExp(i); + this.chartype_.push([regexp, patterns[i]]); 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+ + return this; + } + TinySegmenter.prototype.ctype_ = function(str) { + for (var i in this.chartype_) { + if (str.match(this.chartype_[i][0])) { + return this.chartype_[i][1]; + } + } + return "O"; + } + + TinySegmenter.prototype.ts_ = function(v) { + if (v) { return v; } + return 0; + } + + TinySegmenter.prototype.segment = function(input) { + if (input == null || input == undefined || input == "") { + return []; + } + var result = []; + var seg = ["B3","B2","B1"]; + var ctype = ["O","O","O"]; + var o = input.split(""); + for (i = 0; i < o.length; ++i) { + seg.push(o[i]); + ctype.push(this.ctype_(o[i])) + } + seg.push("E1"); + seg.push("E2"); + seg.push("E3"); + ctype.push("O"); + ctype.push("O"); + ctype.push("O"); + var word = seg[3]; + var p1 = "U"; + var p2 = "U"; + var p3 = "U"; + for (var i = 4; i < seg.length - 3; ++i) { + var score = this.BIAS__; + var w1 = seg[i-3]; + var w2 = seg[i-2]; + var w3 = seg[i-1]; + var w4 = seg[i]; + var w5 = seg[i+1]; + var w6 = seg[i+2]; + var c1 = ctype[i-3]; + var c2 = ctype[i-2]; + var c3 = ctype[i-1]; + var c4 = ctype[i]; + var c5 = ctype[i+1]; + var c6 = ctype[i+2]; + score += this.ts_(this.UP1__[p1]); + score += this.ts_(this.UP2__[p2]); + score += this.ts_(this.UP3__[p3]); + score += this.ts_(this.BP1__[p1 + p2]); + score += this.ts_(this.BP2__[p2 + p3]); + score += this.ts_(this.UW1__[w1]); + score += this.ts_(this.UW2__[w2]); + score += this.ts_(this.UW3__[w3]); + score += this.ts_(this.UW4__[w4]); + score += this.ts_(this.UW5__[w5]); + score += this.ts_(this.UW6__[w6]); + score += this.ts_(this.BW1__[w2 + w3]); + score += this.ts_(this.BW2__[w3 + w4]); + score += this.ts_(this.BW3__[w4 + w5]); + score += this.ts_(this.TW1__[w1 + w2 + w3]); + score += this.ts_(this.TW2__[w2 + w3 + w4]); + score += this.ts_(this.TW3__[w3 + w4 + w5]); + score += this.ts_(this.TW4__[w4 + w5 + w6]); + score += this.ts_(this.UC1__[c1]); + score += this.ts_(this.UC2__[c2]); + score += this.ts_(this.UC3__[c3]); + score += this.ts_(this.UC4__[c4]); + score += this.ts_(this.UC5__[c5]); + score += this.ts_(this.UC6__[c6]); + score += this.ts_(this.BC1__[c2 + c3]); + score += this.ts_(this.BC2__[c3 + c4]); + score += this.ts_(this.BC3__[c4 + c5]); + score += this.ts_(this.TC1__[c1 + c2 + c3]); + score += this.ts_(this.TC2__[c2 + c3 + c4]); + score += this.ts_(this.TC3__[c3 + c4 + c5]); + score += this.ts_(this.TC4__[c4 + c5 + c6]); + // score += this.ts_(this.TC5__[c4 + c5 + c6]); + score += this.ts_(this.UQ1__[p1 + c1]); + score += this.ts_(this.UQ2__[p2 + c2]); + score += this.ts_(this.UQ3__[p3 + c3]); + score += this.ts_(this.BQ1__[p2 + c2 + c3]); + score += this.ts_(this.BQ2__[p2 + c3 + c4]); + score += this.ts_(this.BQ3__[p3 + c2 + c3]); + score += this.ts_(this.BQ4__[p3 + c3 + c4]); + score += this.ts_(this.TQ1__[p2 + c1 + c2 + c3]); + score += this.ts_(this.TQ2__[p2 + c2 + c3 + c4]); + score += this.ts_(this.TQ3__[p3 + c1 + c2 + c3]); + score += this.ts_(this.TQ4__[p3 + c2 + c3 + c4]); + var p = "O"; + if (score > 0) { + result.push(word); + word = ""; + p = "B"; + } + p1 = p2; + p2 = p3; + p3 = p; + word += seg[i]; + } + result.push(word); + + return result; + } + + lunr.TinySegmenter = TinySegmenter; + }; + +})); \ No newline at end of file diff --git a/assets/javascripts/lunr/wordcut.js b/assets/javascripts/lunr/wordcut.js new file mode 100644 index 00000000..146f4b44 --- /dev/null +++ b/assets/javascripts/lunr/wordcut.js @@ -0,0 +1,6708 @@ +(function(f){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=f()}else if(typeof define==="function"&&define.amd){define([],f)}else{var g;if(typeof window!=="undefined"){g=window}else if(typeof global!=="undefined"){g=global}else if(typeof self!=="undefined"){g=self}else{g=this}(g.lunr || (g.lunr = {})).wordcut = f()}})(function(){var define,module,exports;return (function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require=="function"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);var f=new Error("Cannot find module '"+o+"'");throw f.code="MODULE_NOT_FOUND",f}var l=n[o]={exports:{}};t[o][0].call(l.exports,function(e){var n=t[o][1][e];return s(n?n:e)},l,l.exports,e,t,n,r)}return n[o].exports}var i=typeof require=="function"&&require;for(var o=0;o 1; + }) + this.addWords(words, false) + } + if(finalize){ + this.finalizeDict(); + } + }, + + dictSeek: function (l, r, ch, strOffset, pos) { + var ans = null; + while (l <= r) { + var m = Math.floor((l + r) / 2), + dict_item = this.dict[m], + len = dict_item.length; + if (len <= strOffset) { + l = m + 1; + } else { + var ch_ = dict_item[strOffset]; + if (ch_ < ch) { + l = m + 1; + } else if (ch_ > ch) { + r = m - 1; + } else { + ans = m; + if (pos == LEFT) { + r = m - 1; + } else { + l = m + 1; + } + } + } + } + return ans; + }, + + isFinal: function (acceptor) { + return this.dict[acceptor.l].length == acceptor.strOffset; + }, + + createAcceptor: function () { + return { + l: 0, + r: this.dict.length - 1, + strOffset: 0, + isFinal: false, + dict: this, + transit: function (ch) { + return this.dict.transit(this, ch); + }, + isError: false, + tag: "DICT", + w: 1, + type: "DICT" + }; + }, + + transit: function (acceptor, ch) { + var l = this.dictSeek(acceptor.l, + acceptor.r, + ch, + acceptor.strOffset, + LEFT); + if (l !== null) { + var r = this.dictSeek(l, + acceptor.r, + ch, + acceptor.strOffset, + RIGHT); + acceptor.l = l; + acceptor.r = r; + acceptor.strOffset++; + acceptor.isFinal = this.isFinal(acceptor); + } else { + acceptor.isError = true; + } + return acceptor; + }, + + sortuniq: function(a){ + return a.sort().filter(function(item, pos, arr){ + return !pos || item != arr[pos - 1]; + }) + }, + + flatten: function(a){ + //[[1,2],[3]] -> [1,2,3] + return [].concat.apply([], a); + } +}; +module.exports = WordcutDict; + +}).call(this,"/dist/tmp") +},{"glob":16,"path":22}],3:[function(require,module,exports){ +var WordRule = { + createAcceptor: function(tag) { + if (tag["WORD_RULE"]) + return null; + + return {strOffset: 0, + isFinal: false, + transit: function(ch) { + var lch = ch.toLowerCase(); + if (lch >= "a" && lch <= "z") { + this.isFinal = true; + this.strOffset++; + } else { + this.isError = true; + } + return this; + }, + isError: false, + tag: "WORD_RULE", + type: "WORD_RULE", + w: 1}; + } +}; + +var NumberRule = { + createAcceptor: function(tag) { + if (tag["NUMBER_RULE"]) + return null; + + return {strOffset: 0, + isFinal: false, + transit: function(ch) { + if (ch >= "0" && ch <= "9") { + this.isFinal = true; + this.strOffset++; + } else { + this.isError = true; + } + return this; + }, + isError: false, + tag: "NUMBER_RULE", + type: "NUMBER_RULE", + w: 1}; + } +}; + +var SpaceRule = { + tag: "SPACE_RULE", + createAcceptor: function(tag) { + + if (tag["SPACE_RULE"]) + return null; + + return {strOffset: 0, + isFinal: false, + transit: function(ch) { + if (ch == " " || ch == "\t" || ch == "\r" || ch == "\n" || + ch == "\u00A0" || ch=="\u2003"//nbsp and emsp + ) { + this.isFinal = true; + this.strOffset++; + } else { + this.isError = true; + } + return this; + }, + isError: false, + tag: SpaceRule.tag, + w: 1, + type: "SPACE_RULE"}; + } +} + +var SingleSymbolRule = { + tag: "SINSYM", + createAcceptor: function(tag) { + return {strOffset: 0, + isFinal: false, + transit: function(ch) { + if (this.strOffset == 0 && ch.match(/^[\@\(\)\/\,\-\."`]$/)) { + this.isFinal = true; + this.strOffset++; + } else { + this.isError = true; + } + return this; + }, + isError: false, + tag: "SINSYM", + w: 1, + type: "SINSYM"}; + } +} + + +var LatinRules = [WordRule, SpaceRule, SingleSymbolRule, NumberRule]; + +module.exports = LatinRules; + +},{}],4:[function(require,module,exports){ +var _ = require("underscore") + , WordcutCore = require("./wordcut_core"); +var PathInfoBuilder = { + + /* + buildByPartAcceptors: function(path, acceptors, i) { + var + var genInfos = partAcceptors.reduce(function(genInfos, acceptor) { + + }, []); + + return genInfos; + } + */ + + buildByAcceptors: function(path, finalAcceptors, i) { + var self = this; + var infos = finalAcceptors.map(function(acceptor) { + var p = i - acceptor.strOffset + 1 + , _info = path[p]; + + var info = {p: p, + mw: _info.mw + (acceptor.mw === undefined ? 0 : acceptor.mw), + w: acceptor.w + _info.w, + unk: (acceptor.unk ? acceptor.unk : 0) + _info.unk, + type: acceptor.type}; + + if (acceptor.type == "PART") { + for(var j = p + 1; j <= i; j++) { + path[j].merge = p; + } + info.merge = p; + } + + return info; + }); + return infos.filter(function(info) { return info; }); + }, + + fallback: function(path, leftBoundary, text, i) { + var _info = path[leftBoundary]; + if (text[i].match(/[\u0E48-\u0E4E]/)) { + if (leftBoundary != 0) + leftBoundary = path[leftBoundary].p; + return {p: leftBoundary, + mw: 0, + w: 1 + _info.w, + unk: 1 + _info.unk, + type: "UNK"}; +/* } else if(leftBoundary > 0 && path[leftBoundary].type !== "UNK") { + leftBoundary = path[leftBoundary].p; + return {p: leftBoundary, + w: 1 + _info.w, + unk: 1 + _info.unk, + type: "UNK"}; */ + } else { + return {p: leftBoundary, + mw: _info.mw, + w: 1 + _info.w, + unk: 1 + _info.unk, + type: "UNK"}; + } + }, + + build: function(path, finalAcceptors, i, leftBoundary, text) { + var basicPathInfos = this.buildByAcceptors(path, finalAcceptors, i); + if (basicPathInfos.length > 0) { + return basicPathInfos; + } else { + return [this.fallback(path, leftBoundary, text, i)]; + } + } +}; + +module.exports = function() { + return _.clone(PathInfoBuilder); +} + +},{"./wordcut_core":8,"underscore":25}],5:[function(require,module,exports){ +var _ = require("underscore"); + + +var PathSelector = { + selectPath: function(paths) { + var path = paths.reduce(function(selectedPath, path) { + if (selectedPath == null) { + return path; + } else { + if (path.unk < selectedPath.unk) + return path; + if (path.unk == selectedPath.unk) { + if (path.mw < selectedPath.mw) + return path + if (path.mw == selectedPath.mw) { + if (path.w < selectedPath.w) + return path; + } + } + return selectedPath; + } + }, null); + return path; + }, + + createPath: function() { + return [{p:null, w:0, unk:0, type: "INIT", mw:0}]; + } +}; + +module.exports = function() { + return _.clone(PathSelector); +}; + +},{"underscore":25}],6:[function(require,module,exports){ +function isMatch(pat, offset, ch) { + if (pat.length <= offset) + return false; + var _ch = pat[offset]; + return _ch == ch || + (_ch.match(/[กข]/) && ch.match(/[ก-ฮ]/)) || + (_ch.match(/[มบ]/) && ch.match(/[ก-ฮ]/)) || + (_ch.match(/\u0E49/) && ch.match(/[\u0E48-\u0E4B]/)); +} + +var Rule0 = { + pat: "เหก็ม", + createAcceptor: function(tag) { + return {strOffset: 0, + isFinal: false, + transit: function(ch) { + if (isMatch(Rule0.pat, this.strOffset,ch)) { + this.isFinal = (this.strOffset + 1 == Rule0.pat.length); + this.strOffset++; + } else { + this.isError = true; + } + return this; + }, + isError: false, + tag: "THAI_RULE", + type: "THAI_RULE", + w: 1}; + } +}; + +var PartRule = { + createAcceptor: function(tag) { + return {strOffset: 0, + patterns: [ + "แก", "เก", "ก้", "กก์", "กา", "กี", "กิ", "กืก" + ], + isFinal: false, + transit: function(ch) { + var offset = this.strOffset; + this.patterns = this.patterns.filter(function(pat) { + return isMatch(pat, offset, ch); + }); + + if (this.patterns.length > 0) { + var len = 1 + offset; + this.isFinal = this.patterns.some(function(pat) { + return pat.length == len; + }); + this.strOffset++; + } else { + this.isError = true; + } + return this; + }, + isError: false, + tag: "PART", + type: "PART", + unk: 1, + w: 1}; + } +}; + +var ThaiRules = [Rule0, PartRule]; + +module.exports = ThaiRules; + +},{}],7:[function(require,module,exports){ +var sys = require("sys") + , WordcutDict = require("./dict") + , WordcutCore = require("./wordcut_core") + , PathInfoBuilder = require("./path_info_builder") + , PathSelector = require("./path_selector") + , Acceptors = require("./acceptors") + , latinRules = require("./latin_rules") + , thaiRules = require("./thai_rules") + , _ = require("underscore"); + + +var Wordcut = Object.create(WordcutCore); +Wordcut.defaultPathInfoBuilder = PathInfoBuilder; +Wordcut.defaultPathSelector = PathSelector; +Wordcut.defaultAcceptors = Acceptors; +Wordcut.defaultLatinRules = latinRules; +Wordcut.defaultThaiRules = thaiRules; +Wordcut.defaultDict = WordcutDict; + + +Wordcut.initNoDict = function(dict_path) { + var self = this; + self.pathInfoBuilder = new self.defaultPathInfoBuilder; + self.pathSelector = new self.defaultPathSelector; + self.acceptors = new self.defaultAcceptors; + self.defaultLatinRules.forEach(function(rule) { + self.acceptors.creators.push(rule); + }); + self.defaultThaiRules.forEach(function(rule) { + self.acceptors.creators.push(rule); + }); +}; + +Wordcut.init = function(dict_path, withDefault, additionalWords) { + withDefault = withDefault || false; + this.initNoDict(); + var dict = _.clone(this.defaultDict); + dict.init(dict_path, withDefault, additionalWords); + this.acceptors.creators.push(dict); +}; + +module.exports = Wordcut; + +},{"./acceptors":1,"./dict":2,"./latin_rules":3,"./path_info_builder":4,"./path_selector":5,"./thai_rules":6,"./wordcut_core":8,"sys":28,"underscore":25}],8:[function(require,module,exports){ +var WordcutCore = { + + buildPath: function(text) { + var self = this + , path = self.pathSelector.createPath() + , leftBoundary = 0; + self.acceptors.reset(); + for (var i = 0; i < text.length; i++) { + var ch = text[i]; + self.acceptors.transit(ch); + + var possiblePathInfos = self + .pathInfoBuilder + .build(path, + self.acceptors.getFinalAcceptors(), + i, + leftBoundary, + text); + var selectedPath = self.pathSelector.selectPath(possiblePathInfos) + + path.push(selectedPath); + if (selectedPath.type !== "UNK") { + leftBoundary = i; + } + } + return path; + }, + + pathToRanges: function(path) { + var e = path.length - 1 + , ranges = []; + + while (e > 0) { + var info = path[e] + , s = info.p; + + if (info.merge !== undefined && ranges.length > 0) { + var r = ranges[ranges.length - 1]; + r.s = info.merge; + s = r.s; + } else { + ranges.push({s:s, e:e}); + } + e = s; + } + return ranges.reverse(); + }, + + rangesToText: function(text, ranges, delimiter) { + return ranges.map(function(r) { + return text.substring(r.s, r.e); + }).join(delimiter); + }, + + cut: function(text, delimiter) { + var path = this.buildPath(text) + , ranges = this.pathToRanges(path); + return this + .rangesToText(text, ranges, + (delimiter === undefined ? "|" : delimiter)); + }, + + cutIntoRanges: function(text, noText) { + var path = this.buildPath(text) + , ranges = this.pathToRanges(path); + + if (!noText) { + ranges.forEach(function(r) { + r.text = text.substring(r.s, r.e); + }); + } + return ranges; + }, + + cutIntoArray: function(text) { + var path = this.buildPath(text) + , ranges = this.pathToRanges(path); + + return ranges.map(function(r) { + return text.substring(r.s, r.e) + }); + } +}; + +module.exports = WordcutCore; + +},{}],9:[function(require,module,exports){ +// http://wiki.commonjs.org/wiki/Unit_Testing/1.0 +// +// THIS IS NOT TESTED NOR LIKELY TO WORK OUTSIDE V8! +// +// Originally from narwhal.js (http://narwhaljs.org) +// Copyright (c) 2009 Thomas Robinson <280north.com> +// +// Permission is hereby granted, free of charge, to any person obtaining a copy +// of this software and associated documentation files (the 'Software'), to +// deal in the Software without restriction, including without limitation the +// rights to use, copy, modify, merge, publish, distribute, sublicense, and/or +// sell copies of the Software, and to permit persons to whom the Software is +// furnished to do so, subject to the following conditions: +// +// The above copyright notice and this permission notice shall be included in +// all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED 'AS IS', WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +// AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN +// ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION +// WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + +// when used in node, this will actually load the util module we depend on +// versus loading the builtin util module as happens otherwise +// this is a bug in node module loading as far as I am concerned +var util = require('util/'); + +var pSlice = Array.prototype.slice; +var hasOwn = Object.prototype.hasOwnProperty; + +// 1. The assert module provides functions that throw +// AssertionError's when particular conditions are not met. The +// assert module must conform to the following interface. + +var assert = module.exports = ok; + +// 2. The AssertionError is defined in assert. +// new assert.AssertionError({ message: message, +// actual: actual, +// expected: expected }) + +assert.AssertionError = function AssertionError(options) { + this.name = 'AssertionError'; + this.actual = options.actual; + this.expected = options.expected; + this.operator = options.operator; + if (options.message) { + this.message = options.message; + this.generatedMessage = false; + } else { + this.message = getMessage(this); + this.generatedMessage = true; + } + var stackStartFunction = options.stackStartFunction || fail; + + if (Error.captureStackTrace) { + Error.captureStackTrace(this, stackStartFunction); + } + else { + // non v8 browsers so we can have a stacktrace + var err = new Error(); + if (err.stack) { + var out = err.stack; + + // try to strip useless frames + var fn_name = stackStartFunction.name; + var idx = out.indexOf('\n' + fn_name); + if (idx >= 0) { + // once we have located the function frame + // we need to strip out everything before it (and its line) + var next_line = out.indexOf('\n', idx + 1); + out = out.substring(next_line + 1); + } + + this.stack = out; + } + } +}; + +// assert.AssertionError instanceof Error +util.inherits(assert.AssertionError, Error); + +function replacer(key, value) { + if (util.isUndefined(value)) { + return '' + value; + } + if (util.isNumber(value) && !isFinite(value)) { + return value.toString(); + } + if (util.isFunction(value) || util.isRegExp(value)) { + return value.toString(); + } + return value; +} + +function truncate(s, n) { + if (util.isString(s)) { + return s.length < n ? s : s.slice(0, n); + } else { + return s; + } +} + +function getMessage(self) { + return truncate(JSON.stringify(self.actual, replacer), 128) + ' ' + + self.operator + ' ' + + truncate(JSON.stringify(self.expected, replacer), 128); +} + +// At present only the three keys mentioned above are used and +// understood by the spec. Implementations or sub modules can pass +// other keys to the AssertionError's constructor - they will be +// ignored. + +// 3. All of the following functions must throw an AssertionError +// when a corresponding condition is not met, with a message that +// may be undefined if not provided. All assertion methods provide +// both the actual and expected values to the assertion error for +// display purposes. + +function fail(actual, expected, message, operator, stackStartFunction) { + throw new assert.AssertionError({ + message: message, + actual: actual, + expected: expected, + operator: operator, + stackStartFunction: stackStartFunction + }); +} + +// EXTENSION! allows for well behaved errors defined elsewhere. +assert.fail = fail; + +// 4. Pure assertion tests whether a value is truthy, as determined +// by !!guard. +// assert.ok(guard, message_opt); +// This statement is equivalent to assert.equal(true, !!guard, +// message_opt);. To test strictly for the value true, use +// assert.strictEqual(true, guard, message_opt);. + +function ok(value, message) { + if (!value) fail(value, true, message, '==', assert.ok); +} +assert.ok = ok; + +// 5. The equality assertion tests shallow, coercive equality with +// ==. +// assert.equal(actual, expected, message_opt); + +assert.equal = function equal(actual, expected, message) { + if (actual != expected) fail(actual, expected, message, '==', assert.equal); +}; + +// 6. The non-equality assertion tests for whether two objects are not equal +// with != assert.notEqual(actual, expected, message_opt); + +assert.notEqual = function notEqual(actual, expected, message) { + if (actual == expected) { + fail(actual, expected, message, '!=', assert.notEqual); + } +}; + +// 7. The equivalence assertion tests a deep equality relation. +// assert.deepEqual(actual, expected, message_opt); + +assert.deepEqual = function deepEqual(actual, expected, message) { + if (!_deepEqual(actual, expected)) { + fail(actual, expected, message, 'deepEqual', assert.deepEqual); + } +}; + +function _deepEqual(actual, expected) { + // 7.1. All identical values are equivalent, as determined by ===. + if (actual === expected) { + return true; + + } else if (util.isBuffer(actual) && util.isBuffer(expected)) { + if (actual.length != expected.length) return false; + + for (var i = 0; i < actual.length; i++) { + if (actual[i] !== expected[i]) return false; + } + + return true; + + // 7.2. If the expected value is a Date object, the actual value is + // equivalent if it is also a Date object that refers to the same time. + } else if (util.isDate(actual) && util.isDate(expected)) { + return actual.getTime() === expected.getTime(); + + // 7.3 If the expected value is a RegExp object, the actual value is + // equivalent if it is also a RegExp object with the same source and + // properties (`global`, `multiline`, `lastIndex`, `ignoreCase`). + } else if (util.isRegExp(actual) && util.isRegExp(expected)) { + return actual.source === expected.source && + actual.global === expected.global && + actual.multiline === expected.multiline && + actual.lastIndex === expected.lastIndex && + actual.ignoreCase === expected.ignoreCase; + + // 7.4. Other pairs that do not both pass typeof value == 'object', + // equivalence is determined by ==. + } else if (!util.isObject(actual) && !util.isObject(expected)) { + return actual == expected; + + // 7.5 For all other Object pairs, including Array objects, equivalence is + // determined by having the same number of owned properties (as verified + // with Object.prototype.hasOwnProperty.call), the same set of keys + // (although not necessarily the same order), equivalent values for every + // corresponding key, and an identical 'prototype' property. Note: this + // accounts for both named and indexed properties on Arrays. + } else { + return objEquiv(actual, expected); + } +} + +function isArguments(object) { + return Object.prototype.toString.call(object) == '[object Arguments]'; +} + +function objEquiv(a, b) { + if (util.isNullOrUndefined(a) || util.isNullOrUndefined(b)) + return false; + // an identical 'prototype' property. + if (a.prototype !== b.prototype) return false; + // if one is a primitive, the other must be same + if (util.isPrimitive(a) || util.isPrimitive(b)) { + return a === b; + } + var aIsArgs = isArguments(a), + bIsArgs = isArguments(b); + if ((aIsArgs && !bIsArgs) || (!aIsArgs && bIsArgs)) + return false; + if (aIsArgs) { + a = pSlice.call(a); + b = pSlice.call(b); + return _deepEqual(a, b); + } + var ka = objectKeys(a), + kb = objectKeys(b), + key, i; + // having the same number of owned properties (keys incorporates + // hasOwnProperty) + if (ka.length != kb.length) + return false; + //the same set of keys (although not necessarily the same order), + ka.sort(); + kb.sort(); + //~~~cheap key test + for (i = ka.length - 1; i >= 0; i--) { + if (ka[i] != kb[i]) + return false; + } + //equivalent values for every corresponding key, and + //~~~possibly expensive deep test + for (i = ka.length - 1; i >= 0; i--) { + key = ka[i]; + if (!_deepEqual(a[key], b[key])) return false; + } + return true; +} + +// 8. The non-equivalence assertion tests for any deep inequality. +// assert.notDeepEqual(actual, expected, message_opt); + +assert.notDeepEqual = function notDeepEqual(actual, expected, message) { + if (_deepEqual(actual, expected)) { + fail(actual, expected, message, 'notDeepEqual', assert.notDeepEqual); + } +}; + +// 9. The strict equality assertion tests strict equality, as determined by ===. +// assert.strictEqual(actual, expected, message_opt); + +assert.strictEqual = function strictEqual(actual, expected, message) { + if (actual !== expected) { + fail(actual, expected, message, '===', assert.strictEqual); + } +}; + +// 10. The strict non-equality assertion tests for strict inequality, as +// determined by !==. assert.notStrictEqual(actual, expected, message_opt); + +assert.notStrictEqual = function notStrictEqual(actual, expected, message) { + if (actual === expected) { + fail(actual, expected, message, '!==', assert.notStrictEqual); + } +}; + +function expectedException(actual, expected) { + if (!actual || !expected) { + return false; + } + + if (Object.prototype.toString.call(expected) == '[object RegExp]') { + return expected.test(actual); + } else if (actual instanceof expected) { + return true; + } else if (expected.call({}, actual) === true) { + return true; + } + + return false; +} + +function _throws(shouldThrow, block, expected, message) { + var actual; + + if (util.isString(expected)) { + message = expected; + expected = null; + } + + try { + block(); + } catch (e) { + actual = e; + } + + message = (expected && expected.name ? ' (' + expected.name + ').' : '.') + + (message ? ' ' + message : '.'); + + if (shouldThrow && !actual) { + fail(actual, expected, 'Missing expected exception' + message); + } + + if (!shouldThrow && expectedException(actual, expected)) { + fail(actual, expected, 'Got unwanted exception' + message); + } + + if ((shouldThrow && actual && expected && + !expectedException(actual, expected)) || (!shouldThrow && actual)) { + throw actual; + } +} + +// 11. Expected to throw an error: +// assert.throws(block, Error_opt, message_opt); + +assert.throws = function(block, /*optional*/error, /*optional*/message) { + _throws.apply(this, [true].concat(pSlice.call(arguments))); +}; + +// EXTENSION! This is annoying to write outside this module. +assert.doesNotThrow = function(block, /*optional*/message) { + _throws.apply(this, [false].concat(pSlice.call(arguments))); +}; + +assert.ifError = function(err) { if (err) {throw err;}}; + +var objectKeys = Object.keys || function (obj) { + var keys = []; + for (var key in obj) { + if (hasOwn.call(obj, key)) keys.push(key); + } + return keys; +}; + +},{"util/":28}],10:[function(require,module,exports){ +'use strict'; +module.exports = balanced; +function balanced(a, b, str) { + if (a instanceof RegExp) a = maybeMatch(a, str); + if (b instanceof RegExp) b = maybeMatch(b, str); + + var r = range(a, b, str); + + return r && { + start: r[0], + end: r[1], + pre: str.slice(0, r[0]), + body: str.slice(r[0] + a.length, r[1]), + post: str.slice(r[1] + b.length) + }; +} + +function maybeMatch(reg, str) { + var m = str.match(reg); + return m ? m[0] : null; +} + +balanced.range = range; +function range(a, b, str) { + var begs, beg, left, right, result; + var ai = str.indexOf(a); + var bi = str.indexOf(b, ai + 1); + var i = ai; + + if (ai >= 0 && bi > 0) { + begs = []; + left = str.length; + + while (i >= 0 && !result) { + if (i == ai) { + begs.push(i); + ai = str.indexOf(a, i + 1); + } else if (begs.length == 1) { + result = [ begs.pop(), bi ]; + } else { + beg = begs.pop(); + if (beg < left) { + left = beg; + right = bi; + } + + bi = str.indexOf(b, i + 1); + } + + i = ai < bi && ai >= 0 ? ai : bi; + } + + if (begs.length) { + result = [ left, right ]; + } + } + + return result; +} + +},{}],11:[function(require,module,exports){ +var concatMap = require('concat-map'); +var balanced = require('balanced-match'); + +module.exports = expandTop; + +var escSlash = '\0SLASH'+Math.random()+'\0'; +var escOpen = '\0OPEN'+Math.random()+'\0'; +var escClose = '\0CLOSE'+Math.random()+'\0'; +var escComma = '\0COMMA'+Math.random()+'\0'; +var escPeriod = '\0PERIOD'+Math.random()+'\0'; + +function numeric(str) { + return parseInt(str, 10) == str + ? parseInt(str, 10) + : str.charCodeAt(0); +} + +function escapeBraces(str) { + return str.split('\\\\').join(escSlash) + .split('\\{').join(escOpen) + .split('\\}').join(escClose) + .split('\\,').join(escComma) + .split('\\.').join(escPeriod); +} + +function unescapeBraces(str) { + return str.split(escSlash).join('\\') + .split(escOpen).join('{') + .split(escClose).join('}') + .split(escComma).join(',') + .split(escPeriod).join('.'); +} + + +// Basically just str.split(","), but handling cases +// where we have nested braced sections, which should be +// treated as individual members, like {a,{b,c},d} +function parseCommaParts(str) { + if (!str) + return ['']; + + var parts = []; + var m = balanced('{', '}', str); + + if (!m) + return str.split(','); + + var pre = m.pre; + var body = m.body; + var post = m.post; + var p = pre.split(','); + + p[p.length-1] += '{' + body + '}'; + var postParts = parseCommaParts(post); + if (post.length) { + p[p.length-1] += postParts.shift(); + p.push.apply(p, postParts); + } + + parts.push.apply(parts, p); + + return parts; +} + +function expandTop(str) { + if (!str) + return []; + + // I don't know why Bash 4.3 does this, but it does. + // Anything starting with {} will have the first two bytes preserved + // but *only* at the top level, so {},a}b will not expand to anything, + // but a{},b}c will be expanded to [a}c,abc]. + // One could argue that this is a bug in Bash, but since the goal of + // this module is to match Bash's rules, we escape a leading {} + if (str.substr(0, 2) === '{}') { + str = '\\{\\}' + str.substr(2); + } + + return expand(escapeBraces(str), true).map(unescapeBraces); +} + +function identity(e) { + return e; +} + +function embrace(str) { + return '{' + str + '}'; +} +function isPadded(el) { + return /^-?0\d/.test(el); +} + +function lte(i, y) { + return i <= y; +} +function gte(i, y) { + return i >= y; +} + +function expand(str, isTop) { + var expansions = []; + + var m = balanced('{', '}', str); + if (!m || /\$$/.test(m.pre)) return [str]; + + var isNumericSequence = /^-?\d+\.\.-?\d+(?:\.\.-?\d+)?$/.test(m.body); + var isAlphaSequence = /^[a-zA-Z]\.\.[a-zA-Z](?:\.\.-?\d+)?$/.test(m.body); + var isSequence = isNumericSequence || isAlphaSequence; + var isOptions = m.body.indexOf(',') >= 0; + if (!isSequence && !isOptions) { + // {a},b} + if (m.post.match(/,.*\}/)) { + str = m.pre + '{' + m.body + escClose + m.post; + return expand(str); + } + return [str]; + } + + var n; + if (isSequence) { + n = m.body.split(/\.\./); + } else { + n = parseCommaParts(m.body); + if (n.length === 1) { + // x{{a,b}}y ==> x{a}y x{b}y + n = expand(n[0], false).map(embrace); + if (n.length === 1) { + var post = m.post.length + ? expand(m.post, false) + : ['']; + return post.map(function(p) { + return m.pre + n[0] + p; + }); + } + } + } + + // at this point, n is the parts, and we know it's not a comma set + // with a single entry. + + // no need to expand pre, since it is guaranteed to be free of brace-sets + var pre = m.pre; + var post = m.post.length + ? expand(m.post, false) + : ['']; + + var N; + + if (isSequence) { + var x = numeric(n[0]); + var y = numeric(n[1]); + var width = Math.max(n[0].length, n[1].length) + var incr = n.length == 3 + ? Math.abs(numeric(n[2])) + : 1; + var test = lte; + var reverse = y < x; + if (reverse) { + incr *= -1; + test = gte; + } + var pad = n.some(isPadded); + + N = []; + + for (var i = x; test(i, y); i += incr) { + var c; + if (isAlphaSequence) { + c = String.fromCharCode(i); + if (c === '\\') + c = ''; + } else { + c = String(i); + if (pad) { + var need = width - c.length; + if (need > 0) { + var z = new Array(need + 1).join('0'); + if (i < 0) + c = '-' + z + c.slice(1); + else + c = z + c; + } + } + } + N.push(c); + } + } else { + N = concatMap(n, function(el) { return expand(el, false) }); + } + + for (var j = 0; j < N.length; j++) { + for (var k = 0; k < post.length; k++) { + var expansion = pre + N[j] + post[k]; + if (!isTop || isSequence || expansion) + expansions.push(expansion); + } + } + + return expansions; +} + + +},{"balanced-match":10,"concat-map":13}],12:[function(require,module,exports){ + +},{}],13:[function(require,module,exports){ +module.exports = function (xs, fn) { + var res = []; + for (var i = 0; i < xs.length; i++) { + var x = fn(xs[i], i); + if (isArray(x)) res.push.apply(res, x); + else res.push(x); + } + return res; +}; + +var isArray = Array.isArray || function (xs) { + return Object.prototype.toString.call(xs) === '[object Array]'; +}; + +},{}],14:[function(require,module,exports){ +// Copyright Joyent, Inc. and other Node contributors. +// +// Permission is hereby granted, free of charge, to any person obtaining a +// copy of this software and associated documentation files (the +// "Software"), to deal in the Software without restriction, including +// without limitation the rights to use, copy, modify, merge, publish, +// distribute, sublicense, and/or sell copies of the Software, and to permit +// persons to whom the Software is furnished to do so, subject to the +// following conditions: +// +// The above copyright notice and this permission notice shall be included +// in all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS +// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF +// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN +// NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, +// DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR +// OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE +// USE OR OTHER DEALINGS IN THE SOFTWARE. + +function EventEmitter() { + this._events = this._events || {}; + this._maxListeners = this._maxListeners || undefined; +} +module.exports = EventEmitter; + +// Backwards-compat with node 0.10.x +EventEmitter.EventEmitter = EventEmitter; + +EventEmitter.prototype._events = undefined; +EventEmitter.prototype._maxListeners = undefined; + +// By default EventEmitters will print a warning if more than 10 listeners are +// added to it. This is a useful default which helps finding memory leaks. +EventEmitter.defaultMaxListeners = 10; + +// Obviously not all Emitters should be limited to 10. This function allows +// that to be increased. Set to zero for unlimited. +EventEmitter.prototype.setMaxListeners = function(n) { + if (!isNumber(n) || n < 0 || isNaN(n)) + throw TypeError('n must be a positive number'); + this._maxListeners = n; + return this; +}; + +EventEmitter.prototype.emit = function(type) { + var er, handler, len, args, i, listeners; + + if (!this._events) + this._events = {}; + + // If there is no 'error' event listener then throw. + if (type === 'error') { + if (!this._events.error || + (isObject(this._events.error) && !this._events.error.length)) { + er = arguments[1]; + if (er instanceof Error) { + throw er; // Unhandled 'error' event + } + throw TypeError('Uncaught, unspecified "error" event.'); + } + } + + handler = this._events[type]; + + if (isUndefined(handler)) + return false; + + if (isFunction(handler)) { + switch (arguments.length) { + // fast cases + case 1: + handler.call(this); + break; + case 2: + handler.call(this, arguments[1]); + break; + case 3: + handler.call(this, arguments[1], arguments[2]); + break; + // slower + default: + len = arguments.length; + args = new Array(len - 1); + for (i = 1; i < len; i++) + args[i - 1] = arguments[i]; + handler.apply(this, args); + } + } else if (isObject(handler)) { + len = arguments.length; + args = new Array(len - 1); + for (i = 1; i < len; i++) + args[i - 1] = arguments[i]; + + listeners = handler.slice(); + len = listeners.length; + for (i = 0; i < len; i++) + listeners[i].apply(this, args); + } + + return true; +}; + +EventEmitter.prototype.addListener = function(type, listener) { + var m; + + if (!isFunction(listener)) + throw TypeError('listener must be a function'); + + if (!this._events) + this._events = {}; + + // To avoid recursion in the case that type === "newListener"! Before + // adding it to the listeners, first emit "newListener". + if (this._events.newListener) + this.emit('newListener', type, + isFunction(listener.listener) ? + listener.listener : listener); + + if (!this._events[type]) + // Optimize the case of one listener. Don't need the extra array object. + this._events[type] = listener; + else if (isObject(this._events[type])) + // If we've already got an array, just append. + this._events[type].push(listener); + else + // Adding the second element, need to change to array. + this._events[type] = [this._events[type], listener]; + + // Check for listener leak + if (isObject(this._events[type]) && !this._events[type].warned) { + var m; + if (!isUndefined(this._maxListeners)) { + m = this._maxListeners; + } else { + m = EventEmitter.defaultMaxListeners; + } + + if (m && m > 0 && this._events[type].length > m) { + this._events[type].warned = true; + console.error('(node) warning: possible EventEmitter memory ' + + 'leak detected. %d listeners added. ' + + 'Use emitter.setMaxListeners() to increase limit.', + this._events[type].length); + if (typeof console.trace === 'function') { + // not supported in IE 10 + console.trace(); + } + } + } + + return this; +}; + +EventEmitter.prototype.on = EventEmitter.prototype.addListener; + +EventEmitter.prototype.once = function(type, listener) { + if (!isFunction(listener)) + throw TypeError('listener must be a function'); + + var fired = false; + + function g() { + this.removeListener(type, g); + + if (!fired) { + fired = true; + listener.apply(this, arguments); + } + } + + g.listener = listener; + this.on(type, g); + + return this; +}; + +// emits a 'removeListener' event iff the listener was removed +EventEmitter.prototype.removeListener = function(type, listener) { + var list, position, length, i; + + if (!isFunction(listener)) + throw TypeError('listener must be a function'); + + if (!this._events || !this._events[type]) + return this; + + list = this._events[type]; + length = list.length; + position = -1; + + if (list === listener || + (isFunction(list.listener) && list.listener === listener)) { + delete this._events[type]; + if (this._events.removeListener) + this.emit('removeListener', type, listener); + + } else if (isObject(list)) { + for (i = length; i-- > 0;) { + if (list[i] === listener || + (list[i].listener && list[i].listener === listener)) { + position = i; + break; + } + } + + if (position < 0) + return this; + + if (list.length === 1) { + list.length = 0; + delete this._events[type]; + } else { + list.splice(position, 1); + } + + if (this._events.removeListener) + this.emit('removeListener', type, listener); + } + + return this; +}; + +EventEmitter.prototype.removeAllListeners = function(type) { + var key, listeners; + + if (!this._events) + return this; + + // not listening for removeListener, no need to emit + if (!this._events.removeListener) { + if (arguments.length === 0) + this._events = {}; + else if (this._events[type]) + delete this._events[type]; + return this; + } + + // emit removeListener for all listeners on all events + if (arguments.length === 0) { + for (key in this._events) { + if (key === 'removeListener') continue; + this.removeAllListeners(key); + } + this.removeAllListeners('removeListener'); + this._events = {}; + return this; + } + + listeners = this._events[type]; + + if (isFunction(listeners)) { + this.removeListener(type, listeners); + } else { + // LIFO order + while (listeners.length) + this.removeListener(type, listeners[listeners.length - 1]); + } + delete this._events[type]; + + return this; +}; + +EventEmitter.prototype.listeners = function(type) { + var ret; + if (!this._events || !this._events[type]) + ret = []; + else if (isFunction(this._events[type])) + ret = [this._events[type]]; + else + ret = this._events[type].slice(); + return ret; +}; + +EventEmitter.listenerCount = function(emitter, type) { + var ret; + if (!emitter._events || !emitter._events[type]) + ret = 0; + else if (isFunction(emitter._events[type])) + ret = 1; + else + ret = emitter._events[type].length; + return ret; +}; + +function isFunction(arg) { + return typeof arg === 'function'; +} + +function isNumber(arg) { + return typeof arg === 'number'; +} + +function isObject(arg) { + return typeof arg === 'object' && arg !== null; +} + +function isUndefined(arg) { + return arg === void 0; +} + +},{}],15:[function(require,module,exports){ +(function (process){ +exports.alphasort = alphasort +exports.alphasorti = alphasorti +exports.setopts = setopts +exports.ownProp = ownProp +exports.makeAbs = makeAbs +exports.finish = finish +exports.mark = mark +exports.isIgnored = isIgnored +exports.childrenIgnored = childrenIgnored + +function ownProp (obj, field) { + return Object.prototype.hasOwnProperty.call(obj, field) +} + +var path = require("path") +var minimatch = require("minimatch") +var isAbsolute = require("path-is-absolute") +var Minimatch = minimatch.Minimatch + +function alphasorti (a, b) { + return a.toLowerCase().localeCompare(b.toLowerCase()) +} + +function alphasort (a, b) { + return a.localeCompare(b) +} + +function setupIgnores (self, options) { + self.ignore = options.ignore || [] + + if (!Array.isArray(self.ignore)) + self.ignore = [self.ignore] + + if (self.ignore.length) { + self.ignore = self.ignore.map(ignoreMap) + } +} + +function ignoreMap (pattern) { + var gmatcher = null + if (pattern.slice(-3) === '/**') { + var gpattern = pattern.replace(/(\/\*\*)+$/, '') + gmatcher = new Minimatch(gpattern) + } + + return { + matcher: new Minimatch(pattern), + gmatcher: gmatcher + } +} + +function setopts (self, pattern, options) { + if (!options) + options = {} + + // base-matching: just use globstar for that. + if (options.matchBase && -1 === pattern.indexOf("/")) { + if (options.noglobstar) { + throw new Error("base matching requires globstar") + } + pattern = "**/" + pattern + } + + self.silent = !!options.silent + self.pattern = pattern + self.strict = options.strict !== false + self.realpath = !!options.realpath + self.realpathCache = options.realpathCache || Object.create(null) + self.follow = !!options.follow + self.dot = !!options.dot + self.mark = !!options.mark + self.nodir = !!options.nodir + if (self.nodir) + self.mark = true + self.sync = !!options.sync + self.nounique = !!options.nounique + self.nonull = !!options.nonull + self.nosort = !!options.nosort + self.nocase = !!options.nocase + self.stat = !!options.stat + self.noprocess = !!options.noprocess + + self.maxLength = options.maxLength || Infinity + self.cache = options.cache || Object.create(null) + self.statCache = options.statCache || Object.create(null) + self.symlinks = options.symlinks || Object.create(null) + + setupIgnores(self, options) + + self.changedCwd = false + var cwd = process.cwd() + if (!ownProp(options, "cwd")) + self.cwd = cwd + else { + self.cwd = options.cwd + self.changedCwd = path.resolve(options.cwd) !== cwd + } + + self.root = options.root || path.resolve(self.cwd, "/") + self.root = path.resolve(self.root) + if (process.platform === "win32") + self.root = self.root.replace(/\\/g, "/") + + self.nomount = !!options.nomount + + // disable comments and negation unless the user explicitly + // passes in false as the option. + options.nonegate = options.nonegate === false ? false : true + options.nocomment = options.nocomment === false ? false : true + deprecationWarning(options) + + self.minimatch = new Minimatch(pattern, options) + self.options = self.minimatch.options +} + +// TODO(isaacs): remove entirely in v6 +// exported to reset in tests +exports.deprecationWarned +function deprecationWarning(options) { + if (!options.nonegate || !options.nocomment) { + if (process.noDeprecation !== true && !exports.deprecationWarned) { + var msg = 'glob WARNING: comments and negation will be disabled in v6' + if (process.throwDeprecation) + throw new Error(msg) + else if (process.traceDeprecation) + console.trace(msg) + else + console.error(msg) + + exports.deprecationWarned = true + } + } +} + +function finish (self) { + var nou = self.nounique + var all = nou ? [] : Object.create(null) + + for (var i = 0, l = self.matches.length; i < l; i ++) { + var matches = self.matches[i] + if (!matches || Object.keys(matches).length === 0) { + if (self.nonull) { + // do like the shell, and spit out the literal glob + var literal = self.minimatch.globSet[i] + if (nou) + all.push(literal) + else + all[literal] = true + } + } else { + // had matches + var m = Object.keys(matches) + if (nou) + all.push.apply(all, m) + else + m.forEach(function (m) { + all[m] = true + }) + } + } + + if (!nou) + all = Object.keys(all) + + if (!self.nosort) + all = all.sort(self.nocase ? alphasorti : alphasort) + + // at *some* point we statted all of these + if (self.mark) { + for (var i = 0; i < all.length; i++) { + all[i] = self._mark(all[i]) + } + if (self.nodir) { + all = all.filter(function (e) { + return !(/\/$/.test(e)) + }) + } + } + + if (self.ignore.length) + all = all.filter(function(m) { + return !isIgnored(self, m) + }) + + self.found = all +} + +function mark (self, p) { + var abs = makeAbs(self, p) + var c = self.cache[abs] + var m = p + if (c) { + var isDir = c === 'DIR' || Array.isArray(c) + var slash = p.slice(-1) === '/' + + if (isDir && !slash) + m += '/' + else if (!isDir && slash) + m = m.slice(0, -1) + + if (m !== p) { + var mabs = makeAbs(self, m) + self.statCache[mabs] = self.statCache[abs] + self.cache[mabs] = self.cache[abs] + } + } + + return m +} + +// lotta situps... +function makeAbs (self, f) { + var abs = f + if (f.charAt(0) === '/') { + abs = path.join(self.root, f) + } else if (isAbsolute(f) || f === '') { + abs = f + } else if (self.changedCwd) { + abs = path.resolve(self.cwd, f) + } else { + abs = path.resolve(f) + } + return abs +} + + +// Return true, if pattern ends with globstar '**', for the accompanying parent directory. +// Ex:- If node_modules/** is the pattern, add 'node_modules' to ignore list along with it's contents +function isIgnored (self, path) { + if (!self.ignore.length) + return false + + return self.ignore.some(function(item) { + return item.matcher.match(path) || !!(item.gmatcher && item.gmatcher.match(path)) + }) +} + +function childrenIgnored (self, path) { + if (!self.ignore.length) + return false + + return self.ignore.some(function(item) { + return !!(item.gmatcher && item.gmatcher.match(path)) + }) +} + +}).call(this,require('_process')) +},{"_process":24,"minimatch":20,"path":22,"path-is-absolute":23}],16:[function(require,module,exports){ +(function (process){ +// Approach: +// +// 1. Get the minimatch set +// 2. For each pattern in the set, PROCESS(pattern, false) +// 3. Store matches per-set, then uniq them +// +// PROCESS(pattern, inGlobStar) +// Get the first [n] items from pattern that are all strings +// Join these together. This is PREFIX. +// If there is no more remaining, then stat(PREFIX) and +// add to matches if it succeeds. END. +// +// If inGlobStar and PREFIX is symlink and points to dir +// set ENTRIES = [] +// else readdir(PREFIX) as ENTRIES +// If fail, END +// +// with ENTRIES +// If pattern[n] is GLOBSTAR +// // handle the case where the globstar match is empty +// // by pruning it out, and testing the resulting pattern +// PROCESS(pattern[0..n] + pattern[n+1 .. $], false) +// // handle other cases. +// for ENTRY in ENTRIES (not dotfiles) +// // attach globstar + tail onto the entry +// // Mark that this entry is a globstar match +// PROCESS(pattern[0..n] + ENTRY + pattern[n .. $], true) +// +// else // not globstar +// for ENTRY in ENTRIES (not dotfiles, unless pattern[n] is dot) +// Test ENTRY against pattern[n] +// If fails, continue +// If passes, PROCESS(pattern[0..n] + item + pattern[n+1 .. $]) +// +// Caveat: +// Cache all stats and readdirs results to minimize syscall. Since all +// we ever care about is existence and directory-ness, we can just keep +// `true` for files, and [children,...] for directories, or `false` for +// things that don't exist. + +module.exports = glob + +var fs = require('fs') +var minimatch = require('minimatch') +var Minimatch = minimatch.Minimatch +var inherits = require('inherits') +var EE = require('events').EventEmitter +var path = require('path') +var assert = require('assert') +var isAbsolute = require('path-is-absolute') +var globSync = require('./sync.js') +var common = require('./common.js') +var alphasort = common.alphasort +var alphasorti = common.alphasorti +var setopts = common.setopts +var ownProp = common.ownProp +var inflight = require('inflight') +var util = require('util') +var childrenIgnored = common.childrenIgnored +var isIgnored = common.isIgnored + +var once = require('once') + +function glob (pattern, options, cb) { + if (typeof options === 'function') cb = options, options = {} + if (!options) options = {} + + if (options.sync) { + if (cb) + throw new TypeError('callback provided to sync glob') + return globSync(pattern, options) + } + + return new Glob(pattern, options, cb) +} + +glob.sync = globSync +var GlobSync = glob.GlobSync = globSync.GlobSync + +// old api surface +glob.glob = glob + +glob.hasMagic = function (pattern, options_) { + var options = util._extend({}, options_) + options.noprocess = true + + var g = new Glob(pattern, options) + var set = g.minimatch.set + if (set.length > 1) + return true + + for (var j = 0; j < set[0].length; j++) { + if (typeof set[0][j] !== 'string') + return true + } + + return false +} + +glob.Glob = Glob +inherits(Glob, EE) +function Glob (pattern, options, cb) { + if (typeof options === 'function') { + cb = options + options = null + } + + if (options && options.sync) { + if (cb) + throw new TypeError('callback provided to sync glob') + return new GlobSync(pattern, options) + } + + if (!(this instanceof Glob)) + return new Glob(pattern, options, cb) + + setopts(this, pattern, options) + this._didRealPath = false + + // process each pattern in the minimatch set + var n = this.minimatch.set.length + + // The matches are stored as {: true,...} so that + // duplicates are automagically pruned. + // Later, we do an Object.keys() on these. + // Keep them as a list so we can fill in when nonull is set. + this.matches = new Array(n) + + if (typeof cb === 'function') { + cb = once(cb) + this.on('error', cb) + this.on('end', function (matches) { + cb(null, matches) + }) + } + + var self = this + var n = this.minimatch.set.length + this._processing = 0 + this.matches = new Array(n) + + this._emitQueue = [] + this._processQueue = [] + this.paused = false + + if (this.noprocess) + return this + + if (n === 0) + return done() + + for (var i = 0; i < n; i ++) { + this._process(this.minimatch.set[i], i, false, done) + } + + function done () { + --self._processing + if (self._processing <= 0) + self._finish() + } +} + +Glob.prototype._finish = function () { + assert(this instanceof Glob) + if (this.aborted) + return + + if (this.realpath && !this._didRealpath) + return this._realpath() + + common.finish(this) + this.emit('end', this.found) +} + +Glob.prototype._realpath = function () { + if (this._didRealpath) + return + + this._didRealpath = true + + var n = this.matches.length + if (n === 0) + return this._finish() + + var self = this + for (var i = 0; i < this.matches.length; i++) + this._realpathSet(i, next) + + function next () { + if (--n === 0) + self._finish() + } +} + +Glob.prototype._realpathSet = function (index, cb) { + var matchset = this.matches[index] + if (!matchset) + return cb() + + var found = Object.keys(matchset) + var self = this + var n = found.length + + if (n === 0) + return cb() + + var set = this.matches[index] = Object.create(null) + found.forEach(function (p, i) { + // If there's a problem with the stat, then it means that + // one or more of the links in the realpath couldn't be + // resolved. just return the abs value in that case. + p = self._makeAbs(p) + fs.realpath(p, self.realpathCache, function (er, real) { + if (!er) + set[real] = true + else if (er.syscall === 'stat') + set[p] = true + else + self.emit('error', er) // srsly wtf right here + + if (--n === 0) { + self.matches[index] = set + cb() + } + }) + }) +} + +Glob.prototype._mark = function (p) { + return common.mark(this, p) +} + +Glob.prototype._makeAbs = function (f) { + return common.makeAbs(this, f) +} + +Glob.prototype.abort = function () { + this.aborted = true + this.emit('abort') +} + +Glob.prototype.pause = function () { + if (!this.paused) { + this.paused = true + this.emit('pause') + } +} + +Glob.prototype.resume = function () { + if (this.paused) { + this.emit('resume') + this.paused = false + if (this._emitQueue.length) { + var eq = this._emitQueue.slice(0) + this._emitQueue.length = 0 + for (var i = 0; i < eq.length; i ++) { + var e = eq[i] + this._emitMatch(e[0], e[1]) + } + } + if (this._processQueue.length) { + var pq = this._processQueue.slice(0) + this._processQueue.length = 0 + for (var i = 0; i < pq.length; i ++) { + var p = pq[i] + this._processing-- + this._process(p[0], p[1], p[2], p[3]) + } + } + } +} + +Glob.prototype._process = function (pattern, index, inGlobStar, cb) { + assert(this instanceof Glob) + assert(typeof cb === 'function') + + if (this.aborted) + return + + this._processing++ + if (this.paused) { + this._processQueue.push([pattern, index, inGlobStar, cb]) + return + } + + //console.error('PROCESS %d', this._processing, pattern) + + // Get the first [n] parts of pattern that are all strings. + var n = 0 + while (typeof pattern[n] === 'string') { + n ++ + } + // now n is the index of the first one that is *not* a string. + + // see if there's anything else + var prefix + switch (n) { + // if not, then this is rather simple + case pattern.length: + this._processSimple(pattern.join('/'), index, cb) + return + + case 0: + // pattern *starts* with some non-trivial item. + // going to readdir(cwd), but not include the prefix in matches. + prefix = null + break + + default: + // pattern has some string bits in the front. + // whatever it starts with, whether that's 'absolute' like /foo/bar, + // or 'relative' like '../baz' + prefix = pattern.slice(0, n).join('/') + break + } + + var remain = pattern.slice(n) + + // get the list of entries. + var read + if (prefix === null) + read = '.' + else if (isAbsolute(prefix) || isAbsolute(pattern.join('/'))) { + if (!prefix || !isAbsolute(prefix)) + prefix = '/' + prefix + read = prefix + } else + read = prefix + + var abs = this._makeAbs(read) + + //if ignored, skip _processing + if (childrenIgnored(this, read)) + return cb() + + var isGlobStar = remain[0] === minimatch.GLOBSTAR + if (isGlobStar) + this._processGlobStar(prefix, read, abs, remain, index, inGlobStar, cb) + else + this._processReaddir(prefix, read, abs, remain, index, inGlobStar, cb) +} + +Glob.prototype._processReaddir = function (prefix, read, abs, remain, index, inGlobStar, cb) { + var self = this + this._readdir(abs, inGlobStar, function (er, entries) { + return self._processReaddir2(prefix, read, abs, remain, index, inGlobStar, entries, cb) + }) +} + +Glob.prototype._processReaddir2 = function (prefix, read, abs, remain, index, inGlobStar, entries, cb) { + + // if the abs isn't a dir, then nothing can match! + if (!entries) + return cb() + + // It will only match dot entries if it starts with a dot, or if + // dot is set. Stuff like @(.foo|.bar) isn't allowed. + var pn = remain[0] + var negate = !!this.minimatch.negate + var rawGlob = pn._glob + var dotOk = this.dot || rawGlob.charAt(0) === '.' + + var matchedEntries = [] + for (var i = 0; i < entries.length; i++) { + var e = entries[i] + if (e.charAt(0) !== '.' || dotOk) { + var m + if (negate && !prefix) { + m = !e.match(pn) + } else { + m = e.match(pn) + } + if (m) + matchedEntries.push(e) + } + } + + //console.error('prd2', prefix, entries, remain[0]._glob, matchedEntries) + + var len = matchedEntries.length + // If there are no matched entries, then nothing matches. + if (len === 0) + return cb() + + // if this is the last remaining pattern bit, then no need for + // an additional stat *unless* the user has specified mark or + // stat explicitly. We know they exist, since readdir returned + // them. + + if (remain.length === 1 && !this.mark && !this.stat) { + if (!this.matches[index]) + this.matches[index] = Object.create(null) + + for (var i = 0; i < len; i ++) { + var e = matchedEntries[i] + if (prefix) { + if (prefix !== '/') + e = prefix + '/' + e + else + e = prefix + e + } + + if (e.charAt(0) === '/' && !this.nomount) { + e = path.join(this.root, e) + } + this._emitMatch(index, e) + } + // This was the last one, and no stats were needed + return cb() + } + + // now test all matched entries as stand-ins for that part + // of the pattern. + remain.shift() + for (var i = 0; i < len; i ++) { + var e = matchedEntries[i] + var newPattern + if (prefix) { + if (prefix !== '/') + e = prefix + '/' + e + else + e = prefix + e + } + this._process([e].concat(remain), index, inGlobStar, cb) + } + cb() +} + +Glob.prototype._emitMatch = function (index, e) { + if (this.aborted) + return + + if (this.matches[index][e]) + return + + if (isIgnored(this, e)) + return + + if (this.paused) { + this._emitQueue.push([index, e]) + return + } + + var abs = this._makeAbs(e) + + if (this.nodir) { + var c = this.cache[abs] + if (c === 'DIR' || Array.isArray(c)) + return + } + + if (this.mark) + e = this._mark(e) + + this.matches[index][e] = true + + var st = this.statCache[abs] + if (st) + this.emit('stat', e, st) + + this.emit('match', e) +} + +Glob.prototype._readdirInGlobStar = function (abs, cb) { + if (this.aborted) + return + + // follow all symlinked directories forever + // just proceed as if this is a non-globstar situation + if (this.follow) + return this._readdir(abs, false, cb) + + var lstatkey = 'lstat\0' + abs + var self = this + var lstatcb = inflight(lstatkey, lstatcb_) + + if (lstatcb) + fs.lstat(abs, lstatcb) + + function lstatcb_ (er, lstat) { + if (er) + return cb() + + var isSym = lstat.isSymbolicLink() + self.symlinks[abs] = isSym + + // If it's not a symlink or a dir, then it's definitely a regular file. + // don't bother doing a readdir in that case. + if (!isSym && !lstat.isDirectory()) { + self.cache[abs] = 'FILE' + cb() + } else + self._readdir(abs, false, cb) + } +} + +Glob.prototype._readdir = function (abs, inGlobStar, cb) { + if (this.aborted) + return + + cb = inflight('readdir\0'+abs+'\0'+inGlobStar, cb) + if (!cb) + return + + //console.error('RD %j %j', +inGlobStar, abs) + if (inGlobStar && !ownProp(this.symlinks, abs)) + return this._readdirInGlobStar(abs, cb) + + if (ownProp(this.cache, abs)) { + var c = this.cache[abs] + if (!c || c === 'FILE') + return cb() + + if (Array.isArray(c)) + return cb(null, c) + } + + var self = this + fs.readdir(abs, readdirCb(this, abs, cb)) +} + +function readdirCb (self, abs, cb) { + return function (er, entries) { + if (er) + self._readdirError(abs, er, cb) + else + self._readdirEntries(abs, entries, cb) + } +} + +Glob.prototype._readdirEntries = function (abs, entries, cb) { + if (this.aborted) + return + + // if we haven't asked to stat everything, then just + // assume that everything in there exists, so we can avoid + // having to stat it a second time. + if (!this.mark && !this.stat) { + for (var i = 0; i < entries.length; i ++) { + var e = entries[i] + if (abs === '/') + e = abs + e + else + e = abs + '/' + e + this.cache[e] = true + } + } + + this.cache[abs] = entries + return cb(null, entries) +} + +Glob.prototype._readdirError = function (f, er, cb) { + if (this.aborted) + return + + // handle errors, and cache the information + switch (er.code) { + case 'ENOTSUP': // https://github.com/isaacs/node-glob/issues/205 + case 'ENOTDIR': // totally normal. means it *does* exist. + this.cache[this._makeAbs(f)] = 'FILE' + break + + case 'ENOENT': // not terribly unusual + case 'ELOOP': + case 'ENAMETOOLONG': + case 'UNKNOWN': + this.cache[this._makeAbs(f)] = false + break + + default: // some unusual error. Treat as failure. + this.cache[this._makeAbs(f)] = false + if (this.strict) { + this.emit('error', er) + // If the error is handled, then we abort + // if not, we threw out of here + this.abort() + } + if (!this.silent) + console.error('glob error', er) + break + } + + return cb() +} + +Glob.prototype._processGlobStar = function (prefix, read, abs, remain, index, inGlobStar, cb) { + var self = this + this._readdir(abs, inGlobStar, function (er, entries) { + self._processGlobStar2(prefix, read, abs, remain, index, inGlobStar, entries, cb) + }) +} + + +Glob.prototype._processGlobStar2 = function (prefix, read, abs, remain, index, inGlobStar, entries, cb) { + //console.error('pgs2', prefix, remain[0], entries) + + // no entries means not a dir, so it can never have matches + // foo.txt/** doesn't match foo.txt + if (!entries) + return cb() + + // test without the globstar, and with every child both below + // and replacing the globstar. + var remainWithoutGlobStar = remain.slice(1) + var gspref = prefix ? [ prefix ] : [] + var noGlobStar = gspref.concat(remainWithoutGlobStar) + + // the noGlobStar pattern exits the inGlobStar state + this._process(noGlobStar, index, false, cb) + + var isSym = this.symlinks[abs] + var len = entries.length + + // If it's a symlink, and we're in a globstar, then stop + if (isSym && inGlobStar) + return cb() + + for (var i = 0; i < len; i++) { + var e = entries[i] + if (e.charAt(0) === '.' && !this.dot) + continue + + // these two cases enter the inGlobStar state + var instead = gspref.concat(entries[i], remainWithoutGlobStar) + this._process(instead, index, true, cb) + + var below = gspref.concat(entries[i], remain) + this._process(below, index, true, cb) + } + + cb() +} + +Glob.prototype._processSimple = function (prefix, index, cb) { + // XXX review this. Shouldn't it be doing the mounting etc + // before doing stat? kinda weird? + var self = this + this._stat(prefix, function (er, exists) { + self._processSimple2(prefix, index, er, exists, cb) + }) +} +Glob.prototype._processSimple2 = function (prefix, index, er, exists, cb) { + + //console.error('ps2', prefix, exists) + + if (!this.matches[index]) + this.matches[index] = Object.create(null) + + // If it doesn't exist, then just mark the lack of results + if (!exists) + return cb() + + if (prefix && isAbsolute(prefix) && !this.nomount) { + var trail = /[\/\\]$/.test(prefix) + if (prefix.charAt(0) === '/') { + prefix = path.join(this.root, prefix) + } else { + prefix = path.resolve(this.root, prefix) + if (trail) + prefix += '/' + } + } + + if (process.platform === 'win32') + prefix = prefix.replace(/\\/g, '/') + + // Mark this as a match + this._emitMatch(index, prefix) + cb() +} + +// Returns either 'DIR', 'FILE', or false +Glob.prototype._stat = function (f, cb) { + var abs = this._makeAbs(f) + var needDir = f.slice(-1) === '/' + + if (f.length > this.maxLength) + return cb() + + if (!this.stat && ownProp(this.cache, abs)) { + var c = this.cache[abs] + + if (Array.isArray(c)) + c = 'DIR' + + // It exists, but maybe not how we need it + if (!needDir || c === 'DIR') + return cb(null, c) + + if (needDir && c === 'FILE') + return cb() + + // otherwise we have to stat, because maybe c=true + // if we know it exists, but not what it is. + } + + var exists + var stat = this.statCache[abs] + if (stat !== undefined) { + if (stat === false) + return cb(null, stat) + else { + var type = stat.isDirectory() ? 'DIR' : 'FILE' + if (needDir && type === 'FILE') + return cb() + else + return cb(null, type, stat) + } + } + + var self = this + var statcb = inflight('stat\0' + abs, lstatcb_) + if (statcb) + fs.lstat(abs, statcb) + + function lstatcb_ (er, lstat) { + if (lstat && lstat.isSymbolicLink()) { + // If it's a symlink, then treat it as the target, unless + // the target does not exist, then treat it as a file. + return fs.stat(abs, function (er, stat) { + if (er) + self._stat2(f, abs, null, lstat, cb) + else + self._stat2(f, abs, er, stat, cb) + }) + } else { + self._stat2(f, abs, er, lstat, cb) + } + } +} + +Glob.prototype._stat2 = function (f, abs, er, stat, cb) { + if (er) { + this.statCache[abs] = false + return cb() + } + + var needDir = f.slice(-1) === '/' + this.statCache[abs] = stat + + if (abs.slice(-1) === '/' && !stat.isDirectory()) + return cb(null, false, stat) + + var c = stat.isDirectory() ? 'DIR' : 'FILE' + this.cache[abs] = this.cache[abs] || c + + if (needDir && c !== 'DIR') + return cb() + + return cb(null, c, stat) +} + +}).call(this,require('_process')) +},{"./common.js":15,"./sync.js":17,"_process":24,"assert":9,"events":14,"fs":12,"inflight":18,"inherits":19,"minimatch":20,"once":21,"path":22,"path-is-absolute":23,"util":28}],17:[function(require,module,exports){ +(function (process){ +module.exports = globSync +globSync.GlobSync = GlobSync + +var fs = require('fs') +var minimatch = require('minimatch') +var Minimatch = minimatch.Minimatch +var Glob = require('./glob.js').Glob +var util = require('util') +var path = require('path') +var assert = require('assert') +var isAbsolute = require('path-is-absolute') +var common = require('./common.js') +var alphasort = common.alphasort +var alphasorti = common.alphasorti +var setopts = common.setopts +var ownProp = common.ownProp +var childrenIgnored = common.childrenIgnored + +function globSync (pattern, options) { + if (typeof options === 'function' || arguments.length === 3) + throw new TypeError('callback provided to sync glob\n'+ + 'See: https://github.com/isaacs/node-glob/issues/167') + + return new GlobSync(pattern, options).found +} + +function GlobSync (pattern, options) { + if (!pattern) + throw new Error('must provide pattern') + + if (typeof options === 'function' || arguments.length === 3) + throw new TypeError('callback provided to sync glob\n'+ + 'See: https://github.com/isaacs/node-glob/issues/167') + + if (!(this instanceof GlobSync)) + return new GlobSync(pattern, options) + + setopts(this, pattern, options) + + if (this.noprocess) + return this + + var n = this.minimatch.set.length + this.matches = new Array(n) + for (var i = 0; i < n; i ++) { + this._process(this.minimatch.set[i], i, false) + } + this._finish() +} + +GlobSync.prototype._finish = function () { + assert(this instanceof GlobSync) + if (this.realpath) { + var self = this + this.matches.forEach(function (matchset, index) { + var set = self.matches[index] = Object.create(null) + for (var p in matchset) { + try { + p = self._makeAbs(p) + var real = fs.realpathSync(p, self.realpathCache) + set[real] = true + } catch (er) { + if (er.syscall === 'stat') + set[self._makeAbs(p)] = true + else + throw er + } + } + }) + } + common.finish(this) +} + + +GlobSync.prototype._process = function (pattern, index, inGlobStar) { + assert(this instanceof GlobSync) + + // Get the first [n] parts of pattern that are all strings. + var n = 0 + while (typeof pattern[n] === 'string') { + n ++ + } + // now n is the index of the first one that is *not* a string. + + // See if there's anything else + var prefix + switch (n) { + // if not, then this is rather simple + case pattern.length: + this._processSimple(pattern.join('/'), index) + return + + case 0: + // pattern *starts* with some non-trivial item. + // going to readdir(cwd), but not include the prefix in matches. + prefix = null + break + + default: + // pattern has some string bits in the front. + // whatever it starts with, whether that's 'absolute' like /foo/bar, + // or 'relative' like '../baz' + prefix = pattern.slice(0, n).join('/') + break + } + + var remain = pattern.slice(n) + + // get the list of entries. + var read + if (prefix === null) + read = '.' + else if (isAbsolute(prefix) || isAbsolute(pattern.join('/'))) { + if (!prefix || !isAbsolute(prefix)) + prefix = '/' + prefix + read = prefix + } else + read = prefix + + var abs = this._makeAbs(read) + + //if ignored, skip processing + if (childrenIgnored(this, read)) + return + + var isGlobStar = remain[0] === minimatch.GLOBSTAR + if (isGlobStar) + this._processGlobStar(prefix, read, abs, remain, index, inGlobStar) + else + this._processReaddir(prefix, read, abs, remain, index, inGlobStar) +} + + +GlobSync.prototype._processReaddir = function (prefix, read, abs, remain, index, inGlobStar) { + var entries = this._readdir(abs, inGlobStar) + + // if the abs isn't a dir, then nothing can match! + if (!entries) + return + + // It will only match dot entries if it starts with a dot, or if + // dot is set. Stuff like @(.foo|.bar) isn't allowed. + var pn = remain[0] + var negate = !!this.minimatch.negate + var rawGlob = pn._glob + var dotOk = this.dot || rawGlob.charAt(0) === '.' + + var matchedEntries = [] + for (var i = 0; i < entries.length; i++) { + var e = entries[i] + if (e.charAt(0) !== '.' || dotOk) { + var m + if (negate && !prefix) { + m = !e.match(pn) + } else { + m = e.match(pn) + } + if (m) + matchedEntries.push(e) + } + } + + var len = matchedEntries.length + // If there are no matched entries, then nothing matches. + if (len === 0) + return + + // if this is the last remaining pattern bit, then no need for + // an additional stat *unless* the user has specified mark or + // stat explicitly. We know they exist, since readdir returned + // them. + + if (remain.length === 1 && !this.mark && !this.stat) { + if (!this.matches[index]) + this.matches[index] = Object.create(null) + + for (var i = 0; i < len; i ++) { + var e = matchedEntries[i] + if (prefix) { + if (prefix.slice(-1) !== '/') + e = prefix + '/' + e + else + e = prefix + e + } + + if (e.charAt(0) === '/' && !this.nomount) { + e = path.join(this.root, e) + } + this.matches[index][e] = true + } + // This was the last one, and no stats were needed + return + } + + // now test all matched entries as stand-ins for that part + // of the pattern. + remain.shift() + for (var i = 0; i < len; i ++) { + var e = matchedEntries[i] + var newPattern + if (prefix) + newPattern = [prefix, e] + else + newPattern = [e] + this._process(newPattern.concat(remain), index, inGlobStar) + } +} + + +GlobSync.prototype._emitMatch = function (index, e) { + var abs = this._makeAbs(e) + if (this.mark) + e = this._mark(e) + + if (this.matches[index][e]) + return + + if (this.nodir) { + var c = this.cache[this._makeAbs(e)] + if (c === 'DIR' || Array.isArray(c)) + return + } + + this.matches[index][e] = true + if (this.stat) + this._stat(e) +} + + +GlobSync.prototype._readdirInGlobStar = function (abs) { + // follow all symlinked directories forever + // just proceed as if this is a non-globstar situation + if (this.follow) + return this._readdir(abs, false) + + var entries + var lstat + var stat + try { + lstat = fs.lstatSync(abs) + } catch (er) { + // lstat failed, doesn't exist + return null + } + + var isSym = lstat.isSymbolicLink() + this.symlinks[abs] = isSym + + // If it's not a symlink or a dir, then it's definitely a regular file. + // don't bother doing a readdir in that case. + if (!isSym && !lstat.isDirectory()) + this.cache[abs] = 'FILE' + else + entries = this._readdir(abs, false) + + return entries +} + +GlobSync.prototype._readdir = function (abs, inGlobStar) { + var entries + + if (inGlobStar && !ownProp(this.symlinks, abs)) + return this._readdirInGlobStar(abs) + + if (ownProp(this.cache, abs)) { + var c = this.cache[abs] + if (!c || c === 'FILE') + return null + + if (Array.isArray(c)) + return c + } + + try { + return this._readdirEntries(abs, fs.readdirSync(abs)) + } catch (er) { + this._readdirError(abs, er) + return null + } +} + +GlobSync.prototype._readdirEntries = function (abs, entries) { + // if we haven't asked to stat everything, then just + // assume that everything in there exists, so we can avoid + // having to stat it a second time. + if (!this.mark && !this.stat) { + for (var i = 0; i < entries.length; i ++) { + var e = entries[i] + if (abs === '/') + e = abs + e + else + e = abs + '/' + e + this.cache[e] = true + } + } + + this.cache[abs] = entries + + // mark and cache dir-ness + return entries +} + +GlobSync.prototype._readdirError = function (f, er) { + // handle errors, and cache the information + switch (er.code) { + case 'ENOTSUP': // https://github.com/isaacs/node-glob/issues/205 + case 'ENOTDIR': // totally normal. means it *does* exist. + this.cache[this._makeAbs(f)] = 'FILE' + break + + case 'ENOENT': // not terribly unusual + case 'ELOOP': + case 'ENAMETOOLONG': + case 'UNKNOWN': + this.cache[this._makeAbs(f)] = false + break + + default: // some unusual error. Treat as failure. + this.cache[this._makeAbs(f)] = false + if (this.strict) + throw er + if (!this.silent) + console.error('glob error', er) + break + } +} + +GlobSync.prototype._processGlobStar = function (prefix, read, abs, remain, index, inGlobStar) { + + var entries = this._readdir(abs, inGlobStar) + + // no entries means not a dir, so it can never have matches + // foo.txt/** doesn't match foo.txt + if (!entries) + return + + // test without the globstar, and with every child both below + // and replacing the globstar. + var remainWithoutGlobStar = remain.slice(1) + var gspref = prefix ? [ prefix ] : [] + var noGlobStar = gspref.concat(remainWithoutGlobStar) + + // the noGlobStar pattern exits the inGlobStar state + this._process(noGlobStar, index, false) + + var len = entries.length + var isSym = this.symlinks[abs] + + // If it's a symlink, and we're in a globstar, then stop + if (isSym && inGlobStar) + return + + for (var i = 0; i < len; i++) { + var e = entries[i] + if (e.charAt(0) === '.' && !this.dot) + continue + + // these two cases enter the inGlobStar state + var instead = gspref.concat(entries[i], remainWithoutGlobStar) + this._process(instead, index, true) + + var below = gspref.concat(entries[i], remain) + this._process(below, index, true) + } +} + +GlobSync.prototype._processSimple = function (prefix, index) { + // XXX review this. Shouldn't it be doing the mounting etc + // before doing stat? kinda weird? + var exists = this._stat(prefix) + + if (!this.matches[index]) + this.matches[index] = Object.create(null) + + // If it doesn't exist, then just mark the lack of results + if (!exists) + return + + if (prefix && isAbsolute(prefix) && !this.nomount) { + var trail = /[\/\\]$/.test(prefix) + if (prefix.charAt(0) === '/') { + prefix = path.join(this.root, prefix) + } else { + prefix = path.resolve(this.root, prefix) + if (trail) + prefix += '/' + } + } + + if (process.platform === 'win32') + prefix = prefix.replace(/\\/g, '/') + + // Mark this as a match + this.matches[index][prefix] = true +} + +// Returns either 'DIR', 'FILE', or false +GlobSync.prototype._stat = function (f) { + var abs = this._makeAbs(f) + var needDir = f.slice(-1) === '/' + + if (f.length > this.maxLength) + return false + + if (!this.stat && ownProp(this.cache, abs)) { + var c = this.cache[abs] + + if (Array.isArray(c)) + c = 'DIR' + + // It exists, but maybe not how we need it + if (!needDir || c === 'DIR') + return c + + if (needDir && c === 'FILE') + return false + + // otherwise we have to stat, because maybe c=true + // if we know it exists, but not what it is. + } + + var exists + var stat = this.statCache[abs] + if (!stat) { + var lstat + try { + lstat = fs.lstatSync(abs) + } catch (er) { + return false + } + + if (lstat.isSymbolicLink()) { + try { + stat = fs.statSync(abs) + } catch (er) { + stat = lstat + } + } else { + stat = lstat + } + } + + this.statCache[abs] = stat + + var c = stat.isDirectory() ? 'DIR' : 'FILE' + this.cache[abs] = this.cache[abs] || c + + if (needDir && c !== 'DIR') + return false + + return c +} + +GlobSync.prototype._mark = function (p) { + return common.mark(this, p) +} + +GlobSync.prototype._makeAbs = function (f) { + return common.makeAbs(this, f) +} + +}).call(this,require('_process')) +},{"./common.js":15,"./glob.js":16,"_process":24,"assert":9,"fs":12,"minimatch":20,"path":22,"path-is-absolute":23,"util":28}],18:[function(require,module,exports){ +(function (process){ +var wrappy = require('wrappy') +var reqs = Object.create(null) +var once = require('once') + +module.exports = wrappy(inflight) + +function inflight (key, cb) { + if (reqs[key]) { + reqs[key].push(cb) + return null + } else { + reqs[key] = [cb] + return makeres(key) + } +} + +function makeres (key) { + return once(function RES () { + var cbs = reqs[key] + var len = cbs.length + var args = slice(arguments) + + // XXX It's somewhat ambiguous whether a new callback added in this + // pass should be queued for later execution if something in the + // list of callbacks throws, or if it should just be discarded. + // However, it's such an edge case that it hardly matters, and either + // choice is likely as surprising as the other. + // As it happens, we do go ahead and schedule it for later execution. + try { + for (var i = 0; i < len; i++) { + cbs[i].apply(null, args) + } + } finally { + if (cbs.length > len) { + // added more in the interim. + // de-zalgo, just in case, but don't call again. + cbs.splice(0, len) + process.nextTick(function () { + RES.apply(null, args) + }) + } else { + delete reqs[key] + } + } + }) +} + +function slice (args) { + var length = args.length + var array = [] + + for (var i = 0; i < length; i++) array[i] = args[i] + return array +} + +}).call(this,require('_process')) +},{"_process":24,"once":21,"wrappy":29}],19:[function(require,module,exports){ +if (typeof Object.create === 'function') { + // implementation from standard node.js 'util' module + module.exports = function inherits(ctor, superCtor) { + ctor.super_ = superCtor + ctor.prototype = Object.create(superCtor.prototype, { + constructor: { + value: ctor, + enumerable: false, + writable: true, + configurable: true + } + }); + }; +} else { + // old school shim for old browsers + module.exports = function inherits(ctor, superCtor) { + ctor.super_ = superCtor + var TempCtor = function () {} + TempCtor.prototype = superCtor.prototype + ctor.prototype = new TempCtor() + ctor.prototype.constructor = ctor + } +} + +},{}],20:[function(require,module,exports){ +module.exports = minimatch +minimatch.Minimatch = Minimatch + +var path = { sep: '/' } +try { + path = require('path') +} catch (er) {} + +var GLOBSTAR = minimatch.GLOBSTAR = Minimatch.GLOBSTAR = {} +var expand = require('brace-expansion') + +var plTypes = { + '!': { open: '(?:(?!(?:', close: '))[^/]*?)'}, + '?': { open: '(?:', close: ')?' }, + '+': { open: '(?:', close: ')+' }, + '*': { open: '(?:', close: ')*' }, + '@': { open: '(?:', close: ')' } +} + +// any single thing other than / +// don't need to escape / when using new RegExp() +var qmark = '[^/]' + +// * => any number of characters +var star = qmark + '*?' + +// ** when dots are allowed. Anything goes, except .. and . +// not (^ or / followed by one or two dots followed by $ or /), +// followed by anything, any number of times. +var twoStarDot = '(?:(?!(?:\\\/|^)(?:\\.{1,2})($|\\\/)).)*?' + +// not a ^ or / followed by a dot, +// followed by anything, any number of times. +var twoStarNoDot = '(?:(?!(?:\\\/|^)\\.).)*?' + +// characters that need to be escaped in RegExp. +var reSpecials = charSet('().*{}+?[]^$\\!') + +// "abc" -> { a:true, b:true, c:true } +function charSet (s) { + return s.split('').reduce(function (set, c) { + set[c] = true + return set + }, {}) +} + +// normalizes slashes. +var slashSplit = /\/+/ + +minimatch.filter = filter +function filter (pattern, options) { + options = options || {} + return function (p, i, list) { + return minimatch(p, pattern, options) + } +} + +function ext (a, b) { + a = a || {} + b = b || {} + var t = {} + Object.keys(b).forEach(function (k) { + t[k] = b[k] + }) + Object.keys(a).forEach(function (k) { + t[k] = a[k] + }) + return t +} + +minimatch.defaults = function (def) { + if (!def || !Object.keys(def).length) return minimatch + + var orig = minimatch + + var m = function minimatch (p, pattern, options) { + return orig.minimatch(p, pattern, ext(def, options)) + } + + m.Minimatch = function Minimatch (pattern, options) { + return new orig.Minimatch(pattern, ext(def, options)) + } + + return m +} + +Minimatch.defaults = function (def) { + if (!def || !Object.keys(def).length) return Minimatch + return minimatch.defaults(def).Minimatch +} + +function minimatch (p, pattern, options) { + if (typeof pattern !== 'string') { + throw new TypeError('glob pattern string required') + } + + if (!options) options = {} + + // shortcut: comments match nothing. + if (!options.nocomment && pattern.charAt(0) === '#') { + return false + } + + // "" only matches "" + if (pattern.trim() === '') return p === '' + + return new Minimatch(pattern, options).match(p) +} + +function Minimatch (pattern, options) { + if (!(this instanceof Minimatch)) { + return new Minimatch(pattern, options) + } + + if (typeof pattern !== 'string') { + throw new TypeError('glob pattern string required') + } + + if (!options) options = {} + pattern = pattern.trim() + + // windows support: need to use /, not \ + if (path.sep !== '/') { + pattern = pattern.split(path.sep).join('/') + } + + this.options = options + this.set = [] + this.pattern = pattern + this.regexp = null + this.negate = false + this.comment = false + this.empty = false + + // make the set of regexps etc. + this.make() +} + +Minimatch.prototype.debug = function () {} + +Minimatch.prototype.make = make +function make () { + // don't do it more than once. + if (this._made) return + + var pattern = this.pattern + var options = this.options + + // empty patterns and comments match nothing. + if (!options.nocomment && pattern.charAt(0) === '#') { + this.comment = true + return + } + if (!pattern) { + this.empty = true + return + } + + // step 1: figure out negation, etc. + this.parseNegate() + + // step 2: expand braces + var set = this.globSet = this.braceExpand() + + if (options.debug) this.debug = console.error + + this.debug(this.pattern, set) + + // step 3: now we have a set, so turn each one into a series of path-portion + // matching patterns. + // These will be regexps, except in the case of "**", which is + // set to the GLOBSTAR object for globstar behavior, + // and will not contain any / characters + set = this.globParts = set.map(function (s) { + return s.split(slashSplit) + }) + + this.debug(this.pattern, set) + + // glob --> regexps + set = set.map(function (s, si, set) { + return s.map(this.parse, this) + }, this) + + this.debug(this.pattern, set) + + // filter out everything that didn't compile properly. + set = set.filter(function (s) { + return s.indexOf(false) === -1 + }) + + this.debug(this.pattern, set) + + this.set = set +} + +Minimatch.prototype.parseNegate = parseNegate +function parseNegate () { + var pattern = this.pattern + var negate = false + var options = this.options + var negateOffset = 0 + + if (options.nonegate) return + + for (var i = 0, l = pattern.length + ; i < l && pattern.charAt(i) === '!' + ; i++) { + negate = !negate + negateOffset++ + } + + if (negateOffset) this.pattern = pattern.substr(negateOffset) + this.negate = negate +} + +// Brace expansion: +// a{b,c}d -> abd acd +// a{b,}c -> abc ac +// a{0..3}d -> a0d a1d a2d a3d +// a{b,c{d,e}f}g -> abg acdfg acefg +// a{b,c}d{e,f}g -> abdeg acdeg abdeg abdfg +// +// Invalid sets are not expanded. +// a{2..}b -> a{2..}b +// a{b}c -> a{b}c +minimatch.braceExpand = function (pattern, options) { + return braceExpand(pattern, options) +} + +Minimatch.prototype.braceExpand = braceExpand + +function braceExpand (pattern, options) { + if (!options) { + if (this instanceof Minimatch) { + options = this.options + } else { + options = {} + } + } + + pattern = typeof pattern === 'undefined' + ? this.pattern : pattern + + if (typeof pattern === 'undefined') { + throw new TypeError('undefined pattern') + } + + if (options.nobrace || + !pattern.match(/\{.*\}/)) { + // shortcut. no need to expand. + return [pattern] + } + + return expand(pattern) +} + +// parse a component of the expanded set. +// At this point, no pattern may contain "/" in it +// so we're going to return a 2d array, where each entry is the full +// pattern, split on '/', and then turned into a regular expression. +// A regexp is made at the end which joins each array with an +// escaped /, and another full one which joins each regexp with |. +// +// Following the lead of Bash 4.1, note that "**" only has special meaning +// when it is the *only* thing in a path portion. Otherwise, any series +// of * is equivalent to a single *. Globstar behavior is enabled by +// default, and can be disabled by setting options.noglobstar. +Minimatch.prototype.parse = parse +var SUBPARSE = {} +function parse (pattern, isSub) { + if (pattern.length > 1024 * 64) { + throw new TypeError('pattern is too long') + } + + var options = this.options + + // shortcuts + if (!options.noglobstar && pattern === '**') return GLOBSTAR + if (pattern === '') return '' + + var re = '' + var hasMagic = !!options.nocase + var escaping = false + // ? => one single character + var patternListStack = [] + var negativeLists = [] + var stateChar + var inClass = false + var reClassStart = -1 + var classStart = -1 + // . and .. never match anything that doesn't start with ., + // even when options.dot is set. + var patternStart = pattern.charAt(0) === '.' ? '' // anything + // not (start or / followed by . or .. followed by / or end) + : options.dot ? '(?!(?:^|\\\/)\\.{1,2}(?:$|\\\/))' + : '(?!\\.)' + var self = this + + function clearStateChar () { + if (stateChar) { + // we had some state-tracking character + // that wasn't consumed by this pass. + switch (stateChar) { + case '*': + re += star + hasMagic = true + break + case '?': + re += qmark + hasMagic = true + break + default: + re += '\\' + stateChar + break + } + self.debug('clearStateChar %j %j', stateChar, re) + stateChar = false + } + } + + for (var i = 0, len = pattern.length, c + ; (i < len) && (c = pattern.charAt(i)) + ; i++) { + this.debug('%s\t%s %s %j', pattern, i, re, c) + + // skip over any that are escaped. + if (escaping && reSpecials[c]) { + re += '\\' + c + escaping = false + continue + } + + switch (c) { + case '/': + // completely not allowed, even escaped. + // Should already be path-split by now. + return false + + case '\\': + clearStateChar() + escaping = true + continue + + // the various stateChar values + // for the "extglob" stuff. + case '?': + case '*': + case '+': + case '@': + case '!': + this.debug('%s\t%s %s %j <-- stateChar', pattern, i, re, c) + + // all of those are literals inside a class, except that + // the glob [!a] means [^a] in regexp + if (inClass) { + this.debug(' in class') + if (c === '!' && i === classStart + 1) c = '^' + re += c + continue + } + + // if we already have a stateChar, then it means + // that there was something like ** or +? in there. + // Handle the stateChar, then proceed with this one. + self.debug('call clearStateChar %j', stateChar) + clearStateChar() + stateChar = c + // if extglob is disabled, then +(asdf|foo) isn't a thing. + // just clear the statechar *now*, rather than even diving into + // the patternList stuff. + if (options.noext) clearStateChar() + continue + + case '(': + if (inClass) { + re += '(' + continue + } + + if (!stateChar) { + re += '\\(' + continue + } + + patternListStack.push({ + type: stateChar, + start: i - 1, + reStart: re.length, + open: plTypes[stateChar].open, + close: plTypes[stateChar].close + }) + // negation is (?:(?!js)[^/]*) + re += stateChar === '!' ? '(?:(?!(?:' : '(?:' + this.debug('plType %j %j', stateChar, re) + stateChar = false + continue + + case ')': + if (inClass || !patternListStack.length) { + re += '\\)' + continue + } + + clearStateChar() + hasMagic = true + var pl = patternListStack.pop() + // negation is (?:(?!js)[^/]*) + // The others are (?:) + re += pl.close + if (pl.type === '!') { + negativeLists.push(pl) + } + pl.reEnd = re.length + continue + + case '|': + if (inClass || !patternListStack.length || escaping) { + re += '\\|' + escaping = false + continue + } + + clearStateChar() + re += '|' + continue + + // these are mostly the same in regexp and glob + case '[': + // swallow any state-tracking char before the [ + clearStateChar() + + if (inClass) { + re += '\\' + c + continue + } + + inClass = true + classStart = i + reClassStart = re.length + re += c + continue + + case ']': + // a right bracket shall lose its special + // meaning and represent itself in + // a bracket expression if it occurs + // first in the list. -- POSIX.2 2.8.3.2 + if (i === classStart + 1 || !inClass) { + re += '\\' + c + escaping = false + continue + } + + // handle the case where we left a class open. + // "[z-a]" is valid, equivalent to "\[z-a\]" + if (inClass) { + // split where the last [ was, make sure we don't have + // an invalid re. if so, re-walk the contents of the + // would-be class to re-translate any characters that + // were passed through as-is + // TODO: It would probably be faster to determine this + // without a try/catch and a new RegExp, but it's tricky + // to do safely. For now, this is safe and works. + var cs = pattern.substring(classStart + 1, i) + try { + RegExp('[' + cs + ']') + } catch (er) { + // not a valid class! + var sp = this.parse(cs, SUBPARSE) + re = re.substr(0, reClassStart) + '\\[' + sp[0] + '\\]' + hasMagic = hasMagic || sp[1] + inClass = false + continue + } + } + + // finish up the class. + hasMagic = true + inClass = false + re += c + continue + + default: + // swallow any state char that wasn't consumed + clearStateChar() + + if (escaping) { + // no need + escaping = false + } else if (reSpecials[c] + && !(c === '^' && inClass)) { + re += '\\' + } + + re += c + + } // switch + } // for + + // handle the case where we left a class open. + // "[abc" is valid, equivalent to "\[abc" + if (inClass) { + // split where the last [ was, and escape it + // this is a huge pita. We now have to re-walk + // the contents of the would-be class to re-translate + // any characters that were passed through as-is + cs = pattern.substr(classStart + 1) + sp = this.parse(cs, SUBPARSE) + re = re.substr(0, reClassStart) + '\\[' + sp[0] + hasMagic = hasMagic || sp[1] + } + + // handle the case where we had a +( thing at the *end* + // of the pattern. + // each pattern list stack adds 3 chars, and we need to go through + // and escape any | chars that were passed through as-is for the regexp. + // Go through and escape them, taking care not to double-escape any + // | chars that were already escaped. + for (pl = patternListStack.pop(); pl; pl = patternListStack.pop()) { + var tail = re.slice(pl.reStart + pl.open.length) + this.debug('setting tail', re, pl) + // maybe some even number of \, then maybe 1 \, followed by a | + tail = tail.replace(/((?:\\{2}){0,64})(\\?)\|/g, function (_, $1, $2) { + if (!$2) { + // the | isn't already escaped, so escape it. + $2 = '\\' + } + + // need to escape all those slashes *again*, without escaping the + // one that we need for escaping the | character. As it works out, + // escaping an even number of slashes can be done by simply repeating + // it exactly after itself. That's why this trick works. + // + // I am sorry that you have to see this. + return $1 + $1 + $2 + '|' + }) + + this.debug('tail=%j\n %s', tail, tail, pl, re) + var t = pl.type === '*' ? star + : pl.type === '?' ? qmark + : '\\' + pl.type + + hasMagic = true + re = re.slice(0, pl.reStart) + t + '\\(' + tail + } + + // handle trailing things that only matter at the very end. + clearStateChar() + if (escaping) { + // trailing \\ + re += '\\\\' + } + + // only need to apply the nodot start if the re starts with + // something that could conceivably capture a dot + var addPatternStart = false + switch (re.charAt(0)) { + case '.': + case '[': + case '(': addPatternStart = true + } + + // Hack to work around lack of negative lookbehind in JS + // A pattern like: *.!(x).!(y|z) needs to ensure that a name + // like 'a.xyz.yz' doesn't match. So, the first negative + // lookahead, has to look ALL the way ahead, to the end of + // the pattern. + for (var n = negativeLists.length - 1; n > -1; n--) { + var nl = negativeLists[n] + + var nlBefore = re.slice(0, nl.reStart) + var nlFirst = re.slice(nl.reStart, nl.reEnd - 8) + var nlLast = re.slice(nl.reEnd - 8, nl.reEnd) + var nlAfter = re.slice(nl.reEnd) + + nlLast += nlAfter + + // Handle nested stuff like *(*.js|!(*.json)), where open parens + // mean that we should *not* include the ) in the bit that is considered + // "after" the negated section. + var openParensBefore = nlBefore.split('(').length - 1 + var cleanAfter = nlAfter + for (i = 0; i < openParensBefore; i++) { + cleanAfter = cleanAfter.replace(/\)[+*?]?/, '') + } + nlAfter = cleanAfter + + var dollar = '' + if (nlAfter === '' && isSub !== SUBPARSE) { + dollar = '$' + } + var newRe = nlBefore + nlFirst + nlAfter + dollar + nlLast + re = newRe + } + + // if the re is not "" at this point, then we need to make sure + // it doesn't match against an empty path part. + // Otherwise a/* will match a/, which it should not. + if (re !== '' && hasMagic) { + re = '(?=.)' + re + } + + if (addPatternStart) { + re = patternStart + re + } + + // parsing just a piece of a larger pattern. + if (isSub === SUBPARSE) { + return [re, hasMagic] + } + + // skip the regexp for non-magical patterns + // unescape anything in it, though, so that it'll be + // an exact match against a file etc. + if (!hasMagic) { + return globUnescape(pattern) + } + + var flags = options.nocase ? 'i' : '' + try { + var regExp = new RegExp('^' + re + '$', flags) + } catch (er) { + // If it was an invalid regular expression, then it can't match + // anything. This trick looks for a character after the end of + // the string, which is of course impossible, except in multi-line + // mode, but it's not a /m regex. + return new RegExp('$.') + } + + regExp._glob = pattern + regExp._src = re + + return regExp +} + +minimatch.makeRe = function (pattern, options) { + return new Minimatch(pattern, options || {}).makeRe() +} + +Minimatch.prototype.makeRe = makeRe +function makeRe () { + if (this.regexp || this.regexp === false) return this.regexp + + // at this point, this.set is a 2d array of partial + // pattern strings, or "**". + // + // It's better to use .match(). This function shouldn't + // be used, really, but it's pretty convenient sometimes, + // when you just want to work with a regex. + var set = this.set + + if (!set.length) { + this.regexp = false + return this.regexp + } + var options = this.options + + var twoStar = options.noglobstar ? star + : options.dot ? twoStarDot + : twoStarNoDot + var flags = options.nocase ? 'i' : '' + + var re = set.map(function (pattern) { + return pattern.map(function (p) { + return (p === GLOBSTAR) ? twoStar + : (typeof p === 'string') ? regExpEscape(p) + : p._src + }).join('\\\/') + }).join('|') + + // must match entire pattern + // ending in a * or ** will make it less strict. + re = '^(?:' + re + ')$' + + // can match anything, as long as it's not this. + if (this.negate) re = '^(?!' + re + ').*$' + + try { + this.regexp = new RegExp(re, flags) + } catch (ex) { + this.regexp = false + } + return this.regexp +} + +minimatch.match = function (list, pattern, options) { + options = options || {} + var mm = new Minimatch(pattern, options) + list = list.filter(function (f) { + return mm.match(f) + }) + if (mm.options.nonull && !list.length) { + list.push(pattern) + } + return list +} + +Minimatch.prototype.match = match +function match (f, partial) { + this.debug('match', f, this.pattern) + // short-circuit in the case of busted things. + // comments, etc. + if (this.comment) return false + if (this.empty) return f === '' + + if (f === '/' && partial) return true + + var options = this.options + + // windows: need to use /, not \ + if (path.sep !== '/') { + f = f.split(path.sep).join('/') + } + + // treat the test path as a set of pathparts. + f = f.split(slashSplit) + this.debug(this.pattern, 'split', f) + + // just ONE of the pattern sets in this.set needs to match + // in order for it to be valid. If negating, then just one + // match means that we have failed. + // Either way, return on the first hit. + + var set = this.set + this.debug(this.pattern, 'set', set) + + // Find the basename of the path by looking for the last non-empty segment + var filename + var i + for (i = f.length - 1; i >= 0; i--) { + filename = f[i] + if (filename) break + } + + for (i = 0; i < set.length; i++) { + var pattern = set[i] + var file = f + if (options.matchBase && pattern.length === 1) { + file = [filename] + } + var hit = this.matchOne(file, pattern, partial) + if (hit) { + if (options.flipNegate) return true + return !this.negate + } + } + + // didn't get any hits. this is success if it's a negative + // pattern, failure otherwise. + if (options.flipNegate) return false + return this.negate +} + +// set partial to true to test if, for example, +// "/a/b" matches the start of "/*/b/*/d" +// Partial means, if you run out of file before you run +// out of pattern, then that's fine, as long as all +// the parts match. +Minimatch.prototype.matchOne = function (file, pattern, partial) { + var options = this.options + + this.debug('matchOne', + { 'this': this, file: file, pattern: pattern }) + + this.debug('matchOne', file.length, pattern.length) + + for (var fi = 0, + pi = 0, + fl = file.length, + pl = pattern.length + ; (fi < fl) && (pi < pl) + ; fi++, pi++) { + this.debug('matchOne loop') + var p = pattern[pi] + var f = file[fi] + + this.debug(pattern, p, f) + + // should be impossible. + // some invalid regexp stuff in the set. + if (p === false) return false + + if (p === GLOBSTAR) { + this.debug('GLOBSTAR', [pattern, p, f]) + + // "**" + // a/**/b/**/c would match the following: + // a/b/x/y/z/c + // a/x/y/z/b/c + // a/b/x/b/x/c + // a/b/c + // To do this, take the rest of the pattern after + // the **, and see if it would match the file remainder. + // If so, return success. + // If not, the ** "swallows" a segment, and try again. + // This is recursively awful. + // + // a/**/b/**/c matching a/b/x/y/z/c + // - a matches a + // - doublestar + // - matchOne(b/x/y/z/c, b/**/c) + // - b matches b + // - doublestar + // - matchOne(x/y/z/c, c) -> no + // - matchOne(y/z/c, c) -> no + // - matchOne(z/c, c) -> no + // - matchOne(c, c) yes, hit + var fr = fi + var pr = pi + 1 + if (pr === pl) { + this.debug('** at the end') + // a ** at the end will just swallow the rest. + // We have found a match. + // however, it will not swallow /.x, unless + // options.dot is set. + // . and .. are *never* matched by **, for explosively + // exponential reasons. + for (; fi < fl; fi++) { + if (file[fi] === '.' || file[fi] === '..' || + (!options.dot && file[fi].charAt(0) === '.')) return false + } + return true + } + + // ok, let's see if we can swallow whatever we can. + while (fr < fl) { + var swallowee = file[fr] + + this.debug('\nglobstar while', file, fr, pattern, pr, swallowee) + + // XXX remove this slice. Just pass the start index. + if (this.matchOne(file.slice(fr), pattern.slice(pr), partial)) { + this.debug('globstar found match!', fr, fl, swallowee) + // found a match. + return true + } else { + // can't swallow "." or ".." ever. + // can only swallow ".foo" when explicitly asked. + if (swallowee === '.' || swallowee === '..' || + (!options.dot && swallowee.charAt(0) === '.')) { + this.debug('dot detected!', file, fr, pattern, pr) + break + } + + // ** swallows a segment, and continue. + this.debug('globstar swallow a segment, and continue') + fr++ + } + } + + // no match was found. + // However, in partial mode, we can't say this is necessarily over. + // If there's more *pattern* left, then + if (partial) { + // ran out of file + this.debug('\n>>> no match, partial?', file, fr, pattern, pr) + if (fr === fl) return true + } + return false + } + + // something other than ** + // non-magic patterns just have to match exactly + // patterns with magic have been turned into regexps. + var hit + if (typeof p === 'string') { + if (options.nocase) { + hit = f.toLowerCase() === p.toLowerCase() + } else { + hit = f === p + } + this.debug('string match', p, f, hit) + } else { + hit = f.match(p) + this.debug('pattern match', p, f, hit) + } + + if (!hit) return false + } + + // Note: ending in / means that we'll get a final "" + // at the end of the pattern. This can only match a + // corresponding "" at the end of the file. + // If the file ends in /, then it can only match a + // a pattern that ends in /, unless the pattern just + // doesn't have any more for it. But, a/b/ should *not* + // match "a/b/*", even though "" matches against the + // [^/]*? pattern, except in partial mode, where it might + // simply not be reached yet. + // However, a/b/ should still satisfy a/* + + // now either we fell off the end of the pattern, or we're done. + if (fi === fl && pi === pl) { + // ran out of pattern and filename at the same time. + // an exact hit! + return true + } else if (fi === fl) { + // ran out of file, but still had pattern left. + // this is ok if we're doing the match as part of + // a glob fs traversal. + return partial + } else if (pi === pl) { + // ran out of pattern, still have file left. + // this is only acceptable if we're on the very last + // empty segment of a file with a trailing slash. + // a/* should match a/b/ + var emptyFileEnd = (fi === fl - 1) && (file[fi] === '') + return emptyFileEnd + } + + // should be unreachable. + throw new Error('wtf?') +} + +// replace stuff like \* with * +function globUnescape (s) { + return s.replace(/\\(.)/g, '$1') +} + +function regExpEscape (s) { + return s.replace(/[-[\]{}()*+?.,\\^$|#\s]/g, '\\$&') +} + +},{"brace-expansion":11,"path":22}],21:[function(require,module,exports){ +var wrappy = require('wrappy') +module.exports = wrappy(once) +module.exports.strict = wrappy(onceStrict) + +once.proto = once(function () { + Object.defineProperty(Function.prototype, 'once', { + value: function () { + return once(this) + }, + configurable: true + }) + + Object.defineProperty(Function.prototype, 'onceStrict', { + value: function () { + return onceStrict(this) + }, + configurable: true + }) +}) + +function once (fn) { + var f = function () { + if (f.called) return f.value + f.called = true + return f.value = fn.apply(this, arguments) + } + f.called = false + return f +} + +function onceStrict (fn) { + var f = function () { + if (f.called) + throw new Error(f.onceError) + f.called = true + return f.value = fn.apply(this, arguments) + } + var name = fn.name || 'Function wrapped with `once`' + f.onceError = name + " shouldn't be called more than once" + f.called = false + return f +} + +},{"wrappy":29}],22:[function(require,module,exports){ +(function (process){ +// Copyright Joyent, Inc. and other Node contributors. +// +// Permission is hereby granted, free of charge, to any person obtaining a +// copy of this software and associated documentation files (the +// "Software"), to deal in the Software without restriction, including +// without limitation the rights to use, copy, modify, merge, publish, +// distribute, sublicense, and/or sell copies of the Software, and to permit +// persons to whom the Software is furnished to do so, subject to the +// following conditions: +// +// The above copyright notice and this permission notice shall be included +// in all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS +// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF +// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN +// NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, +// DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR +// OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE +// USE OR OTHER DEALINGS IN THE SOFTWARE. + +// resolves . and .. elements in a path array with directory names there +// must be no slashes, empty elements, or device names (c:\) in the array +// (so also no leading and trailing slashes - it does not distinguish +// relative and absolute paths) +function normalizeArray(parts, allowAboveRoot) { + // if the path tries to go above the root, `up` ends up > 0 + var up = 0; + for (var i = parts.length - 1; i >= 0; i--) { + var last = parts[i]; + if (last === '.') { + parts.splice(i, 1); + } else if (last === '..') { + parts.splice(i, 1); + up++; + } else if (up) { + parts.splice(i, 1); + up--; + } + } + + // if the path is allowed to go above the root, restore leading ..s + if (allowAboveRoot) { + for (; up--; up) { + parts.unshift('..'); + } + } + + return parts; +} + +// Split a filename into [root, dir, basename, ext], unix version +// 'root' is just a slash, or nothing. +var splitPathRe = + /^(\/?|)([\s\S]*?)((?:\.{1,2}|[^\/]+?|)(\.[^.\/]*|))(?:[\/]*)$/; +var splitPath = function(filename) { + return splitPathRe.exec(filename).slice(1); +}; + +// path.resolve([from ...], to) +// posix version +exports.resolve = function() { + var resolvedPath = '', + resolvedAbsolute = false; + + for (var i = arguments.length - 1; i >= -1 && !resolvedAbsolute; i--) { + var path = (i >= 0) ? arguments[i] : process.cwd(); + + // Skip empty and invalid entries + if (typeof path !== 'string') { + throw new TypeError('Arguments to path.resolve must be strings'); + } else if (!path) { + continue; + } + + resolvedPath = path + '/' + resolvedPath; + resolvedAbsolute = path.charAt(0) === '/'; + } + + // At this point the path should be resolved to a full absolute path, but + // handle relative paths to be safe (might happen when process.cwd() fails) + + // Normalize the path + resolvedPath = normalizeArray(filter(resolvedPath.split('/'), function(p) { + return !!p; + }), !resolvedAbsolute).join('/'); + + return ((resolvedAbsolute ? '/' : '') + resolvedPath) || '.'; +}; + +// path.normalize(path) +// posix version +exports.normalize = function(path) { + var isAbsolute = exports.isAbsolute(path), + trailingSlash = substr(path, -1) === '/'; + + // Normalize the path + path = normalizeArray(filter(path.split('/'), function(p) { + return !!p; + }), !isAbsolute).join('/'); + + if (!path && !isAbsolute) { + path = '.'; + } + if (path && trailingSlash) { + path += '/'; + } + + return (isAbsolute ? '/' : '') + path; +}; + +// posix version +exports.isAbsolute = function(path) { + return path.charAt(0) === '/'; +}; + +// posix version +exports.join = function() { + var paths = Array.prototype.slice.call(arguments, 0); + return exports.normalize(filter(paths, function(p, index) { + if (typeof p !== 'string') { + throw new TypeError('Arguments to path.join must be strings'); + } + return p; + }).join('/')); +}; + + +// path.relative(from, to) +// posix version +exports.relative = function(from, to) { + from = exports.resolve(from).substr(1); + to = exports.resolve(to).substr(1); + + function trim(arr) { + var start = 0; + for (; start < arr.length; start++) { + if (arr[start] !== '') break; + } + + var end = arr.length - 1; + for (; end >= 0; end--) { + if (arr[end] !== '') break; + } + + if (start > end) return []; + return arr.slice(start, end - start + 1); + } + + var fromParts = trim(from.split('/')); + var toParts = trim(to.split('/')); + + var length = Math.min(fromParts.length, toParts.length); + var samePartsLength = length; + for (var i = 0; i < length; i++) { + if (fromParts[i] !== toParts[i]) { + samePartsLength = i; + break; + } + } + + var outputParts = []; + for (var i = samePartsLength; i < fromParts.length; i++) { + outputParts.push('..'); + } + + outputParts = outputParts.concat(toParts.slice(samePartsLength)); + + return outputParts.join('/'); +}; + +exports.sep = '/'; +exports.delimiter = ':'; + +exports.dirname = function(path) { + var result = splitPath(path), + root = result[0], + dir = result[1]; + + if (!root && !dir) { + // No dirname whatsoever + return '.'; + } + + if (dir) { + // It has a dirname, strip trailing slash + dir = dir.substr(0, dir.length - 1); + } + + return root + dir; +}; + + +exports.basename = function(path, ext) { + var f = splitPath(path)[2]; + // TODO: make this comparison case-insensitive on windows? + if (ext && f.substr(-1 * ext.length) === ext) { + f = f.substr(0, f.length - ext.length); + } + return f; +}; + + +exports.extname = function(path) { + return splitPath(path)[3]; +}; + +function filter (xs, f) { + if (xs.filter) return xs.filter(f); + var res = []; + for (var i = 0; i < xs.length; i++) { + if (f(xs[i], i, xs)) res.push(xs[i]); + } + return res; +} + +// String.prototype.substr - negative index don't work in IE8 +var substr = 'ab'.substr(-1) === 'b' + ? function (str, start, len) { return str.substr(start, len) } + : function (str, start, len) { + if (start < 0) start = str.length + start; + return str.substr(start, len); + } +; + +}).call(this,require('_process')) +},{"_process":24}],23:[function(require,module,exports){ +(function (process){ +'use strict'; + +function posix(path) { + return path.charAt(0) === '/'; +} + +function win32(path) { + // https://github.com/nodejs/node/blob/b3fcc245fb25539909ef1d5eaa01dbf92e168633/lib/path.js#L56 + var splitDeviceRe = /^([a-zA-Z]:|[\\\/]{2}[^\\\/]+[\\\/]+[^\\\/]+)?([\\\/])?([\s\S]*?)$/; + var result = splitDeviceRe.exec(path); + var device = result[1] || ''; + var isUnc = Boolean(device && device.charAt(1) !== ':'); + + // UNC paths are always absolute + return Boolean(result[2] || isUnc); +} + +module.exports = process.platform === 'win32' ? win32 : posix; +module.exports.posix = posix; +module.exports.win32 = win32; + +}).call(this,require('_process')) +},{"_process":24}],24:[function(require,module,exports){ +// shim for using process in browser +var process = module.exports = {}; + +// cached from whatever global is present so that test runners that stub it +// don't break things. But we need to wrap it in a try catch in case it is +// wrapped in strict mode code which doesn't define any globals. It's inside a +// function because try/catches deoptimize in certain engines. + +var cachedSetTimeout; +var cachedClearTimeout; + +function defaultSetTimout() { + throw new Error('setTimeout has not been defined'); +} +function defaultClearTimeout () { + throw new Error('clearTimeout has not been defined'); +} +(function () { + try { + if (typeof setTimeout === 'function') { + cachedSetTimeout = setTimeout; + } else { + cachedSetTimeout = defaultSetTimout; + } + } catch (e) { + cachedSetTimeout = defaultSetTimout; + } + try { + if (typeof clearTimeout === 'function') { + cachedClearTimeout = clearTimeout; + } else { + cachedClearTimeout = defaultClearTimeout; + } + } catch (e) { + cachedClearTimeout = defaultClearTimeout; + } +} ()) +function runTimeout(fun) { + if (cachedSetTimeout === setTimeout) { + //normal enviroments in sane situations + return setTimeout(fun, 0); + } + // if setTimeout wasn't available but was latter defined + if ((cachedSetTimeout === defaultSetTimout || !cachedSetTimeout) && setTimeout) { + cachedSetTimeout = setTimeout; + return setTimeout(fun, 0); + } + try { + // when when somebody has screwed with setTimeout but no I.E. maddness + return cachedSetTimeout(fun, 0); + } catch(e){ + try { + // When we are in I.E. but the script has been evaled so I.E. doesn't trust the global object when called normally + return cachedSetTimeout.call(null, fun, 0); + } catch(e){ + // same as above but when it's a version of I.E. that must have the global object for 'this', hopfully our context correct otherwise it will throw a global error + return cachedSetTimeout.call(this, fun, 0); + } + } + + +} +function runClearTimeout(marker) { + if (cachedClearTimeout === clearTimeout) { + //normal enviroments in sane situations + return clearTimeout(marker); + } + // if clearTimeout wasn't available but was latter defined + if ((cachedClearTimeout === defaultClearTimeout || !cachedClearTimeout) && clearTimeout) { + cachedClearTimeout = clearTimeout; + return clearTimeout(marker); + } + try { + // when when somebody has screwed with setTimeout but no I.E. maddness + return cachedClearTimeout(marker); + } catch (e){ + try { + // When we are in I.E. but the script has been evaled so I.E. doesn't trust the global object when called normally + return cachedClearTimeout.call(null, marker); + } catch (e){ + // same as above but when it's a version of I.E. that must have the global object for 'this', hopfully our context correct otherwise it will throw a global error. + // Some versions of I.E. have different rules for clearTimeout vs setTimeout + return cachedClearTimeout.call(this, marker); + } + } + + + +} +var queue = []; +var draining = false; +var currentQueue; +var queueIndex = -1; + +function cleanUpNextTick() { + if (!draining || !currentQueue) { + return; + } + draining = false; + if (currentQueue.length) { + queue = currentQueue.concat(queue); + } else { + queueIndex = -1; + } + if (queue.length) { + drainQueue(); + } +} + +function drainQueue() { + if (draining) { + return; + } + var timeout = runTimeout(cleanUpNextTick); + draining = true; + + var len = queue.length; + while(len) { + currentQueue = queue; + queue = []; + while (++queueIndex < len) { + if (currentQueue) { + currentQueue[queueIndex].run(); + } + } + queueIndex = -1; + len = queue.length; + } + currentQueue = null; + draining = false; + runClearTimeout(timeout); +} + +process.nextTick = function (fun) { + var args = new Array(arguments.length - 1); + if (arguments.length > 1) { + for (var i = 1; i < arguments.length; i++) { + args[i - 1] = arguments[i]; + } + } + queue.push(new Item(fun, args)); + if (queue.length === 1 && !draining) { + runTimeout(drainQueue); + } +}; + +// v8 likes predictible objects +function Item(fun, array) { + this.fun = fun; + this.array = array; +} +Item.prototype.run = function () { + this.fun.apply(null, this.array); +}; +process.title = 'browser'; +process.browser = true; +process.env = {}; +process.argv = []; +process.version = ''; // empty string to avoid regexp issues +process.versions = {}; + +function noop() {} + +process.on = noop; +process.addListener = noop; +process.once = noop; +process.off = noop; +process.removeListener = noop; +process.removeAllListeners = noop; +process.emit = noop; +process.prependListener = noop; +process.prependOnceListener = noop; + +process.listeners = function (name) { return [] } + +process.binding = function (name) { + throw new Error('process.binding is not supported'); +}; + +process.cwd = function () { return '/' }; +process.chdir = function (dir) { + throw new Error('process.chdir is not supported'); +}; +process.umask = function() { return 0; }; + +},{}],25:[function(require,module,exports){ +// Underscore.js 1.8.3 +// http://underscorejs.org +// (c) 2009-2015 Jeremy Ashkenas, DocumentCloud and Investigative Reporters & Editors +// Underscore may be freely distributed under the MIT license. + +(function() { + + // Baseline setup + // -------------- + + // Establish the root object, `window` in the browser, or `exports` on the server. + var root = this; + + // Save the previous value of the `_` variable. + var previousUnderscore = root._; + + // Save bytes in the minified (but not gzipped) version: + var ArrayProto = Array.prototype, ObjProto = Object.prototype, FuncProto = Function.prototype; + + // Create quick reference variables for speed access to core prototypes. + var + push = ArrayProto.push, + slice = ArrayProto.slice, + toString = ObjProto.toString, + hasOwnProperty = ObjProto.hasOwnProperty; + + // All **ECMAScript 5** native function implementations that we hope to use + // are declared here. + var + nativeIsArray = Array.isArray, + nativeKeys = Object.keys, + nativeBind = FuncProto.bind, + nativeCreate = Object.create; + + // Naked function reference for surrogate-prototype-swapping. + var Ctor = function(){}; + + // Create a safe reference to the Underscore object for use below. + var _ = function(obj) { + if (obj instanceof _) return obj; + if (!(this instanceof _)) return new _(obj); + this._wrapped = obj; + }; + + // Export the Underscore object for **Node.js**, with + // backwards-compatibility for the old `require()` API. If we're in + // the browser, add `_` as a global object. + if (typeof exports !== 'undefined') { + if (typeof module !== 'undefined' && module.exports) { + exports = module.exports = _; + } + exports._ = _; + } else { + root._ = _; + } + + // Current version. + _.VERSION = '1.8.3'; + + // Internal function that returns an efficient (for current engines) version + // of the passed-in callback, to be repeatedly applied in other Underscore + // functions. + var optimizeCb = function(func, context, argCount) { + if (context === void 0) return func; + switch (argCount == null ? 3 : argCount) { + case 1: return function(value) { + return func.call(context, value); + }; + case 2: return function(value, other) { + return func.call(context, value, other); + }; + case 3: return function(value, index, collection) { + return func.call(context, value, index, collection); + }; + case 4: return function(accumulator, value, index, collection) { + return func.call(context, accumulator, value, index, collection); + }; + } + return function() { + return func.apply(context, arguments); + }; + }; + + // A mostly-internal function to generate callbacks that can be applied + // to each element in a collection, returning the desired result — either + // identity, an arbitrary callback, a property matcher, or a property accessor. + var cb = function(value, context, argCount) { + if (value == null) return _.identity; + if (_.isFunction(value)) return optimizeCb(value, context, argCount); + if (_.isObject(value)) return _.matcher(value); + return _.property(value); + }; + _.iteratee = function(value, context) { + return cb(value, context, Infinity); + }; + + // An internal function for creating assigner functions. + var createAssigner = function(keysFunc, undefinedOnly) { + return function(obj) { + var length = arguments.length; + if (length < 2 || obj == null) return obj; + for (var index = 1; index < length; index++) { + var source = arguments[index], + keys = keysFunc(source), + l = keys.length; + for (var i = 0; i < l; i++) { + var key = keys[i]; + if (!undefinedOnly || obj[key] === void 0) obj[key] = source[key]; + } + } + return obj; + }; + }; + + // An internal function for creating a new object that inherits from another. + var baseCreate = function(prototype) { + if (!_.isObject(prototype)) return {}; + if (nativeCreate) return nativeCreate(prototype); + Ctor.prototype = prototype; + var result = new Ctor; + Ctor.prototype = null; + return result; + }; + + var property = function(key) { + return function(obj) { + return obj == null ? void 0 : obj[key]; + }; + }; + + // Helper for collection methods to determine whether a collection + // should be iterated as an array or as an object + // Related: http://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength + // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094 + var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1; + var getLength = property('length'); + var isArrayLike = function(collection) { + var length = getLength(collection); + return typeof length == 'number' && length >= 0 && length <= MAX_ARRAY_INDEX; + }; + + // Collection Functions + // -------------------- + + // The cornerstone, an `each` implementation, aka `forEach`. + // Handles raw objects in addition to array-likes. Treats all + // sparse array-likes as if they were dense. + _.each = _.forEach = function(obj, iteratee, context) { + iteratee = optimizeCb(iteratee, context); + var i, length; + if (isArrayLike(obj)) { + for (i = 0, length = obj.length; i < length; i++) { + iteratee(obj[i], i, obj); + } + } else { + var keys = _.keys(obj); + for (i = 0, length = keys.length; i < length; i++) { + iteratee(obj[keys[i]], keys[i], obj); + } + } + return obj; + }; + + // Return the results of applying the iteratee to each element. + _.map = _.collect = function(obj, iteratee, context) { + iteratee = cb(iteratee, context); + var keys = !isArrayLike(obj) && _.keys(obj), + length = (keys || obj).length, + results = Array(length); + for (var index = 0; index < length; index++) { + var currentKey = keys ? keys[index] : index; + results[index] = iteratee(obj[currentKey], currentKey, obj); + } + return results; + }; + + // Create a reducing function iterating left or right. + function createReduce(dir) { + // Optimized iterator function as using arguments.length + // in the main function will deoptimize the, see #1991. + function iterator(obj, iteratee, memo, keys, index, length) { + for (; index >= 0 && index < length; index += dir) { + var currentKey = keys ? keys[index] : index; + memo = iteratee(memo, obj[currentKey], currentKey, obj); + } + return memo; + } + + return function(obj, iteratee, memo, context) { + iteratee = optimizeCb(iteratee, context, 4); + var keys = !isArrayLike(obj) && _.keys(obj), + length = (keys || obj).length, + index = dir > 0 ? 0 : length - 1; + // Determine the initial value if none is provided. + if (arguments.length < 3) { + memo = obj[keys ? keys[index] : index]; + index += dir; + } + return iterator(obj, iteratee, memo, keys, index, length); + }; + } + + // **Reduce** builds up a single result from a list of values, aka `inject`, + // or `foldl`. + _.reduce = _.foldl = _.inject = createReduce(1); + + // The right-associative version of reduce, also known as `foldr`. + _.reduceRight = _.foldr = createReduce(-1); + + // Return the first value which passes a truth test. Aliased as `detect`. + _.find = _.detect = function(obj, predicate, context) { + var key; + if (isArrayLike(obj)) { + key = _.findIndex(obj, predicate, context); + } else { + key = _.findKey(obj, predicate, context); + } + if (key !== void 0 && key !== -1) return obj[key]; + }; + + // Return all the elements that pass a truth test. + // Aliased as `select`. + _.filter = _.select = function(obj, predicate, context) { + var results = []; + predicate = cb(predicate, context); + _.each(obj, function(value, index, list) { + if (predicate(value, index, list)) results.push(value); + }); + return results; + }; + + // Return all the elements for which a truth test fails. + _.reject = function(obj, predicate, context) { + return _.filter(obj, _.negate(cb(predicate)), context); + }; + + // Determine whether all of the elements match a truth test. + // Aliased as `all`. + _.every = _.all = function(obj, predicate, context) { + predicate = cb(predicate, context); + var keys = !isArrayLike(obj) && _.keys(obj), + length = (keys || obj).length; + for (var index = 0; index < length; index++) { + var currentKey = keys ? keys[index] : index; + if (!predicate(obj[currentKey], currentKey, obj)) return false; + } + return true; + }; + + // Determine if at least one element in the object matches a truth test. + // Aliased as `any`. + _.some = _.any = function(obj, predicate, context) { + predicate = cb(predicate, context); + var keys = !isArrayLike(obj) && _.keys(obj), + length = (keys || obj).length; + for (var index = 0; index < length; index++) { + var currentKey = keys ? keys[index] : index; + if (predicate(obj[currentKey], currentKey, obj)) return true; + } + return false; + }; + + // Determine if the array or object contains a given item (using `===`). + // Aliased as `includes` and `include`. + _.contains = _.includes = _.include = function(obj, item, fromIndex, guard) { + if (!isArrayLike(obj)) obj = _.values(obj); + if (typeof fromIndex != 'number' || guard) fromIndex = 0; + return _.indexOf(obj, item, fromIndex) >= 0; + }; + + // Invoke a method (with arguments) on every item in a collection. + _.invoke = function(obj, method) { + var args = slice.call(arguments, 2); + var isFunc = _.isFunction(method); + return _.map(obj, function(value) { + var func = isFunc ? method : value[method]; + return func == null ? func : func.apply(value, args); + }); + }; + + // Convenience version of a common use case of `map`: fetching a property. + _.pluck = function(obj, key) { + return _.map(obj, _.property(key)); + }; + + // Convenience version of a common use case of `filter`: selecting only objects + // containing specific `key:value` pairs. + _.where = function(obj, attrs) { + return _.filter(obj, _.matcher(attrs)); + }; + + // Convenience version of a common use case of `find`: getting the first object + // containing specific `key:value` pairs. + _.findWhere = function(obj, attrs) { + return _.find(obj, _.matcher(attrs)); + }; + + // Return the maximum element (or element-based computation). + _.max = function(obj, iteratee, context) { + var result = -Infinity, lastComputed = -Infinity, + value, computed; + if (iteratee == null && obj != null) { + obj = isArrayLike(obj) ? obj : _.values(obj); + for (var i = 0, length = obj.length; i < length; i++) { + value = obj[i]; + if (value > result) { + result = value; + } + } + } else { + iteratee = cb(iteratee, context); + _.each(obj, function(value, index, list) { + computed = iteratee(value, index, list); + if (computed > lastComputed || computed === -Infinity && result === -Infinity) { + result = value; + lastComputed = computed; + } + }); + } + return result; + }; + + // Return the minimum element (or element-based computation). + _.min = function(obj, iteratee, context) { + var result = Infinity, lastComputed = Infinity, + value, computed; + if (iteratee == null && obj != null) { + obj = isArrayLike(obj) ? obj : _.values(obj); + for (var i = 0, length = obj.length; i < length; i++) { + value = obj[i]; + if (value < result) { + result = value; + } + } + } else { + iteratee = cb(iteratee, context); + _.each(obj, function(value, index, list) { + computed = iteratee(value, index, list); + if (computed < lastComputed || computed === Infinity && result === Infinity) { + result = value; + lastComputed = computed; + } + }); + } + return result; + }; + + // Shuffle a collection, using the modern version of the + // [Fisher-Yates shuffle](http://en.wikipedia.org/wiki/Fisher–Yates_shuffle). + _.shuffle = function(obj) { + var set = isArrayLike(obj) ? obj : _.values(obj); + var length = set.length; + var shuffled = Array(length); + for (var index = 0, rand; index < length; index++) { + rand = _.random(0, index); + if (rand !== index) shuffled[index] = shuffled[rand]; + shuffled[rand] = set[index]; + } + return shuffled; + }; + + // Sample **n** random values from a collection. + // If **n** is not specified, returns a single random element. + // The internal `guard` argument allows it to work with `map`. + _.sample = function(obj, n, guard) { + if (n == null || guard) { + if (!isArrayLike(obj)) obj = _.values(obj); + return obj[_.random(obj.length - 1)]; + } + return _.shuffle(obj).slice(0, Math.max(0, n)); + }; + + // Sort the object's values by a criterion produced by an iteratee. + _.sortBy = function(obj, iteratee, context) { + iteratee = cb(iteratee, context); + return _.pluck(_.map(obj, function(value, index, list) { + return { + value: value, + index: index, + criteria: iteratee(value, index, list) + }; + }).sort(function(left, right) { + var a = left.criteria; + var b = right.criteria; + if (a !== b) { + if (a > b || a === void 0) return 1; + if (a < b || b === void 0) return -1; + } + return left.index - right.index; + }), 'value'); + }; + + // An internal function used for aggregate "group by" operations. + var group = function(behavior) { + return function(obj, iteratee, context) { + var result = {}; + iteratee = cb(iteratee, context); + _.each(obj, function(value, index) { + var key = iteratee(value, index, obj); + behavior(result, value, key); + }); + return result; + }; + }; + + // Groups the object's values by a criterion. Pass either a string attribute + // to group by, or a function that returns the criterion. + _.groupBy = group(function(result, value, key) { + if (_.has(result, key)) result[key].push(value); else result[key] = [value]; + }); + + // Indexes the object's values by a criterion, similar to `groupBy`, but for + // when you know that your index values will be unique. + _.indexBy = group(function(result, value, key) { + result[key] = value; + }); + + // Counts instances of an object that group by a certain criterion. Pass + // either a string attribute to count by, or a function that returns the + // criterion. + _.countBy = group(function(result, value, key) { + if (_.has(result, key)) result[key]++; else result[key] = 1; + }); + + // Safely create a real, live array from anything iterable. + _.toArray = function(obj) { + if (!obj) return []; + if (_.isArray(obj)) return slice.call(obj); + if (isArrayLike(obj)) return _.map(obj, _.identity); + return _.values(obj); + }; + + // Return the number of elements in an object. + _.size = function(obj) { + if (obj == null) return 0; + return isArrayLike(obj) ? obj.length : _.keys(obj).length; + }; + + // Split a collection into two arrays: one whose elements all satisfy the given + // predicate, and one whose elements all do not satisfy the predicate. + _.partition = function(obj, predicate, context) { + predicate = cb(predicate, context); + var pass = [], fail = []; + _.each(obj, function(value, key, obj) { + (predicate(value, key, obj) ? pass : fail).push(value); + }); + return [pass, fail]; + }; + + // Array Functions + // --------------- + + // Get the first element of an array. Passing **n** will return the first N + // values in the array. Aliased as `head` and `take`. The **guard** check + // allows it to work with `_.map`. + _.first = _.head = _.take = function(array, n, guard) { + if (array == null) return void 0; + if (n == null || guard) return array[0]; + return _.initial(array, array.length - n); + }; + + // Returns everything but the last entry of the array. Especially useful on + // the arguments object. Passing **n** will return all the values in + // the array, excluding the last N. + _.initial = function(array, n, guard) { + return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n))); + }; + + // Get the last element of an array. Passing **n** will return the last N + // values in the array. + _.last = function(array, n, guard) { + if (array == null) return void 0; + if (n == null || guard) return array[array.length - 1]; + return _.rest(array, Math.max(0, array.length - n)); + }; + + // Returns everything but the first entry of the array. Aliased as `tail` and `drop`. + // Especially useful on the arguments object. Passing an **n** will return + // the rest N values in the array. + _.rest = _.tail = _.drop = function(array, n, guard) { + return slice.call(array, n == null || guard ? 1 : n); + }; + + // Trim out all falsy values from an array. + _.compact = function(array) { + return _.filter(array, _.identity); + }; + + // Internal implementation of a recursive `flatten` function. + var flatten = function(input, shallow, strict, startIndex) { + var output = [], idx = 0; + for (var i = startIndex || 0, length = getLength(input); i < length; i++) { + var value = input[i]; + if (isArrayLike(value) && (_.isArray(value) || _.isArguments(value))) { + //flatten current level of array or arguments object + if (!shallow) value = flatten(value, shallow, strict); + var j = 0, len = value.length; + output.length += len; + while (j < len) { + output[idx++] = value[j++]; + } + } else if (!strict) { + output[idx++] = value; + } + } + return output; + }; + + // Flatten out an array, either recursively (by default), or just one level. + _.flatten = function(array, shallow) { + return flatten(array, shallow, false); + }; + + // Return a version of the array that does not contain the specified value(s). + _.without = function(array) { + return _.difference(array, slice.call(arguments, 1)); + }; + + // Produce a duplicate-free version of the array. If the array has already + // been sorted, you have the option of using a faster algorithm. + // Aliased as `unique`. + _.uniq = _.unique = function(array, isSorted, iteratee, context) { + if (!_.isBoolean(isSorted)) { + context = iteratee; + iteratee = isSorted; + isSorted = false; + } + if (iteratee != null) iteratee = cb(iteratee, context); + var result = []; + var seen = []; + for (var i = 0, length = getLength(array); i < length; i++) { + var value = array[i], + computed = iteratee ? iteratee(value, i, array) : value; + if (isSorted) { + if (!i || seen !== computed) result.push(value); + seen = computed; + } else if (iteratee) { + if (!_.contains(seen, computed)) { + seen.push(computed); + result.push(value); + } + } else if (!_.contains(result, value)) { + result.push(value); + } + } + return result; + }; + + // Produce an array that contains the union: each distinct element from all of + // the passed-in arrays. + _.union = function() { + return _.uniq(flatten(arguments, true, true)); + }; + + // Produce an array that contains every item shared between all the + // passed-in arrays. + _.intersection = function(array) { + var result = []; + var argsLength = arguments.length; + for (var i = 0, length = getLength(array); i < length; i++) { + var item = array[i]; + if (_.contains(result, item)) continue; + for (var j = 1; j < argsLength; j++) { + if (!_.contains(arguments[j], item)) break; + } + if (j === argsLength) result.push(item); + } + return result; + }; + + // Take the difference between one array and a number of other arrays. + // Only the elements present in just the first array will remain. + _.difference = function(array) { + var rest = flatten(arguments, true, true, 1); + return _.filter(array, function(value){ + return !_.contains(rest, value); + }); + }; + + // Zip together multiple lists into a single array -- elements that share + // an index go together. + _.zip = function() { + return _.unzip(arguments); + }; + + // Complement of _.zip. Unzip accepts an array of arrays and groups + // each array's elements on shared indices + _.unzip = function(array) { + var length = array && _.max(array, getLength).length || 0; + var result = Array(length); + + for (var index = 0; index < length; index++) { + result[index] = _.pluck(array, index); + } + return result; + }; + + // Converts lists into objects. Pass either a single array of `[key, value]` + // pairs, or two parallel arrays of the same length -- one of keys, and one of + // the corresponding values. + _.object = function(list, values) { + var result = {}; + for (var i = 0, length = getLength(list); i < length; i++) { + if (values) { + result[list[i]] = values[i]; + } else { + result[list[i][0]] = list[i][1]; + } + } + return result; + }; + + // Generator function to create the findIndex and findLastIndex functions + function createPredicateIndexFinder(dir) { + return function(array, predicate, context) { + predicate = cb(predicate, context); + var length = getLength(array); + var index = dir > 0 ? 0 : length - 1; + for (; index >= 0 && index < length; index += dir) { + if (predicate(array[index], index, array)) return index; + } + return -1; + }; + } + + // Returns the first index on an array-like that passes a predicate test + _.findIndex = createPredicateIndexFinder(1); + _.findLastIndex = createPredicateIndexFinder(-1); + + // Use a comparator function to figure out the smallest index at which + // an object should be inserted so as to maintain order. Uses binary search. + _.sortedIndex = function(array, obj, iteratee, context) { + iteratee = cb(iteratee, context, 1); + var value = iteratee(obj); + var low = 0, high = getLength(array); + while (low < high) { + var mid = Math.floor((low + high) / 2); + if (iteratee(array[mid]) < value) low = mid + 1; else high = mid; + } + return low; + }; + + // Generator function to create the indexOf and lastIndexOf functions + function createIndexFinder(dir, predicateFind, sortedIndex) { + return function(array, item, idx) { + var i = 0, length = getLength(array); + if (typeof idx == 'number') { + if (dir > 0) { + i = idx >= 0 ? idx : Math.max(idx + length, i); + } else { + length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1; + } + } else if (sortedIndex && idx && length) { + idx = sortedIndex(array, item); + return array[idx] === item ? idx : -1; + } + if (item !== item) { + idx = predicateFind(slice.call(array, i, length), _.isNaN); + return idx >= 0 ? idx + i : -1; + } + for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) { + if (array[idx] === item) return idx; + } + return -1; + }; + } + + // Return the position of the first occurrence of an item in an array, + // or -1 if the item is not included in the array. + // If the array is large and already in sort order, pass `true` + // for **isSorted** to use binary search. + _.indexOf = createIndexFinder(1, _.findIndex, _.sortedIndex); + _.lastIndexOf = createIndexFinder(-1, _.findLastIndex); + + // Generate an integer Array containing an arithmetic progression. A port of + // the native Python `range()` function. See + // [the Python documentation](http://docs.python.org/library/functions.html#range). + _.range = function(start, stop, step) { + if (stop == null) { + stop = start || 0; + start = 0; + } + step = step || 1; + + var length = Math.max(Math.ceil((stop - start) / step), 0); + var range = Array(length); + + for (var idx = 0; idx < length; idx++, start += step) { + range[idx] = start; + } + + return range; + }; + + // Function (ahem) Functions + // ------------------ + + // Determines whether to execute a function as a constructor + // or a normal function with the provided arguments + var executeBound = function(sourceFunc, boundFunc, context, callingContext, args) { + if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args); + var self = baseCreate(sourceFunc.prototype); + var result = sourceFunc.apply(self, args); + if (_.isObject(result)) return result; + return self; + }; + + // Create a function bound to a given object (assigning `this`, and arguments, + // optionally). Delegates to **ECMAScript 5**'s native `Function.bind` if + // available. + _.bind = function(func, context) { + if (nativeBind && func.bind === nativeBind) return nativeBind.apply(func, slice.call(arguments, 1)); + if (!_.isFunction(func)) throw new TypeError('Bind must be called on a function'); + var args = slice.call(arguments, 2); + var bound = function() { + return executeBound(func, bound, context, this, args.concat(slice.call(arguments))); + }; + return bound; + }; + + // Partially apply a function by creating a version that has had some of its + // arguments pre-filled, without changing its dynamic `this` context. _ acts + // as a placeholder, allowing any combination of arguments to be pre-filled. + _.partial = function(func) { + var boundArgs = slice.call(arguments, 1); + var bound = function() { + var position = 0, length = boundArgs.length; + var args = Array(length); + for (var i = 0; i < length; i++) { + args[i] = boundArgs[i] === _ ? arguments[position++] : boundArgs[i]; + } + while (position < arguments.length) args.push(arguments[position++]); + return executeBound(func, bound, this, this, args); + }; + return bound; + }; + + // Bind a number of an object's methods to that object. Remaining arguments + // are the method names to be bound. Useful for ensuring that all callbacks + // defined on an object belong to it. + _.bindAll = function(obj) { + var i, length = arguments.length, key; + if (length <= 1) throw new Error('bindAll must be passed function names'); + for (i = 1; i < length; i++) { + key = arguments[i]; + obj[key] = _.bind(obj[key], obj); + } + return obj; + }; + + // Memoize an expensive function by storing its results. + _.memoize = function(func, hasher) { + var memoize = function(key) { + var cache = memoize.cache; + var address = '' + (hasher ? hasher.apply(this, arguments) : key); + if (!_.has(cache, address)) cache[address] = func.apply(this, arguments); + return cache[address]; + }; + memoize.cache = {}; + return memoize; + }; + + // Delays a function for the given number of milliseconds, and then calls + // it with the arguments supplied. + _.delay = function(func, wait) { + var args = slice.call(arguments, 2); + return setTimeout(function(){ + return func.apply(null, args); + }, wait); + }; + + // Defers a function, scheduling it to run after the current call stack has + // cleared. + _.defer = _.partial(_.delay, _, 1); + + // Returns a function, that, when invoked, will only be triggered at most once + // during a given window of time. Normally, the throttled function will run + // as much as it can, without ever going more than once per `wait` duration; + // but if you'd like to disable the execution on the leading edge, pass + // `{leading: false}`. To disable execution on the trailing edge, ditto. + _.throttle = function(func, wait, options) { + var context, args, result; + var timeout = null; + var previous = 0; + if (!options) options = {}; + var later = function() { + previous = options.leading === false ? 0 : _.now(); + timeout = null; + result = func.apply(context, args); + if (!timeout) context = args = null; + }; + return function() { + var now = _.now(); + if (!previous && options.leading === false) previous = now; + var remaining = wait - (now - previous); + context = this; + args = arguments; + if (remaining <= 0 || remaining > wait) { + if (timeout) { + clearTimeout(timeout); + timeout = null; + } + previous = now; + result = func.apply(context, args); + if (!timeout) context = args = null; + } else if (!timeout && options.trailing !== false) { + timeout = setTimeout(later, remaining); + } + return result; + }; + }; + + // Returns a function, that, as long as it continues to be invoked, will not + // be triggered. The function will be called after it stops being called for + // N milliseconds. If `immediate` is passed, trigger the function on the + // leading edge, instead of the trailing. + _.debounce = function(func, wait, immediate) { + var timeout, args, context, timestamp, result; + + var later = function() { + var last = _.now() - timestamp; + + if (last < wait && last >= 0) { + timeout = setTimeout(later, wait - last); + } else { + timeout = null; + if (!immediate) { + result = func.apply(context, args); + if (!timeout) context = args = null; + } + } + }; + + return function() { + context = this; + args = arguments; + timestamp = _.now(); + var callNow = immediate && !timeout; + if (!timeout) timeout = setTimeout(later, wait); + if (callNow) { + result = func.apply(context, args); + context = args = null; + } + + return result; + }; + }; + + // Returns the first function passed as an argument to the second, + // allowing you to adjust arguments, run code before and after, and + // conditionally execute the original function. + _.wrap = function(func, wrapper) { + return _.partial(wrapper, func); + }; + + // Returns a negated version of the passed-in predicate. + _.negate = function(predicate) { + return function() { + return !predicate.apply(this, arguments); + }; + }; + + // Returns a function that is the composition of a list of functions, each + // consuming the return value of the function that follows. + _.compose = function() { + var args = arguments; + var start = args.length - 1; + return function() { + var i = start; + var result = args[start].apply(this, arguments); + while (i--) result = args[i].call(this, result); + return result; + }; + }; + + // Returns a function that will only be executed on and after the Nth call. + _.after = function(times, func) { + return function() { + if (--times < 1) { + return func.apply(this, arguments); + } + }; + }; + + // Returns a function that will only be executed up to (but not including) the Nth call. + _.before = function(times, func) { + var memo; + return function() { + if (--times > 0) { + memo = func.apply(this, arguments); + } + if (times <= 1) func = null; + return memo; + }; + }; + + // Returns a function that will be executed at most one time, no matter how + // often you call it. Useful for lazy initialization. + _.once = _.partial(_.before, 2); + + // Object Functions + // ---------------- + + // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed. + var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString'); + var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString', + 'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString']; + + function collectNonEnumProps(obj, keys) { + var nonEnumIdx = nonEnumerableProps.length; + var constructor = obj.constructor; + var proto = (_.isFunction(constructor) && constructor.prototype) || ObjProto; + + // Constructor is a special case. + var prop = 'constructor'; + if (_.has(obj, prop) && !_.contains(keys, prop)) keys.push(prop); + + while (nonEnumIdx--) { + prop = nonEnumerableProps[nonEnumIdx]; + if (prop in obj && obj[prop] !== proto[prop] && !_.contains(keys, prop)) { + keys.push(prop); + } + } + } + + // Retrieve the names of an object's own properties. + // Delegates to **ECMAScript 5**'s native `Object.keys` + _.keys = function(obj) { + if (!_.isObject(obj)) return []; + if (nativeKeys) return nativeKeys(obj); + var keys = []; + for (var key in obj) if (_.has(obj, key)) keys.push(key); + // Ahem, IE < 9. + if (hasEnumBug) collectNonEnumProps(obj, keys); + return keys; + }; + + // Retrieve all the property names of an object. + _.allKeys = function(obj) { + if (!_.isObject(obj)) return []; + var keys = []; + for (var key in obj) keys.push(key); + // Ahem, IE < 9. + if (hasEnumBug) collectNonEnumProps(obj, keys); + return keys; + }; + + // Retrieve the values of an object's properties. + _.values = function(obj) { + var keys = _.keys(obj); + var length = keys.length; + var values = Array(length); + for (var i = 0; i < length; i++) { + values[i] = obj[keys[i]]; + } + return values; + }; + + // Returns the results of applying the iteratee to each element of the object + // In contrast to _.map it returns an object + _.mapObject = function(obj, iteratee, context) { + iteratee = cb(iteratee, context); + var keys = _.keys(obj), + length = keys.length, + results = {}, + currentKey; + for (var index = 0; index < length; index++) { + currentKey = keys[index]; + results[currentKey] = iteratee(obj[currentKey], currentKey, obj); + } + return results; + }; + + // Convert an object into a list of `[key, value]` pairs. + _.pairs = function(obj) { + var keys = _.keys(obj); + var length = keys.length; + var pairs = Array(length); + for (var i = 0; i < length; i++) { + pairs[i] = [keys[i], obj[keys[i]]]; + } + return pairs; + }; + + // Invert the keys and values of an object. The values must be serializable. + _.invert = function(obj) { + var result = {}; + var keys = _.keys(obj); + for (var i = 0, length = keys.length; i < length; i++) { + result[obj[keys[i]]] = keys[i]; + } + return result; + }; + + // Return a sorted list of the function names available on the object. + // Aliased as `methods` + _.functions = _.methods = function(obj) { + var names = []; + for (var key in obj) { + if (_.isFunction(obj[key])) names.push(key); + } + return names.sort(); + }; + + // Extend a given object with all the properties in passed-in object(s). + _.extend = createAssigner(_.allKeys); + + // Assigns a given object with all the own properties in the passed-in object(s) + // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign) + _.extendOwn = _.assign = createAssigner(_.keys); + + // Returns the first key on an object that passes a predicate test + _.findKey = function(obj, predicate, context) { + predicate = cb(predicate, context); + var keys = _.keys(obj), key; + for (var i = 0, length = keys.length; i < length; i++) { + key = keys[i]; + if (predicate(obj[key], key, obj)) return key; + } + }; + + // Return a copy of the object only containing the whitelisted properties. + _.pick = function(object, oiteratee, context) { + var result = {}, obj = object, iteratee, keys; + if (obj == null) return result; + if (_.isFunction(oiteratee)) { + keys = _.allKeys(obj); + iteratee = optimizeCb(oiteratee, context); + } else { + keys = flatten(arguments, false, false, 1); + iteratee = function(value, key, obj) { return key in obj; }; + obj = Object(obj); + } + for (var i = 0, length = keys.length; i < length; i++) { + var key = keys[i]; + var value = obj[key]; + if (iteratee(value, key, obj)) result[key] = value; + } + return result; + }; + + // Return a copy of the object without the blacklisted properties. + _.omit = function(obj, iteratee, context) { + if (_.isFunction(iteratee)) { + iteratee = _.negate(iteratee); + } else { + var keys = _.map(flatten(arguments, false, false, 1), String); + iteratee = function(value, key) { + return !_.contains(keys, key); + }; + } + return _.pick(obj, iteratee, context); + }; + + // Fill in a given object with default properties. + _.defaults = createAssigner(_.allKeys, true); + + // Creates an object that inherits from the given prototype object. + // If additional properties are provided then they will be added to the + // created object. + _.create = function(prototype, props) { + var result = baseCreate(prototype); + if (props) _.extendOwn(result, props); + return result; + }; + + // Create a (shallow-cloned) duplicate of an object. + _.clone = function(obj) { + if (!_.isObject(obj)) return obj; + return _.isArray(obj) ? obj.slice() : _.extend({}, obj); + }; + + // Invokes interceptor with the obj, and then returns obj. + // The primary purpose of this method is to "tap into" a method chain, in + // order to perform operations on intermediate results within the chain. + _.tap = function(obj, interceptor) { + interceptor(obj); + return obj; + }; + + // Returns whether an object has a given set of `key:value` pairs. + _.isMatch = function(object, attrs) { + var keys = _.keys(attrs), length = keys.length; + if (object == null) return !length; + var obj = Object(object); + for (var i = 0; i < length; i++) { + var key = keys[i]; + if (attrs[key] !== obj[key] || !(key in obj)) return false; + } + return true; + }; + + + // Internal recursive comparison function for `isEqual`. + var eq = function(a, b, aStack, bStack) { + // Identical objects are equal. `0 === -0`, but they aren't identical. + // See the [Harmony `egal` proposal](http://wiki.ecmascript.org/doku.php?id=harmony:egal). + if (a === b) return a !== 0 || 1 / a === 1 / b; + // A strict comparison is necessary because `null == undefined`. + if (a == null || b == null) return a === b; + // Unwrap any wrapped objects. + if (a instanceof _) a = a._wrapped; + if (b instanceof _) b = b._wrapped; + // Compare `[[Class]]` names. + var className = toString.call(a); + if (className !== toString.call(b)) return false; + switch (className) { + // Strings, numbers, regular expressions, dates, and booleans are compared by value. + case '[object RegExp]': + // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i') + case '[object String]': + // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is + // equivalent to `new String("5")`. + return '' + a === '' + b; + case '[object Number]': + // `NaN`s are equivalent, but non-reflexive. + // Object(NaN) is equivalent to NaN + if (+a !== +a) return +b !== +b; + // An `egal` comparison is performed for other numeric values. + return +a === 0 ? 1 / +a === 1 / b : +a === +b; + case '[object Date]': + case '[object Boolean]': + // Coerce dates and booleans to numeric primitive values. Dates are compared by their + // millisecond representations. Note that invalid dates with millisecond representations + // of `NaN` are not equivalent. + return +a === +b; + } + + var areArrays = className === '[object Array]'; + if (!areArrays) { + if (typeof a != 'object' || typeof b != 'object') return false; + + // Objects with different constructors are not equivalent, but `Object`s or `Array`s + // from different frames are. + var aCtor = a.constructor, bCtor = b.constructor; + if (aCtor !== bCtor && !(_.isFunction(aCtor) && aCtor instanceof aCtor && + _.isFunction(bCtor) && bCtor instanceof bCtor) + && ('constructor' in a && 'constructor' in b)) { + return false; + } + } + // Assume equality for cyclic structures. The algorithm for detecting cyclic + // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`. + + // Initializing stack of traversed objects. + // It's done here since we only need them for objects and arrays comparison. + aStack = aStack || []; + bStack = bStack || []; + var length = aStack.length; + while (length--) { + // Linear search. Performance is inversely proportional to the number of + // unique nested structures. + if (aStack[length] === a) return bStack[length] === b; + } + + // Add the first object to the stack of traversed objects. + aStack.push(a); + bStack.push(b); + + // Recursively compare objects and arrays. + if (areArrays) { + // Compare array lengths to determine if a deep comparison is necessary. + length = a.length; + if (length !== b.length) return false; + // Deep compare the contents, ignoring non-numeric properties. + while (length--) { + if (!eq(a[length], b[length], aStack, bStack)) return false; + } + } else { + // Deep compare objects. + var keys = _.keys(a), key; + length = keys.length; + // Ensure that both objects contain the same number of properties before comparing deep equality. + if (_.keys(b).length !== length) return false; + while (length--) { + // Deep compare each member + key = keys[length]; + if (!(_.has(b, key) && eq(a[key], b[key], aStack, bStack))) return false; + } + } + // Remove the first object from the stack of traversed objects. + aStack.pop(); + bStack.pop(); + return true; + }; + + // Perform a deep comparison to check if two objects are equal. + _.isEqual = function(a, b) { + return eq(a, b); + }; + + // Is a given array, string, or object empty? + // An "empty" object has no enumerable own-properties. + _.isEmpty = function(obj) { + if (obj == null) return true; + if (isArrayLike(obj) && (_.isArray(obj) || _.isString(obj) || _.isArguments(obj))) return obj.length === 0; + return _.keys(obj).length === 0; + }; + + // Is a given value a DOM element? + _.isElement = function(obj) { + return !!(obj && obj.nodeType === 1); + }; + + // Is a given value an array? + // Delegates to ECMA5's native Array.isArray + _.isArray = nativeIsArray || function(obj) { + return toString.call(obj) === '[object Array]'; + }; + + // Is a given variable an object? + _.isObject = function(obj) { + var type = typeof obj; + return type === 'function' || type === 'object' && !!obj; + }; + + // Add some isType methods: isArguments, isFunction, isString, isNumber, isDate, isRegExp, isError. + _.each(['Arguments', 'Function', 'String', 'Number', 'Date', 'RegExp', 'Error'], function(name) { + _['is' + name] = function(obj) { + return toString.call(obj) === '[object ' + name + ']'; + }; + }); + + // Define a fallback version of the method in browsers (ahem, IE < 9), where + // there isn't any inspectable "Arguments" type. + if (!_.isArguments(arguments)) { + _.isArguments = function(obj) { + return _.has(obj, 'callee'); + }; + } + + // Optimize `isFunction` if appropriate. Work around some typeof bugs in old v8, + // IE 11 (#1621), and in Safari 8 (#1929). + if (typeof /./ != 'function' && typeof Int8Array != 'object') { + _.isFunction = function(obj) { + return typeof obj == 'function' || false; + }; + } + + // Is a given object a finite number? + _.isFinite = function(obj) { + return isFinite(obj) && !isNaN(parseFloat(obj)); + }; + + // Is the given value `NaN`? (NaN is the only number which does not equal itself). + _.isNaN = function(obj) { + return _.isNumber(obj) && obj !== +obj; + }; + + // Is a given value a boolean? + _.isBoolean = function(obj) { + return obj === true || obj === false || toString.call(obj) === '[object Boolean]'; + }; + + // Is a given value equal to null? + _.isNull = function(obj) { + return obj === null; + }; + + // Is a given variable undefined? + _.isUndefined = function(obj) { + return obj === void 0; + }; + + // Shortcut function for checking if an object has a given property directly + // on itself (in other words, not on a prototype). + _.has = function(obj, key) { + return obj != null && hasOwnProperty.call(obj, key); + }; + + // Utility Functions + // ----------------- + + // Run Underscore.js in *noConflict* mode, returning the `_` variable to its + // previous owner. Returns a reference to the Underscore object. + _.noConflict = function() { + root._ = previousUnderscore; + return this; + }; + + // Keep the identity function around for default iteratees. + _.identity = function(value) { + return value; + }; + + // Predicate-generating functions. Often useful outside of Underscore. + _.constant = function(value) { + return function() { + return value; + }; + }; + + _.noop = function(){}; + + _.property = property; + + // Generates a function for a given object that returns a given property. + _.propertyOf = function(obj) { + return obj == null ? function(){} : function(key) { + return obj[key]; + }; + }; + + // Returns a predicate for checking whether an object has a given set of + // `key:value` pairs. + _.matcher = _.matches = function(attrs) { + attrs = _.extendOwn({}, attrs); + return function(obj) { + return _.isMatch(obj, attrs); + }; + }; + + // Run a function **n** times. + _.times = function(n, iteratee, context) { + var accum = Array(Math.max(0, n)); + iteratee = optimizeCb(iteratee, context, 1); + for (var i = 0; i < n; i++) accum[i] = iteratee(i); + return accum; + }; + + // Return a random integer between min and max (inclusive). + _.random = function(min, max) { + if (max == null) { + max = min; + min = 0; + } + return min + Math.floor(Math.random() * (max - min + 1)); + }; + + // A (possibly faster) way to get the current timestamp as an integer. + _.now = Date.now || function() { + return new Date().getTime(); + }; + + // List of HTML entities for escaping. + var escapeMap = { + '&': '&', + '<': '<', + '>': '>', + '"': '"', + "'": ''', + '`': '`' + }; + var unescapeMap = _.invert(escapeMap); + + // Functions for escaping and unescaping strings to/from HTML interpolation. + var createEscaper = function(map) { + var escaper = function(match) { + return map[match]; + }; + // Regexes for identifying a key that needs to be escaped + var source = '(?:' + _.keys(map).join('|') + ')'; + var testRegexp = RegExp(source); + var replaceRegexp = RegExp(source, 'g'); + return function(string) { + string = string == null ? '' : '' + string; + return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string; + }; + }; + _.escape = createEscaper(escapeMap); + _.unescape = createEscaper(unescapeMap); + + // If the value of the named `property` is a function then invoke it with the + // `object` as context; otherwise, return it. + _.result = function(object, property, fallback) { + var value = object == null ? void 0 : object[property]; + if (value === void 0) { + value = fallback; + } + return _.isFunction(value) ? value.call(object) : value; + }; + + // Generate a unique integer id (unique within the entire client session). + // Useful for temporary DOM ids. + var idCounter = 0; + _.uniqueId = function(prefix) { + var id = ++idCounter + ''; + return prefix ? prefix + id : id; + }; + + // By default, Underscore uses ERB-style template delimiters, change the + // following template settings to use alternative delimiters. + _.templateSettings = { + evaluate : /<%([\s\S]+?)%>/g, + interpolate : /<%=([\s\S]+?)%>/g, + escape : /<%-([\s\S]+?)%>/g + }; + + // When customizing `templateSettings`, if you don't want to define an + // interpolation, evaluation or escaping regex, we need one that is + // guaranteed not to match. + var noMatch = /(.)^/; + + // Certain characters need to be escaped so that they can be put into a + // string literal. + var escapes = { + "'": "'", + '\\': '\\', + '\r': 'r', + '\n': 'n', + '\u2028': 'u2028', + '\u2029': 'u2029' + }; + + var escaper = /\\|'|\r|\n|\u2028|\u2029/g; + + var escapeChar = function(match) { + return '\\' + escapes[match]; + }; + + // JavaScript micro-templating, similar to John Resig's implementation. + // Underscore templating handles arbitrary delimiters, preserves whitespace, + // and correctly escapes quotes within interpolated code. + // NB: `oldSettings` only exists for backwards compatibility. + _.template = function(text, settings, oldSettings) { + if (!settings && oldSettings) settings = oldSettings; + settings = _.defaults({}, settings, _.templateSettings); + + // Combine delimiters into one regular expression via alternation. + var matcher = RegExp([ + (settings.escape || noMatch).source, + (settings.interpolate || noMatch).source, + (settings.evaluate || noMatch).source + ].join('|') + '|$', 'g'); + + // Compile the template source, escaping string literals appropriately. + var index = 0; + var source = "__p+='"; + text.replace(matcher, function(match, escape, interpolate, evaluate, offset) { + source += text.slice(index, offset).replace(escaper, escapeChar); + index = offset + match.length; + + if (escape) { + source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'"; + } else if (interpolate) { + source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'"; + } else if (evaluate) { + source += "';\n" + evaluate + "\n__p+='"; + } + + // Adobe VMs need the match returned to produce the correct offest. + return match; + }); + source += "';\n"; + + // If a variable is not specified, place data values in local scope. + if (!settings.variable) source = 'with(obj||{}){\n' + source + '}\n'; + + source = "var __t,__p='',__j=Array.prototype.join," + + "print=function(){__p+=__j.call(arguments,'');};\n" + + source + 'return __p;\n'; + + try { + var render = new Function(settings.variable || 'obj', '_', source); + } catch (e) { + e.source = source; + throw e; + } + + var template = function(data) { + return render.call(this, data, _); + }; + + // Provide the compiled source as a convenience for precompilation. + var argument = settings.variable || 'obj'; + template.source = 'function(' + argument + '){\n' + source + '}'; + + return template; + }; + + // Add a "chain" function. Start chaining a wrapped Underscore object. + _.chain = function(obj) { + var instance = _(obj); + instance._chain = true; + return instance; + }; + + // OOP + // --------------- + // If Underscore is called as a function, it returns a wrapped object that + // can be used OO-style. This wrapper holds altered versions of all the + // underscore functions. Wrapped objects may be chained. + + // Helper function to continue chaining intermediate results. + var result = function(instance, obj) { + return instance._chain ? _(obj).chain() : obj; + }; + + // Add your own custom functions to the Underscore object. + _.mixin = function(obj) { + _.each(_.functions(obj), function(name) { + var func = _[name] = obj[name]; + _.prototype[name] = function() { + var args = [this._wrapped]; + push.apply(args, arguments); + return result(this, func.apply(_, args)); + }; + }); + }; + + // Add all of the Underscore functions to the wrapper object. + _.mixin(_); + + // Add all mutator Array functions to the wrapper. + _.each(['pop', 'push', 'reverse', 'shift', 'sort', 'splice', 'unshift'], function(name) { + var method = ArrayProto[name]; + _.prototype[name] = function() { + var obj = this._wrapped; + method.apply(obj, arguments); + if ((name === 'shift' || name === 'splice') && obj.length === 0) delete obj[0]; + return result(this, obj); + }; + }); + + // Add all accessor Array functions to the wrapper. + _.each(['concat', 'join', 'slice'], function(name) { + var method = ArrayProto[name]; + _.prototype[name] = function() { + return result(this, method.apply(this._wrapped, arguments)); + }; + }); + + // Extracts the result from a wrapped and chained object. + _.prototype.value = function() { + return this._wrapped; + }; + + // Provide unwrapping proxy for some methods used in engine operations + // such as arithmetic and JSON stringification. + _.prototype.valueOf = _.prototype.toJSON = _.prototype.value; + + _.prototype.toString = function() { + return '' + this._wrapped; + }; + + // AMD registration happens at the end for compatibility with AMD loaders + // that may not enforce next-turn semantics on modules. Even though general + // practice for AMD registration is to be anonymous, underscore registers + // as a named module because, like jQuery, it is a base library that is + // popular enough to be bundled in a third party lib, but not be part of + // an AMD load request. Those cases could generate an error when an + // anonymous define() is called outside of a loader request. + if (typeof define === 'function' && define.amd) { + define('underscore', [], function() { + return _; + }); + } +}.call(this)); + +},{}],26:[function(require,module,exports){ +arguments[4][19][0].apply(exports,arguments) +},{"dup":19}],27:[function(require,module,exports){ +module.exports = function isBuffer(arg) { + return arg && typeof arg === 'object' + && typeof arg.copy === 'function' + && typeof arg.fill === 'function' + && typeof arg.readUInt8 === 'function'; +} +},{}],28:[function(require,module,exports){ +(function (process,global){ +// Copyright Joyent, Inc. and other Node contributors. +// +// Permission is hereby granted, free of charge, to any person obtaining a +// copy of this software and associated documentation files (the +// "Software"), to deal in the Software without restriction, including +// without limitation the rights to use, copy, modify, merge, publish, +// distribute, sublicense, and/or sell copies of the Software, and to permit +// persons to whom the Software is furnished to do so, subject to the +// following conditions: +// +// The above copyright notice and this permission notice shall be included +// in all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS +// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF +// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN +// NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, +// DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR +// OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE +// USE OR OTHER DEALINGS IN THE SOFTWARE. + +var formatRegExp = /%[sdj%]/g; +exports.format = function(f) { + if (!isString(f)) { + var objects = []; + for (var i = 0; i < arguments.length; i++) { + objects.push(inspect(arguments[i])); + } + return objects.join(' '); + } + + var i = 1; + var args = arguments; + var len = args.length; + var str = String(f).replace(formatRegExp, function(x) { + if (x === '%%') return '%'; + if (i >= len) return x; + switch (x) { + case '%s': return String(args[i++]); + case '%d': return Number(args[i++]); + case '%j': + try { + return JSON.stringify(args[i++]); + } catch (_) { + return '[Circular]'; + } + default: + return x; + } + }); + for (var x = args[i]; i < len; x = args[++i]) { + if (isNull(x) || !isObject(x)) { + str += ' ' + x; + } else { + str += ' ' + inspect(x); + } + } + return str; +}; + + +// Mark that a method should not be used. +// Returns a modified function which warns once by default. +// If --no-deprecation is set, then it is a no-op. +exports.deprecate = function(fn, msg) { + // Allow for deprecating things in the process of starting up. + if (isUndefined(global.process)) { + return function() { + return exports.deprecate(fn, msg).apply(this, arguments); + }; + } + + if (process.noDeprecation === true) { + return fn; + } + + var warned = false; + function deprecated() { + if (!warned) { + if (process.throwDeprecation) { + throw new Error(msg); + } else if (process.traceDeprecation) { + console.trace(msg); + } else { + console.error(msg); + } + warned = true; + } + return fn.apply(this, arguments); + } + + return deprecated; +}; + + +var debugs = {}; +var debugEnviron; +exports.debuglog = function(set) { + if (isUndefined(debugEnviron)) + debugEnviron = process.env.NODE_DEBUG || ''; + set = set.toUpperCase(); + if (!debugs[set]) { + if (new RegExp('\\b' + set + '\\b', 'i').test(debugEnviron)) { + var pid = process.pid; + debugs[set] = function() { + var msg = exports.format.apply(exports, arguments); + console.error('%s %d: %s', set, pid, msg); + }; + } else { + debugs[set] = function() {}; + } + } + return debugs[set]; +}; + + +/** + * Echos the value of a value. Trys to print the value out + * in the best way possible given the different types. + * + * @param {Object} obj The object to print out. + * @param {Object} opts Optional options object that alters the output. + */ +/* legacy: obj, showHidden, depth, colors*/ +function inspect(obj, opts) { + // default options + var ctx = { + seen: [], + stylize: stylizeNoColor + }; + // legacy... + if (arguments.length >= 3) ctx.depth = arguments[2]; + if (arguments.length >= 4) ctx.colors = arguments[3]; + if (isBoolean(opts)) { + // legacy... + ctx.showHidden = opts; + } else if (opts) { + // got an "options" object + exports._extend(ctx, opts); + } + // set default options + if (isUndefined(ctx.showHidden)) ctx.showHidden = false; + if (isUndefined(ctx.depth)) ctx.depth = 2; + if (isUndefined(ctx.colors)) ctx.colors = false; + if (isUndefined(ctx.customInspect)) ctx.customInspect = true; + if (ctx.colors) ctx.stylize = stylizeWithColor; + return formatValue(ctx, obj, ctx.depth); +} +exports.inspect = inspect; + + +// http://en.wikipedia.org/wiki/ANSI_escape_code#graphics +inspect.colors = { + 'bold' : [1, 22], + 'italic' : [3, 23], + 'underline' : [4, 24], + 'inverse' : [7, 27], + 'white' : [37, 39], + 'grey' : [90, 39], + 'black' : [30, 39], + 'blue' : [34, 39], + 'cyan' : [36, 39], + 'green' : [32, 39], + 'magenta' : [35, 39], + 'red' : [31, 39], + 'yellow' : [33, 39] +}; + +// Don't use 'blue' not visible on cmd.exe +inspect.styles = { + 'special': 'cyan', + 'number': 'yellow', + 'boolean': 'yellow', + 'undefined': 'grey', + 'null': 'bold', + 'string': 'green', + 'date': 'magenta', + // "name": intentionally not styling + 'regexp': 'red' +}; + + +function stylizeWithColor(str, styleType) { + var style = inspect.styles[styleType]; + + if (style) { + return '\u001b[' + inspect.colors[style][0] + 'm' + str + + '\u001b[' + inspect.colors[style][1] + 'm'; + } else { + return str; + } +} + + +function stylizeNoColor(str, styleType) { + return str; +} + + +function arrayToHash(array) { + var hash = {}; + + array.forEach(function(val, idx) { + hash[val] = true; + }); + + return hash; +} + + +function formatValue(ctx, value, recurseTimes) { + // Provide a hook for user-specified inspect functions. + // Check that value is an object with an inspect function on it + if (ctx.customInspect && + value && + isFunction(value.inspect) && + // Filter out the util module, it's inspect function is special + value.inspect !== exports.inspect && + // Also filter out any prototype objects using the circular check. + !(value.constructor && value.constructor.prototype === value)) { + var ret = value.inspect(recurseTimes, ctx); + if (!isString(ret)) { + ret = formatValue(ctx, ret, recurseTimes); + } + return ret; + } + + // Primitive types cannot have properties + var primitive = formatPrimitive(ctx, value); + if (primitive) { + return primitive; + } + + // Look up the keys of the object. + var keys = Object.keys(value); + var visibleKeys = arrayToHash(keys); + + if (ctx.showHidden) { + keys = Object.getOwnPropertyNames(value); + } + + // IE doesn't make error fields non-enumerable + // http://msdn.microsoft.com/en-us/library/ie/dww52sbt(v=vs.94).aspx + if (isError(value) + && (keys.indexOf('message') >= 0 || keys.indexOf('description') >= 0)) { + return formatError(value); + } + + // Some type of object without properties can be shortcutted. + if (keys.length === 0) { + if (isFunction(value)) { + var name = value.name ? ': ' + value.name : ''; + return ctx.stylize('[Function' + name + ']', 'special'); + } + if (isRegExp(value)) { + return ctx.stylize(RegExp.prototype.toString.call(value), 'regexp'); + } + if (isDate(value)) { + return ctx.stylize(Date.prototype.toString.call(value), 'date'); + } + if (isError(value)) { + return formatError(value); + } + } + + var base = '', array = false, braces = ['{', '}']; + + // Make Array say that they are Array + if (isArray(value)) { + array = true; + braces = ['[', ']']; + } + + // Make functions say that they are functions + if (isFunction(value)) { + var n = value.name ? ': ' + value.name : ''; + base = ' [Function' + n + ']'; + } + + // Make RegExps say that they are RegExps + if (isRegExp(value)) { + base = ' ' + RegExp.prototype.toString.call(value); + } + + // Make dates with properties first say the date + if (isDate(value)) { + base = ' ' + Date.prototype.toUTCString.call(value); + } + + // Make error with message first say the error + if (isError(value)) { + base = ' ' + formatError(value); + } + + if (keys.length === 0 && (!array || value.length == 0)) { + return braces[0] + base + braces[1]; + } + + if (recurseTimes < 0) { + if (isRegExp(value)) { + return ctx.stylize(RegExp.prototype.toString.call(value), 'regexp'); + } else { + return ctx.stylize('[Object]', 'special'); + } + } + + ctx.seen.push(value); + + var output; + if (array) { + output = formatArray(ctx, value, recurseTimes, visibleKeys, keys); + } else { + output = keys.map(function(key) { + return formatProperty(ctx, value, recurseTimes, visibleKeys, key, array); + }); + } + + ctx.seen.pop(); + + return reduceToSingleString(output, base, braces); +} + + +function formatPrimitive(ctx, value) { + if (isUndefined(value)) + return ctx.stylize('undefined', 'undefined'); + if (isString(value)) { + var simple = '\'' + JSON.stringify(value).replace(/^"|"$/g, '') + .replace(/'/g, "\\'") + .replace(/\\"/g, '"') + '\''; + return ctx.stylize(simple, 'string'); + } + if (isNumber(value)) + return ctx.stylize('' + value, 'number'); + if (isBoolean(value)) + return ctx.stylize('' + value, 'boolean'); + // For some reason typeof null is "object", so special case here. + if (isNull(value)) + return ctx.stylize('null', 'null'); +} + + +function formatError(value) { + return '[' + Error.prototype.toString.call(value) + ']'; +} + + +function formatArray(ctx, value, recurseTimes, visibleKeys, keys) { + var output = []; + for (var i = 0, l = value.length; i < l; ++i) { + if (hasOwnProperty(value, String(i))) { + output.push(formatProperty(ctx, value, recurseTimes, visibleKeys, + String(i), true)); + } else { + output.push(''); + } + } + keys.forEach(function(key) { + if (!key.match(/^\d+$/)) { + output.push(formatProperty(ctx, value, recurseTimes, visibleKeys, + key, true)); + } + }); + return output; +} + + +function formatProperty(ctx, value, recurseTimes, visibleKeys, key, array) { + var name, str, desc; + desc = Object.getOwnPropertyDescriptor(value, key) || { value: value[key] }; + if (desc.get) { + if (desc.set) { + str = ctx.stylize('[Getter/Setter]', 'special'); + } else { + str = ctx.stylize('[Getter]', 'special'); + } + } else { + if (desc.set) { + str = ctx.stylize('[Setter]', 'special'); + } + } + if (!hasOwnProperty(visibleKeys, key)) { + name = '[' + key + ']'; + } + if (!str) { + if (ctx.seen.indexOf(desc.value) < 0) { + if (isNull(recurseTimes)) { + str = formatValue(ctx, desc.value, null); + } else { + str = formatValue(ctx, desc.value, recurseTimes - 1); + } + if (str.indexOf('\n') > -1) { + if (array) { + str = str.split('\n').map(function(line) { + return ' ' + line; + }).join('\n').substr(2); + } else { + str = '\n' + str.split('\n').map(function(line) { + return ' ' + line; + }).join('\n'); + } + } + } else { + str = ctx.stylize('[Circular]', 'special'); + } + } + if (isUndefined(name)) { + if (array && key.match(/^\d+$/)) { + return str; + } + name = JSON.stringify('' + key); + if (name.match(/^"([a-zA-Z_][a-zA-Z_0-9]*)"$/)) { + name = name.substr(1, name.length - 2); + name = ctx.stylize(name, 'name'); + } else { + name = name.replace(/'/g, "\\'") + .replace(/\\"/g, '"') + .replace(/(^"|"$)/g, "'"); + name = ctx.stylize(name, 'string'); + } + } + + return name + ': ' + str; +} + + +function reduceToSingleString(output, base, braces) { + var numLinesEst = 0; + var length = output.reduce(function(prev, cur) { + numLinesEst++; + if (cur.indexOf('\n') >= 0) numLinesEst++; + return prev + cur.replace(/\u001b\[\d\d?m/g, '').length + 1; + }, 0); + + if (length > 60) { + return braces[0] + + (base === '' ? '' : base + '\n ') + + ' ' + + output.join(',\n ') + + ' ' + + braces[1]; + } + + return braces[0] + base + ' ' + output.join(', ') + ' ' + braces[1]; +} + + +// NOTE: These type checking functions intentionally don't use `instanceof` +// because it is fragile and can be easily faked with `Object.create()`. +function isArray(ar) { + return Array.isArray(ar); +} +exports.isArray = isArray; + +function isBoolean(arg) { + return typeof arg === 'boolean'; +} +exports.isBoolean = isBoolean; + +function isNull(arg) { + return arg === null; +} +exports.isNull = isNull; + +function isNullOrUndefined(arg) { + return arg == null; +} +exports.isNullOrUndefined = isNullOrUndefined; + +function isNumber(arg) { + return typeof arg === 'number'; +} +exports.isNumber = isNumber; + +function isString(arg) { + return typeof arg === 'string'; +} +exports.isString = isString; + +function isSymbol(arg) { + return typeof arg === 'symbol'; +} +exports.isSymbol = isSymbol; + +function isUndefined(arg) { + return arg === void 0; +} +exports.isUndefined = isUndefined; + +function isRegExp(re) { + return isObject(re) && objectToString(re) === '[object RegExp]'; +} +exports.isRegExp = isRegExp; + +function isObject(arg) { + return typeof arg === 'object' && arg !== null; +} +exports.isObject = isObject; + +function isDate(d) { + return isObject(d) && objectToString(d) === '[object Date]'; +} +exports.isDate = isDate; + +function isError(e) { + return isObject(e) && + (objectToString(e) === '[object Error]' || e instanceof Error); +} +exports.isError = isError; + +function isFunction(arg) { + return typeof arg === 'function'; +} +exports.isFunction = isFunction; + +function isPrimitive(arg) { + return arg === null || + typeof arg === 'boolean' || + typeof arg === 'number' || + typeof arg === 'string' || + typeof arg === 'symbol' || // ES6 symbol + typeof arg === 'undefined'; +} +exports.isPrimitive = isPrimitive; + +exports.isBuffer = require('./support/isBuffer'); + +function objectToString(o) { + return Object.prototype.toString.call(o); +} + + +function pad(n) { + return n < 10 ? '0' + n.toString(10) : n.toString(10); +} + + +var months = ['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep', + 'Oct', 'Nov', 'Dec']; + +// 26 Feb 16:19:34 +function timestamp() { + var d = new Date(); + var time = [pad(d.getHours()), + pad(d.getMinutes()), + pad(d.getSeconds())].join(':'); + return [d.getDate(), months[d.getMonth()], time].join(' '); +} + + +// log is just a thin wrapper to console.log that prepends a timestamp +exports.log = function() { + console.log('%s - %s', timestamp(), exports.format.apply(exports, arguments)); +}; + + +/** + * Inherit the prototype methods from one constructor into another. + * + * The Function.prototype.inherits from lang.js rewritten as a standalone + * function (not on Function.prototype). NOTE: If this file is to be loaded + * during bootstrapping this function needs to be rewritten using some native + * functions as prototype setup using normal JavaScript does not work as + * expected during bootstrapping (see mirror.js in r114903). + * + * @param {function} ctor Constructor function which needs to inherit the + * prototype. + * @param {function} superCtor Constructor function to inherit prototype from. + */ +exports.inherits = require('inherits'); + +exports._extend = function(origin, add) { + // Don't do anything if add isn't an object + if (!add || !isObject(add)) return origin; + + var keys = Object.keys(add); + var i = keys.length; + while (i--) { + origin[keys[i]] = add[keys[i]]; + } + return origin; +}; + +function hasOwnProperty(obj, prop) { + return Object.prototype.hasOwnProperty.call(obj, prop); +} + +}).call(this,require('_process'),typeof global !== "undefined" ? global : typeof self !== "undefined" ? self : typeof window !== "undefined" ? window : {}) +},{"./support/isBuffer":27,"_process":24,"inherits":26}],29:[function(require,module,exports){ +// Returns a wrapper function that returns a wrapped callback +// The wrapper function should do some stuff, and return a +// presumably different callback function. +// This makes sure that own properties are retained, so that +// decorations and such are not lost along the way. +module.exports = wrappy +function wrappy (fn, cb) { + if (fn && cb) return wrappy(fn)(cb) + + if (typeof fn !== 'function') + throw new TypeError('need wrapper function') + + Object.keys(fn).forEach(function (k) { + wrapper[k] = fn[k] + }) + + return wrapper + + function wrapper() { + var args = new Array(arguments.length) + for (var i = 0; i < args.length; i++) { + args[i] = arguments[i] + } + var ret = fn.apply(this, args) + var cb = args[args.length-1] + if (typeof ret === 'function' && ret !== cb) { + Object.keys(cb).forEach(function (k) { + ret[k] = cb[k] + }) + } + return ret + } +} + +},{}]},{},[7])(7) +}); \ No newline at end of file diff --git a/assets/javascripts/workers/search.5bf1dace.min.js b/assets/javascripts/workers/search.5bf1dace.min.js new file mode 100644 index 00000000..5b80d43c --- /dev/null +++ b/assets/javascripts/workers/search.5bf1dace.min.js @@ -0,0 +1,48 @@ +"use strict";(()=>{var ge=Object.create;var W=Object.defineProperty,ye=Object.defineProperties,me=Object.getOwnPropertyDescriptor,ve=Object.getOwnPropertyDescriptors,xe=Object.getOwnPropertyNames,G=Object.getOwnPropertySymbols,Se=Object.getPrototypeOf,X=Object.prototype.hasOwnProperty,Qe=Object.prototype.propertyIsEnumerable;var J=(t,e,r)=>e in t?W(t,e,{enumerable:!0,configurable:!0,writable:!0,value:r}):t[e]=r,M=(t,e)=>{for(var r in e||(e={}))X.call(e,r)&&J(t,r,e[r]);if(G)for(var r of G(e))Qe.call(e,r)&&J(t,r,e[r]);return t},Z=(t,e)=>ye(t,ve(e));var K=(t,e)=>()=>(e||t((e={exports:{}}).exports,e),e.exports);var be=(t,e,r,n)=>{if(e&&typeof e=="object"||typeof e=="function")for(let i of xe(e))!X.call(t,i)&&i!==r&&W(t,i,{get:()=>e[i],enumerable:!(n=me(e,i))||n.enumerable});return t};var H=(t,e,r)=>(r=t!=null?ge(Se(t)):{},be(e||!t||!t.__esModule?W(r,"default",{value:t,enumerable:!0}):r,t));var z=(t,e,r)=>new Promise((n,i)=>{var s=u=>{try{a(r.next(u))}catch(c){i(c)}},o=u=>{try{a(r.throw(u))}catch(c){i(c)}},a=u=>u.done?n(u.value):Promise.resolve(u.value).then(s,o);a((r=r.apply(t,e)).next())});var re=K((ee,te)=>{/** + * lunr - http://lunrjs.com - A bit like Solr, but much smaller and not as bright - 2.3.9 + * Copyright (C) 2020 Oliver Nightingale + * @license MIT + */(function(){var t=function(e){var r=new t.Builder;return r.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),r.searchPipeline.add(t.stemmer),e.call(r,r),r.build()};t.version="2.3.9";/*! + * lunr.utils + * Copyright (C) 2020 Oliver Nightingale + */t.utils={},t.utils.warn=function(e){return function(r){e.console&&console.warn&&console.warn(r)}}(this),t.utils.asString=function(e){return e==null?"":e.toString()},t.utils.clone=function(e){if(e==null)return e;for(var r=Object.create(null),n=Object.keys(e),i=0;i0){var h=t.utils.clone(r)||{};h.position=[a,c],h.index=s.length,s.push(new t.Token(n.slice(a,o),h))}a=o+1}}return s},t.tokenizer.separator=/[\s\-]+/;/*! + * lunr.Pipeline + * Copyright (C) 2020 Oliver Nightingale + */t.Pipeline=function(){this._stack=[]},t.Pipeline.registeredFunctions=Object.create(null),t.Pipeline.registerFunction=function(e,r){r in this.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+r),e.label=r,t.Pipeline.registeredFunctions[e.label]=e},t.Pipeline.warnIfFunctionNotRegistered=function(e){var r=e.label&&e.label in this.registeredFunctions;r||t.utils.warn(`Function is not registered with pipeline. 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r=this.highlight(e),n=ue(e).filter(o=>o.presence!==lunr.Query.presence.PROHIBITED),i=this.index.search(`${e}*`).reduce((o,{ref:a,score:u,matchData:c})=>{let h=this.documents.get(a);if(typeof h!="undefined"){let{location:y,title:g,text:b,tags:m,parent:Q}=h,p=ce(n,Object.keys(c.metadata)),d=+!Q+ +Object.values(p).every(w=>w);o.push(Z(M({location:y,title:r(g),text:r(b)},m&&{tags:m.map(r)}),{score:u*(1+d),terms:p}))}return o},[]).sort((o,a)=>a.score-o.score).reduce((o,a)=>{let u=this.documents.get(a.location);if(typeof u!="undefined"){let c="parent"in u?u.parent.location:u.location;o.set(c,[...o.get(c)||[],a])}return o},new Map),s;if(this.options.suggestions){let o=this.index.query(a=>{for(let u of n)a.term(u.term,{fields:["title"],presence:lunr.Query.presence.REQUIRED,wildcard:lunr.Query.wildcard.TRAILING})});s=o.length?Object.keys(o[0].matchData.metadata):[]}return M({items:[...i.values()]},typeof s!="undefined"&&{suggestions:s})}catch(r){console.warn(`Invalid query: ${e} \u2013 see https://bit.ly/2s3ChXG`)}return{items:[]}}};var Y;function ke(t){return z(this,null,function*(){let e="../lunr";if(typeof parent!="undefined"&&"IFrameWorker"in parent){let n=document.querySelector("script[src]"),[i]=n.src.split("/worker");e=e.replace("..",i)}let r=[];for(let n of t.lang){switch(n){case"ja":r.push(`${e}/tinyseg.js`);break;case"hi":case"th":r.push(`${e}/wordcut.js`);break}n!=="en"&&r.push(`${e}/min/lunr.${n}.min.js`)}t.lang.length>1&&r.push(`${e}/min/lunr.multi.min.js`),r.length&&(yield importScripts(`${e}/min/lunr.stemmer.support.min.js`,...r))})}function Te(t){return z(this,null,function*(){switch(t.type){case 0:return yield ke(t.data.config),Y=new U(t.data),{type:1};case 2:return{type:3,data:Y?Y.search(t.data):{items:[]}};default:throw new TypeError("Invalid message type")}})}self.lunr=le.default;addEventListener("message",t=>z(void 0,null,function*(){postMessage(yield Te(t.data))}));})(); +//# sourceMappingURL=search.5bf1dace.min.js.map + diff --git a/assets/javascripts/workers/search.5bf1dace.min.js.map b/assets/javascripts/workers/search.5bf1dace.min.js.map new file mode 100644 index 00000000..1df8be0e --- /dev/null +++ b/assets/javascripts/workers/search.5bf1dace.min.js.map @@ -0,0 +1,8 @@ +{ + "version": 3, + "sources": ["node_modules/lunr/lunr.js", "node_modules/escape-html/index.js", "src/assets/javascripts/integrations/search/worker/main/index.ts", "src/assets/javascripts/polyfills/index.ts", "src/assets/javascripts/integrations/search/document/index.ts", "src/assets/javascripts/integrations/search/highlighter/index.ts", "src/assets/javascripts/integrations/search/query/_/index.ts", "src/assets/javascripts/integrations/search/_/index.ts"], + "sourceRoot": "../../../..", + "sourcesContent": ["/**\n * lunr - http://lunrjs.com - A bit like Solr, but much smaller and not as bright - 2.3.9\n * Copyright (C) 2020 Oliver Nightingale\n * @license MIT\n */\n\n;(function(){\n\n/**\n * A convenience function for configuring and constructing\n * a new lunr Index.\n *\n * A lunr.Builder instance is created and the pipeline setup\n * with a trimmer, stop word filter and stemmer.\n *\n * This builder object is yielded to the configuration function\n * that is passed as a parameter, allowing the list of fields\n * and other builder parameters to be customised.\n *\n * All documents _must_ be added within the passed config function.\n *\n * @example\n * var idx = lunr(function () {\n * this.field('title')\n * this.field('body')\n * this.ref('id')\n *\n * documents.forEach(function (doc) {\n * this.add(doc)\n * }, this)\n * })\n *\n * @see {@link lunr.Builder}\n * @see {@link lunr.Pipeline}\n * @see {@link lunr.trimmer}\n * @see {@link lunr.stopWordFilter}\n * @see {@link lunr.stemmer}\n * @namespace {function} lunr\n */\nvar lunr = function (config) {\n var builder = new lunr.Builder\n\n builder.pipeline.add(\n lunr.trimmer,\n lunr.stopWordFilter,\n lunr.stemmer\n )\n\n builder.searchPipeline.add(\n lunr.stemmer\n )\n\n config.call(builder, builder)\n return builder.build()\n}\n\nlunr.version = \"2.3.9\"\n/*!\n * lunr.utils\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A namespace containing utils for the rest of the lunr library\n * @namespace lunr.utils\n */\nlunr.utils = {}\n\n/**\n * Print a warning message to the console.\n *\n * @param {String} message The message to be printed.\n * @memberOf lunr.utils\n * @function\n */\nlunr.utils.warn = (function (global) {\n /* eslint-disable no-console */\n return function (message) {\n if (global.console && console.warn) {\n console.warn(message)\n }\n }\n /* eslint-enable no-console */\n})(this)\n\n/**\n * Convert an object to a string.\n *\n * In the case of `null` and `undefined` the function returns\n * the empty string, in all other cases the result of calling\n * `toString` on the passed object is returned.\n *\n * @param {Any} obj The object to convert to a string.\n * @return {String} string representation of the passed object.\n * @memberOf lunr.utils\n */\nlunr.utils.asString = function (obj) {\n if (obj === void 0 || obj === null) {\n return \"\"\n } else {\n return obj.toString()\n }\n}\n\n/**\n * Clones an object.\n *\n * Will create a copy of an existing object such that any mutations\n * on the copy cannot affect the original.\n *\n * Only shallow objects are supported, passing a nested object to this\n * function will cause a TypeError.\n *\n * Objects with primitives, and arrays of primitives are supported.\n *\n * @param {Object} obj The object to clone.\n * @return {Object} a clone of the passed object.\n * @throws {TypeError} when a nested object is passed.\n * @memberOf Utils\n */\nlunr.utils.clone = function (obj) {\n if (obj === null || obj === undefined) {\n return obj\n }\n\n var clone = Object.create(null),\n keys = Object.keys(obj)\n\n for (var i = 0; i < keys.length; i++) {\n var key = keys[i],\n val = obj[key]\n\n if (Array.isArray(val)) {\n clone[key] = val.slice()\n continue\n }\n\n if (typeof val === 'string' ||\n typeof val === 'number' ||\n typeof val === 'boolean') {\n clone[key] = val\n continue\n }\n\n throw new TypeError(\"clone is not deep and does not support nested objects\")\n }\n\n return clone\n}\nlunr.FieldRef = function (docRef, fieldName, stringValue) {\n this.docRef = docRef\n this.fieldName = fieldName\n this._stringValue = stringValue\n}\n\nlunr.FieldRef.joiner = \"/\"\n\nlunr.FieldRef.fromString = function (s) {\n var n = s.indexOf(lunr.FieldRef.joiner)\n\n if (n === -1) {\n throw \"malformed field ref string\"\n }\n\n var fieldRef = s.slice(0, n),\n docRef = s.slice(n + 1)\n\n return new lunr.FieldRef (docRef, fieldRef, s)\n}\n\nlunr.FieldRef.prototype.toString = function () {\n if (this._stringValue == undefined) {\n this._stringValue = this.fieldName + lunr.FieldRef.joiner + this.docRef\n }\n\n return this._stringValue\n}\n/*!\n * lunr.Set\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A lunr set.\n *\n * @constructor\n */\nlunr.Set = function (elements) {\n this.elements = Object.create(null)\n\n if (elements) {\n this.length = elements.length\n\n for (var i = 0; i < this.length; i++) {\n this.elements[elements[i]] = true\n }\n } else {\n this.length = 0\n }\n}\n\n/**\n * A complete set that contains all elements.\n *\n * @static\n * @readonly\n * @type {lunr.Set}\n */\nlunr.Set.complete = {\n intersect: function (other) {\n return other\n },\n\n union: function () {\n return this\n },\n\n contains: function () {\n return true\n }\n}\n\n/**\n * An empty set that contains no elements.\n *\n * @static\n * @readonly\n * @type {lunr.Set}\n */\nlunr.Set.empty = {\n intersect: function () {\n return this\n },\n\n union: function (other) {\n return other\n },\n\n contains: function () {\n return false\n }\n}\n\n/**\n * Returns true if this set contains the specified object.\n *\n * @param {object} object - Object whose presence in this set is to be tested.\n * @returns {boolean} - True if this set contains the specified object.\n */\nlunr.Set.prototype.contains = function (object) {\n return !!this.elements[object]\n}\n\n/**\n * Returns a new set containing only the elements that are present in both\n * this set and the specified set.\n *\n * @param {lunr.Set} other - set to intersect with this set.\n * @returns {lunr.Set} a new set that is the intersection of this and the specified set.\n */\n\nlunr.Set.prototype.intersect = function (other) {\n var a, b, elements, intersection = []\n\n if (other === lunr.Set.complete) {\n return this\n }\n\n if (other === lunr.Set.empty) {\n return other\n }\n\n if (this.length < other.length) {\n a = this\n b = other\n } else {\n a = other\n b = this\n }\n\n elements = Object.keys(a.elements)\n\n for (var i = 0; i < elements.length; i++) {\n var element = elements[i]\n if (element in b.elements) {\n intersection.push(element)\n }\n }\n\n return new lunr.Set (intersection)\n}\n\n/**\n * Returns a new set combining the elements of this and the specified set.\n *\n * @param {lunr.Set} other - set to union with this set.\n * @return {lunr.Set} a new set that is the union of this and the specified set.\n */\n\nlunr.Set.prototype.union = function (other) {\n if (other === lunr.Set.complete) {\n return lunr.Set.complete\n }\n\n if (other === lunr.Set.empty) {\n return this\n }\n\n return new lunr.Set(Object.keys(this.elements).concat(Object.keys(other.elements)))\n}\n/**\n * A function to calculate the inverse document frequency for\n * a posting. This is shared between the builder and the index\n *\n * @private\n * @param {object} posting - The posting for a given term\n * @param {number} documentCount - The total number of documents.\n */\nlunr.idf = function (posting, documentCount) {\n var documentsWithTerm = 0\n\n for (var fieldName in posting) {\n if (fieldName == '_index') continue // Ignore the term index, its not a field\n documentsWithTerm += Object.keys(posting[fieldName]).length\n }\n\n var x = (documentCount - documentsWithTerm + 0.5) / (documentsWithTerm + 0.5)\n\n return Math.log(1 + Math.abs(x))\n}\n\n/**\n * A token wraps a string representation of a token\n * as it is passed through the text processing pipeline.\n *\n * @constructor\n * @param {string} [str=''] - The string token being wrapped.\n * @param {object} [metadata={}] - Metadata associated with this token.\n */\nlunr.Token = function (str, metadata) {\n this.str = str || \"\"\n this.metadata = metadata || {}\n}\n\n/**\n * Returns the token string that is being wrapped by this object.\n *\n * @returns {string}\n */\nlunr.Token.prototype.toString = function () {\n return this.str\n}\n\n/**\n * A token update function is used when updating or optionally\n * when cloning a token.\n *\n * @callback lunr.Token~updateFunction\n * @param {string} str - The string representation of the token.\n * @param {Object} metadata - All metadata associated with this token.\n */\n\n/**\n * Applies the given function to the wrapped string token.\n *\n * @example\n * token.update(function (str, metadata) {\n * return str.toUpperCase()\n * })\n *\n * @param {lunr.Token~updateFunction} fn - A function to apply to the token string.\n * @returns {lunr.Token}\n */\nlunr.Token.prototype.update = function (fn) {\n this.str = fn(this.str, this.metadata)\n return this\n}\n\n/**\n * Creates a clone of this token. Optionally a function can be\n * applied to the cloned token.\n *\n * @param {lunr.Token~updateFunction} [fn] - An optional function to apply to the cloned token.\n * @returns {lunr.Token}\n */\nlunr.Token.prototype.clone = function (fn) {\n fn = fn || function (s) { return s }\n return new lunr.Token (fn(this.str, this.metadata), this.metadata)\n}\n/*!\n * lunr.tokenizer\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A function for splitting a string into tokens ready to be inserted into\n * the search index. Uses `lunr.tokenizer.separator` to split strings, change\n * the value of this property to change how strings are split into tokens.\n *\n * This tokenizer will convert its parameter to a string by calling `toString` and\n * then will split this string on the character in `lunr.tokenizer.separator`.\n * Arrays will have their elements converted to strings and wrapped in a lunr.Token.\n *\n * Optional metadata can be passed to the tokenizer, this metadata will be cloned and\n * added as metadata to every token that is created from the object to be tokenized.\n *\n * @static\n * @param {?(string|object|object[])} obj - The object to convert into tokens\n * @param {?object} metadata - Optional metadata to associate with every token\n * @returns {lunr.Token[]}\n * @see {@link lunr.Pipeline}\n */\nlunr.tokenizer = function (obj, metadata) {\n if (obj == null || obj == undefined) {\n return []\n }\n\n if (Array.isArray(obj)) {\n return obj.map(function (t) {\n return new lunr.Token(\n lunr.utils.asString(t).toLowerCase(),\n lunr.utils.clone(metadata)\n )\n })\n }\n\n var str = obj.toString().toLowerCase(),\n len = str.length,\n tokens = []\n\n for (var sliceEnd = 0, sliceStart = 0; sliceEnd <= len; sliceEnd++) {\n var char = str.charAt(sliceEnd),\n sliceLength = sliceEnd - sliceStart\n\n if ((char.match(lunr.tokenizer.separator) || sliceEnd == len)) {\n\n if (sliceLength > 0) {\n var tokenMetadata = lunr.utils.clone(metadata) || {}\n tokenMetadata[\"position\"] = [sliceStart, sliceLength]\n tokenMetadata[\"index\"] = tokens.length\n\n tokens.push(\n new lunr.Token (\n str.slice(sliceStart, sliceEnd),\n tokenMetadata\n )\n )\n }\n\n sliceStart = sliceEnd + 1\n }\n\n }\n\n return tokens\n}\n\n/**\n * The separator used to split a string into tokens. Override this property to change the behaviour of\n * `lunr.tokenizer` behaviour when tokenizing strings. By default this splits on whitespace and hyphens.\n *\n * @static\n * @see lunr.tokenizer\n */\nlunr.tokenizer.separator = /[\\s\\-]+/\n/*!\n * lunr.Pipeline\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.Pipelines maintain an ordered list of functions to be applied to all\n * tokens in documents entering the search index and queries being ran against\n * the index.\n *\n * An instance of lunr.Index created with the lunr shortcut will contain a\n * pipeline with a stop word filter and an English language stemmer. Extra\n * functions can be added before or after either of these functions or these\n * default functions can be removed.\n *\n * When run the pipeline will call each function in turn, passing a token, the\n * index of that token in the original list of all tokens and finally a list of\n * all the original tokens.\n *\n * The output of functions in the pipeline will be passed to the next function\n * in the pipeline. To exclude a token from entering the index the function\n * should return undefined, the rest of the pipeline will not be called with\n * this token.\n *\n * For serialisation of pipelines to work, all functions used in an instance of\n * a pipeline should be registered with lunr.Pipeline. Registered functions can\n * then be loaded. If trying to load a serialised pipeline that uses functions\n * that are not registered an error will be thrown.\n *\n * If not planning on serialising the pipeline then registering pipeline functions\n * is not necessary.\n *\n * @constructor\n */\nlunr.Pipeline = function () {\n this._stack = []\n}\n\nlunr.Pipeline.registeredFunctions = Object.create(null)\n\n/**\n * A pipeline function maps lunr.Token to lunr.Token. A lunr.Token contains the token\n * string as well as all known metadata. A pipeline function can mutate the token string\n * or mutate (or add) metadata for a given token.\n *\n * A pipeline function can indicate that the passed token should be discarded by returning\n * null, undefined or an empty string. This token will not be passed to any downstream pipeline\n * functions and will not be added to the index.\n *\n * Multiple tokens can be returned by returning an array of tokens. Each token will be passed\n * to any downstream pipeline functions and all will returned tokens will be added to the index.\n *\n * Any number of pipeline functions may be chained together using a lunr.Pipeline.\n *\n * @interface lunr.PipelineFunction\n * @param {lunr.Token} token - A token from the document being processed.\n * @param {number} i - The index of this token in the complete list of tokens for this document/field.\n * @param {lunr.Token[]} tokens - All tokens for this document/field.\n * @returns {(?lunr.Token|lunr.Token[])}\n */\n\n/**\n * Register a function with the pipeline.\n *\n * Functions that are used in the pipeline should be registered if the pipeline\n * needs to be serialised, or a serialised pipeline needs to be loaded.\n *\n * Registering a function does not add it to a pipeline, functions must still be\n * added to instances of the pipeline for them to be used when running a pipeline.\n *\n * @param {lunr.PipelineFunction} fn - The function to check for.\n * @param {String} label - The label to register this function with\n */\nlunr.Pipeline.registerFunction = function (fn, label) {\n if (label in this.registeredFunctions) {\n lunr.utils.warn('Overwriting existing registered function: ' + label)\n }\n\n fn.label = label\n lunr.Pipeline.registeredFunctions[fn.label] = fn\n}\n\n/**\n * Warns if the function is not registered as a Pipeline function.\n *\n * @param {lunr.PipelineFunction} fn - The function to check for.\n * @private\n */\nlunr.Pipeline.warnIfFunctionNotRegistered = function (fn) {\n var isRegistered = fn.label && (fn.label in this.registeredFunctions)\n\n if (!isRegistered) {\n lunr.utils.warn('Function is not registered with pipeline. This may cause problems when serialising the index.\\n', fn)\n }\n}\n\n/**\n * Loads a previously serialised pipeline.\n *\n * All functions to be loaded must already be registered with lunr.Pipeline.\n * If any function from the serialised data has not been registered then an\n * error will be thrown.\n *\n * @param {Object} serialised - The serialised pipeline to load.\n * @returns {lunr.Pipeline}\n */\nlunr.Pipeline.load = function (serialised) {\n var pipeline = new lunr.Pipeline\n\n serialised.forEach(function (fnName) {\n var fn = lunr.Pipeline.registeredFunctions[fnName]\n\n if (fn) {\n pipeline.add(fn)\n } else {\n throw new Error('Cannot load unregistered function: ' + fnName)\n }\n })\n\n return pipeline\n}\n\n/**\n * Adds new functions to the end of the pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction[]} functions - Any number of functions to add to the pipeline.\n */\nlunr.Pipeline.prototype.add = function () {\n var fns = Array.prototype.slice.call(arguments)\n\n fns.forEach(function (fn) {\n lunr.Pipeline.warnIfFunctionNotRegistered(fn)\n this._stack.push(fn)\n }, this)\n}\n\n/**\n * Adds a single function after a function that already exists in the\n * pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline.\n * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline.\n */\nlunr.Pipeline.prototype.after = function (existingFn, newFn) {\n lunr.Pipeline.warnIfFunctionNotRegistered(newFn)\n\n var pos = this._stack.indexOf(existingFn)\n if (pos == -1) {\n throw new Error('Cannot find existingFn')\n }\n\n pos = pos + 1\n this._stack.splice(pos, 0, newFn)\n}\n\n/**\n * Adds a single function before a function that already exists in the\n * pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline.\n * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline.\n */\nlunr.Pipeline.prototype.before = function (existingFn, newFn) {\n lunr.Pipeline.warnIfFunctionNotRegistered(newFn)\n\n var pos = this._stack.indexOf(existingFn)\n if (pos == -1) {\n throw new Error('Cannot find existingFn')\n }\n\n this._stack.splice(pos, 0, newFn)\n}\n\n/**\n * Removes a function from the pipeline.\n *\n * @param {lunr.PipelineFunction} fn The function to remove from the pipeline.\n */\nlunr.Pipeline.prototype.remove = function (fn) {\n var pos = this._stack.indexOf(fn)\n if (pos == -1) {\n return\n }\n\n this._stack.splice(pos, 1)\n}\n\n/**\n * Runs the current list of functions that make up the pipeline against the\n * passed tokens.\n *\n * @param {Array} tokens The tokens to run through the pipeline.\n * @returns {Array}\n */\nlunr.Pipeline.prototype.run = function (tokens) {\n var stackLength = this._stack.length\n\n for (var i = 0; i < stackLength; i++) {\n var fn = this._stack[i]\n var memo = []\n\n for (var j = 0; j < tokens.length; j++) {\n var result = fn(tokens[j], j, tokens)\n\n if (result === null || result === void 0 || result === '') continue\n\n if (Array.isArray(result)) {\n for (var k = 0; k < result.length; k++) {\n memo.push(result[k])\n }\n } else {\n memo.push(result)\n }\n }\n\n tokens = memo\n }\n\n return tokens\n}\n\n/**\n * Convenience method for passing a string through a pipeline and getting\n * strings out. This method takes care of wrapping the passed string in a\n * token and mapping the resulting tokens back to strings.\n *\n * @param {string} str - The string to pass through the pipeline.\n * @param {?object} metadata - Optional metadata to associate with the token\n * passed to the pipeline.\n * @returns {string[]}\n */\nlunr.Pipeline.prototype.runString = function (str, metadata) {\n var token = new lunr.Token (str, metadata)\n\n return this.run([token]).map(function (t) {\n return t.toString()\n })\n}\n\n/**\n * Resets the pipeline by removing any existing processors.\n *\n */\nlunr.Pipeline.prototype.reset = function () {\n this._stack = []\n}\n\n/**\n * Returns a representation of the pipeline ready for serialisation.\n *\n * Logs a warning if the function has not been registered.\n *\n * @returns {Array}\n */\nlunr.Pipeline.prototype.toJSON = function () {\n return this._stack.map(function (fn) {\n lunr.Pipeline.warnIfFunctionNotRegistered(fn)\n\n return fn.label\n })\n}\n/*!\n * lunr.Vector\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A vector is used to construct the vector space of documents and queries. These\n * vectors support operations to determine the similarity between two documents or\n * a document and a query.\n *\n * Normally no parameters are required for initializing a vector, but in the case of\n * loading a previously dumped vector the raw elements can be provided to the constructor.\n *\n * For performance reasons vectors are implemented with a flat array, where an elements\n * index is immediately followed by its value. E.g. [index, value, index, value]. This\n * allows the underlying array to be as sparse as possible and still offer decent\n * performance when being used for vector calculations.\n *\n * @constructor\n * @param {Number[]} [elements] - The flat list of element index and element value pairs.\n */\nlunr.Vector = function (elements) {\n this._magnitude = 0\n this.elements = elements || []\n}\n\n\n/**\n * Calculates the position within the vector to insert a given index.\n *\n * This is used internally by insert and upsert. If there are duplicate indexes then\n * the position is returned as if the value for that index were to be updated, but it\n * is the callers responsibility to check whether there is a duplicate at that index\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @returns {Number}\n */\nlunr.Vector.prototype.positionForIndex = function (index) {\n // For an empty vector the tuple can be inserted at the beginning\n if (this.elements.length == 0) {\n return 0\n }\n\n var start = 0,\n end = this.elements.length / 2,\n sliceLength = end - start,\n pivotPoint = Math.floor(sliceLength / 2),\n pivotIndex = this.elements[pivotPoint * 2]\n\n while (sliceLength > 1) {\n if (pivotIndex < index) {\n start = pivotPoint\n }\n\n if (pivotIndex > index) {\n end = pivotPoint\n }\n\n if (pivotIndex == index) {\n break\n }\n\n sliceLength = end - start\n pivotPoint = start + Math.floor(sliceLength / 2)\n pivotIndex = this.elements[pivotPoint * 2]\n }\n\n if (pivotIndex == index) {\n return pivotPoint * 2\n }\n\n if (pivotIndex > index) {\n return pivotPoint * 2\n }\n\n if (pivotIndex < index) {\n return (pivotPoint + 1) * 2\n }\n}\n\n/**\n * Inserts an element at an index within the vector.\n *\n * Does not allow duplicates, will throw an error if there is already an entry\n * for this index.\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @param {Number} val - The value to be inserted into the vector.\n */\nlunr.Vector.prototype.insert = function (insertIdx, val) {\n this.upsert(insertIdx, val, function () {\n throw \"duplicate index\"\n })\n}\n\n/**\n * Inserts or updates an existing index within the vector.\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @param {Number} val - The value to be inserted into the vector.\n * @param {function} fn - A function that is called for updates, the existing value and the\n * requested value are passed as arguments\n */\nlunr.Vector.prototype.upsert = function (insertIdx, val, fn) {\n this._magnitude = 0\n var position = this.positionForIndex(insertIdx)\n\n if (this.elements[position] == insertIdx) {\n this.elements[position + 1] = fn(this.elements[position + 1], val)\n } else {\n this.elements.splice(position, 0, insertIdx, val)\n }\n}\n\n/**\n * Calculates the magnitude of this vector.\n *\n * @returns {Number}\n */\nlunr.Vector.prototype.magnitude = function () {\n if (this._magnitude) return this._magnitude\n\n var sumOfSquares = 0,\n elementsLength = this.elements.length\n\n for (var i = 1; i < elementsLength; i += 2) {\n var val = this.elements[i]\n sumOfSquares += val * val\n }\n\n return this._magnitude = Math.sqrt(sumOfSquares)\n}\n\n/**\n * Calculates the dot product of this vector and another vector.\n *\n * @param {lunr.Vector} otherVector - The vector to compute the dot product with.\n * @returns {Number}\n */\nlunr.Vector.prototype.dot = function (otherVector) {\n var dotProduct = 0,\n a = this.elements, b = otherVector.elements,\n aLen = a.length, bLen = b.length,\n aVal = 0, bVal = 0,\n i = 0, j = 0\n\n while (i < aLen && j < bLen) {\n aVal = a[i], bVal = b[j]\n if (aVal < bVal) {\n i += 2\n } else if (aVal > bVal) {\n j += 2\n } else if (aVal == bVal) {\n dotProduct += a[i + 1] * b[j + 1]\n i += 2\n j += 2\n }\n }\n\n return dotProduct\n}\n\n/**\n * Calculates the similarity between this vector and another vector.\n *\n * @param {lunr.Vector} otherVector - The other vector to calculate the\n * similarity with.\n * @returns {Number}\n */\nlunr.Vector.prototype.similarity = function (otherVector) {\n return this.dot(otherVector) / this.magnitude() || 0\n}\n\n/**\n * Converts the vector to an array of the elements within the vector.\n *\n * @returns {Number[]}\n */\nlunr.Vector.prototype.toArray = function () {\n var output = new Array (this.elements.length / 2)\n\n for (var i = 1, j = 0; i < this.elements.length; i += 2, j++) {\n output[j] = this.elements[i]\n }\n\n return output\n}\n\n/**\n * A JSON serializable representation of the vector.\n *\n * @returns {Number[]}\n */\nlunr.Vector.prototype.toJSON = function () {\n return this.elements\n}\n/* eslint-disable */\n/*!\n * lunr.stemmer\n * Copyright (C) 2020 Oliver Nightingale\n * Includes code from - http://tartarus.org/~martin/PorterStemmer/js.txt\n */\n\n/**\n * lunr.stemmer is an english language stemmer, this is a JavaScript\n * implementation of the PorterStemmer taken from http://tartarus.org/~martin\n *\n * @static\n * @implements {lunr.PipelineFunction}\n * @param {lunr.Token} token - The string to stem\n * @returns {lunr.Token}\n * @see {@link lunr.Pipeline}\n * @function\n */\nlunr.stemmer = (function(){\n var step2list = {\n \"ational\" : \"ate\",\n \"tional\" : \"tion\",\n \"enci\" : \"ence\",\n \"anci\" : \"ance\",\n \"izer\" : \"ize\",\n \"bli\" : \"ble\",\n \"alli\" : \"al\",\n \"entli\" : \"ent\",\n \"eli\" : \"e\",\n \"ousli\" : \"ous\",\n \"ization\" : \"ize\",\n \"ation\" : \"ate\",\n \"ator\" : \"ate\",\n \"alism\" : \"al\",\n \"iveness\" : \"ive\",\n \"fulness\" : \"ful\",\n \"ousness\" : \"ous\",\n \"aliti\" : \"al\",\n \"iviti\" : \"ive\",\n \"biliti\" : \"ble\",\n \"logi\" : \"log\"\n },\n\n step3list = {\n \"icate\" : \"ic\",\n \"ative\" : \"\",\n \"alize\" : \"al\",\n \"iciti\" : \"ic\",\n \"ical\" : \"ic\",\n \"ful\" : \"\",\n \"ness\" : \"\"\n },\n\n c = \"[^aeiou]\", // consonant\n v = \"[aeiouy]\", // vowel\n C = c + \"[^aeiouy]*\", // consonant sequence\n V = v + \"[aeiou]*\", // vowel sequence\n\n mgr0 = \"^(\" + C + \")?\" + V + C, // [C]VC... is m>0\n meq1 = \"^(\" + C + \")?\" + V + C + \"(\" + V + \")?$\", // [C]VC[V] is m=1\n mgr1 = \"^(\" + C + \")?\" + V + C + V + C, // [C]VCVC... is m>1\n s_v = \"^(\" + C + \")?\" + v; // vowel in stem\n\n var re_mgr0 = new RegExp(mgr0);\n var re_mgr1 = new RegExp(mgr1);\n var re_meq1 = new RegExp(meq1);\n var re_s_v = new RegExp(s_v);\n\n var re_1a = /^(.+?)(ss|i)es$/;\n var re2_1a = /^(.+?)([^s])s$/;\n var re_1b = /^(.+?)eed$/;\n var re2_1b = /^(.+?)(ed|ing)$/;\n var re_1b_2 = /.$/;\n var re2_1b_2 = /(at|bl|iz)$/;\n var re3_1b_2 = new RegExp(\"([^aeiouylsz])\\\\1$\");\n var re4_1b_2 = new RegExp(\"^\" + C + v + \"[^aeiouwxy]$\");\n\n var re_1c = /^(.+?[^aeiou])y$/;\n var re_2 = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;\n\n var re_3 = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;\n\n var re_4 = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;\n var re2_4 = /^(.+?)(s|t)(ion)$/;\n\n var re_5 = /^(.+?)e$/;\n var re_5_1 = /ll$/;\n var re3_5 = new RegExp(\"^\" + C + v + \"[^aeiouwxy]$\");\n\n var porterStemmer = function porterStemmer(w) {\n var stem,\n suffix,\n firstch,\n re,\n re2,\n re3,\n re4;\n\n if (w.length < 3) { return w; }\n\n firstch = w.substr(0,1);\n if (firstch == \"y\") {\n w = firstch.toUpperCase() + w.substr(1);\n }\n\n // Step 1a\n re = re_1a\n re2 = re2_1a;\n\n if (re.test(w)) { w = w.replace(re,\"$1$2\"); }\n else if (re2.test(w)) { w = w.replace(re2,\"$1$2\"); }\n\n // Step 1b\n re = re_1b;\n re2 = re2_1b;\n if (re.test(w)) {\n var fp = re.exec(w);\n re = re_mgr0;\n if (re.test(fp[1])) {\n re = re_1b_2;\n w = w.replace(re,\"\");\n }\n } else if (re2.test(w)) {\n var fp = re2.exec(w);\n stem = fp[1];\n re2 = re_s_v;\n if (re2.test(stem)) {\n w = stem;\n re2 = re2_1b_2;\n re3 = re3_1b_2;\n re4 = re4_1b_2;\n if (re2.test(w)) { w = w + \"e\"; }\n else if (re3.test(w)) { re = re_1b_2; w = w.replace(re,\"\"); }\n else if (re4.test(w)) { w = w + \"e\"; }\n }\n }\n\n // Step 1c - replace suffix y or Y by i if preceded by a non-vowel which is not the first letter of the word (so cry -> cri, by -> by, say -> say)\n re = re_1c;\n if (re.test(w)) {\n var fp = re.exec(w);\n stem = fp[1];\n w = stem + \"i\";\n }\n\n // Step 2\n re = re_2;\n if (re.test(w)) {\n var fp = re.exec(w);\n stem = fp[1];\n suffix = fp[2];\n re = re_mgr0;\n if (re.test(stem)) {\n w = stem + step2list[suffix];\n }\n }\n\n // Step 3\n re = re_3;\n if (re.test(w)) {\n var fp = re.exec(w);\n stem = fp[1];\n suffix = fp[2];\n re = re_mgr0;\n if (re.test(stem)) {\n w = stem + step3list[suffix];\n }\n }\n\n // Step 4\n re = re_4;\n re2 = re2_4;\n if (re.test(w)) {\n var fp = re.exec(w);\n stem = fp[1];\n re = re_mgr1;\n if (re.test(stem)) {\n w = stem;\n }\n } else if (re2.test(w)) {\n var fp = re2.exec(w);\n stem = fp[1] + fp[2];\n re2 = re_mgr1;\n if (re2.test(stem)) {\n w = stem;\n }\n }\n\n // Step 5\n re = re_5;\n if (re.test(w)) {\n var fp = re.exec(w);\n stem = fp[1];\n re = re_mgr1;\n re2 = re_meq1;\n re3 = re3_5;\n if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) {\n w = stem;\n }\n }\n\n re = re_5_1;\n re2 = re_mgr1;\n if (re.test(w) && re2.test(w)) {\n re = re_1b_2;\n w = w.replace(re,\"\");\n }\n\n // and turn initial Y back to y\n\n if (firstch == \"y\") {\n w = firstch.toLowerCase() + w.substr(1);\n }\n\n return w;\n };\n\n return function (token) {\n return token.update(porterStemmer);\n }\n})();\n\nlunr.Pipeline.registerFunction(lunr.stemmer, 'stemmer')\n/*!\n * lunr.stopWordFilter\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.generateStopWordFilter builds a stopWordFilter function from the provided\n * list of stop words.\n *\n * The built in lunr.stopWordFilter is built using this generator and can be used\n * to generate custom stopWordFilters for applications or non English languages.\n *\n * @function\n * @param {Array} token The token to pass through the filter\n * @returns {lunr.PipelineFunction}\n * @see lunr.Pipeline\n * @see lunr.stopWordFilter\n */\nlunr.generateStopWordFilter = function (stopWords) {\n var words = stopWords.reduce(function (memo, stopWord) {\n memo[stopWord] = stopWord\n return memo\n }, {})\n\n return function (token) {\n if (token && words[token.toString()] !== token.toString()) return token\n }\n}\n\n/**\n * lunr.stopWordFilter is an English language stop word list filter, any words\n * contained in the list will not be passed through the filter.\n *\n * This is intended to be used in the Pipeline. If the token does not pass the\n * filter then undefined will be returned.\n *\n * @function\n * @implements {lunr.PipelineFunction}\n * @params {lunr.Token} token - A token to check for being a stop word.\n * @returns {lunr.Token}\n * @see {@link lunr.Pipeline}\n */\nlunr.stopWordFilter = lunr.generateStopWordFilter([\n 'a',\n 'able',\n 'about',\n 'across',\n 'after',\n 'all',\n 'almost',\n 'also',\n 'am',\n 'among',\n 'an',\n 'and',\n 'any',\n 'are',\n 'as',\n 'at',\n 'be',\n 'because',\n 'been',\n 'but',\n 'by',\n 'can',\n 'cannot',\n 'could',\n 'dear',\n 'did',\n 'do',\n 'does',\n 'either',\n 'else',\n 'ever',\n 'every',\n 'for',\n 'from',\n 'get',\n 'got',\n 'had',\n 'has',\n 'have',\n 'he',\n 'her',\n 'hers',\n 'him',\n 'his',\n 'how',\n 'however',\n 'i',\n 'if',\n 'in',\n 'into',\n 'is',\n 'it',\n 'its',\n 'just',\n 'least',\n 'let',\n 'like',\n 'likely',\n 'may',\n 'me',\n 'might',\n 'most',\n 'must',\n 'my',\n 'neither',\n 'no',\n 'nor',\n 'not',\n 'of',\n 'off',\n 'often',\n 'on',\n 'only',\n 'or',\n 'other',\n 'our',\n 'own',\n 'rather',\n 'said',\n 'say',\n 'says',\n 'she',\n 'should',\n 'since',\n 'so',\n 'some',\n 'than',\n 'that',\n 'the',\n 'their',\n 'them',\n 'then',\n 'there',\n 'these',\n 'they',\n 'this',\n 'tis',\n 'to',\n 'too',\n 'twas',\n 'us',\n 'wants',\n 'was',\n 'we',\n 'were',\n 'what',\n 'when',\n 'where',\n 'which',\n 'while',\n 'who',\n 'whom',\n 'why',\n 'will',\n 'with',\n 'would',\n 'yet',\n 'you',\n 'your'\n])\n\nlunr.Pipeline.registerFunction(lunr.stopWordFilter, 'stopWordFilter')\n/*!\n * lunr.trimmer\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.trimmer is a pipeline function for trimming non word\n * characters from the beginning and end of tokens before they\n * enter the index.\n *\n * This implementation may not work correctly for non latin\n * characters and should either be removed or adapted for use\n * with languages with non-latin characters.\n *\n * @static\n * @implements {lunr.PipelineFunction}\n * @param {lunr.Token} token The token to pass through the filter\n * @returns {lunr.Token}\n * @see lunr.Pipeline\n */\nlunr.trimmer = function (token) {\n return token.update(function (s) {\n return s.replace(/^\\W+/, '').replace(/\\W+$/, '')\n })\n}\n\nlunr.Pipeline.registerFunction(lunr.trimmer, 'trimmer')\n/*!\n * lunr.TokenSet\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A token set is used to store the unique list of all tokens\n * within an index. Token sets are also used to represent an\n * incoming query to the index, this query token set and index\n * token set are then intersected to find which tokens to look\n * up in the inverted index.\n *\n * A token set can hold multiple tokens, as in the case of the\n * index token set, or it can hold a single token as in the\n * case of a simple query token set.\n *\n * Additionally token sets are used to perform wildcard matching.\n * Leading, contained and trailing wildcards are supported, and\n * from this edit distance matching can also be provided.\n *\n * Token sets are implemented as a minimal finite state automata,\n * where both common prefixes and suffixes are shared between tokens.\n * This helps to reduce the space used for storing the token set.\n *\n * @constructor\n */\nlunr.TokenSet = function () {\n this.final = false\n this.edges = {}\n this.id = lunr.TokenSet._nextId\n lunr.TokenSet._nextId += 1\n}\n\n/**\n * Keeps track of the next, auto increment, identifier to assign\n * to a new tokenSet.\n *\n * TokenSets require a unique identifier to be correctly minimised.\n *\n * @private\n */\nlunr.TokenSet._nextId = 1\n\n/**\n * Creates a TokenSet instance from the given sorted array of words.\n *\n * @param {String[]} arr - A sorted array of strings to create the set from.\n * @returns {lunr.TokenSet}\n * @throws Will throw an error if the input array is not sorted.\n */\nlunr.TokenSet.fromArray = function (arr) {\n var builder = new lunr.TokenSet.Builder\n\n for (var i = 0, len = arr.length; i < len; i++) {\n builder.insert(arr[i])\n }\n\n builder.finish()\n return builder.root\n}\n\n/**\n * Creates a token set from a query clause.\n *\n * @private\n * @param {Object} clause - A single clause from lunr.Query.\n * @param {string} clause.term - The query clause term.\n * @param {number} [clause.editDistance] - The optional edit distance for the term.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.fromClause = function (clause) {\n if ('editDistance' in clause) {\n return lunr.TokenSet.fromFuzzyString(clause.term, clause.editDistance)\n } else {\n return lunr.TokenSet.fromString(clause.term)\n }\n}\n\n/**\n * Creates a token set representing a single string with a specified\n * edit distance.\n *\n * Insertions, deletions, substitutions and transpositions are each\n * treated as an edit distance of 1.\n *\n * Increasing the allowed edit distance will have a dramatic impact\n * on the performance of both creating and intersecting these TokenSets.\n * It is advised to keep the edit distance less than 3.\n *\n * @param {string} str - The string to create the token set from.\n * @param {number} editDistance - The allowed edit distance to match.\n * @returns {lunr.Vector}\n */\nlunr.TokenSet.fromFuzzyString = function (str, editDistance) {\n var root = new lunr.TokenSet\n\n var stack = [{\n node: root,\n editsRemaining: editDistance,\n str: str\n }]\n\n while (stack.length) {\n var frame = stack.pop()\n\n // no edit\n if (frame.str.length > 0) {\n var char = frame.str.charAt(0),\n noEditNode\n\n if (char in frame.node.edges) {\n noEditNode = frame.node.edges[char]\n } else {\n noEditNode = new lunr.TokenSet\n frame.node.edges[char] = noEditNode\n }\n\n if (frame.str.length == 1) {\n noEditNode.final = true\n }\n\n stack.push({\n node: noEditNode,\n editsRemaining: frame.editsRemaining,\n str: frame.str.slice(1)\n })\n }\n\n if (frame.editsRemaining == 0) {\n continue\n }\n\n // insertion\n if (\"*\" in frame.node.edges) {\n var insertionNode = frame.node.edges[\"*\"]\n } else {\n var insertionNode = new lunr.TokenSet\n frame.node.edges[\"*\"] = insertionNode\n }\n\n if (frame.str.length == 0) {\n insertionNode.final = true\n }\n\n stack.push({\n node: insertionNode,\n editsRemaining: frame.editsRemaining - 1,\n str: frame.str\n })\n\n // deletion\n // can only do a deletion if we have enough edits remaining\n // and if there are characters left to delete in the string\n if (frame.str.length > 1) {\n stack.push({\n node: frame.node,\n editsRemaining: frame.editsRemaining - 1,\n str: frame.str.slice(1)\n })\n }\n\n // deletion\n // just removing the last character from the str\n if (frame.str.length == 1) {\n frame.node.final = true\n }\n\n // substitution\n // can only do a substitution if we have enough edits remaining\n // and if there are characters left to substitute\n if (frame.str.length >= 1) {\n if (\"*\" in frame.node.edges) {\n var substitutionNode = frame.node.edges[\"*\"]\n } else {\n var substitutionNode = new lunr.TokenSet\n frame.node.edges[\"*\"] = substitutionNode\n }\n\n if (frame.str.length == 1) {\n substitutionNode.final = true\n }\n\n stack.push({\n node: substitutionNode,\n editsRemaining: frame.editsRemaining - 1,\n str: frame.str.slice(1)\n })\n }\n\n // transposition\n // can only do a transposition if there are edits remaining\n // and there are enough characters to transpose\n if (frame.str.length > 1) {\n var charA = frame.str.charAt(0),\n charB = frame.str.charAt(1),\n transposeNode\n\n if (charB in frame.node.edges) {\n transposeNode = frame.node.edges[charB]\n } else {\n transposeNode = new lunr.TokenSet\n frame.node.edges[charB] = transposeNode\n }\n\n if (frame.str.length == 1) {\n transposeNode.final = true\n }\n\n stack.push({\n node: transposeNode,\n editsRemaining: frame.editsRemaining - 1,\n str: charA + frame.str.slice(2)\n })\n }\n }\n\n return root\n}\n\n/**\n * Creates a TokenSet from a string.\n *\n * The string may contain one or more wildcard characters (*)\n * that will allow wildcard matching when intersecting with\n * another TokenSet.\n *\n * @param {string} str - The string to create a TokenSet from.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.fromString = function (str) {\n var node = new lunr.TokenSet,\n root = node\n\n /*\n * Iterates through all characters within the passed string\n * appending a node for each character.\n *\n * When a wildcard character is found then a self\n * referencing edge is introduced to continually match\n * any number of any characters.\n */\n for (var i = 0, len = str.length; i < len; i++) {\n var char = str[i],\n final = (i == len - 1)\n\n if (char == \"*\") {\n node.edges[char] = node\n node.final = final\n\n } else {\n var next = new lunr.TokenSet\n next.final = final\n\n node.edges[char] = next\n node = next\n }\n }\n\n return root\n}\n\n/**\n * Converts this TokenSet into an array of strings\n * contained within the TokenSet.\n *\n * This is not intended to be used on a TokenSet that\n * contains wildcards, in these cases the results are\n * undefined and are likely to cause an infinite loop.\n *\n * @returns {string[]}\n */\nlunr.TokenSet.prototype.toArray = function () {\n var words = []\n\n var stack = [{\n prefix: \"\",\n node: this\n }]\n\n while (stack.length) {\n var frame = stack.pop(),\n edges = Object.keys(frame.node.edges),\n len = edges.length\n\n if (frame.node.final) {\n /* In Safari, at this point the prefix is sometimes corrupted, see:\n * https://github.com/olivernn/lunr.js/issues/279 Calling any\n * String.prototype method forces Safari to \"cast\" this string to what\n * it's supposed to be, fixing the bug. */\n frame.prefix.charAt(0)\n words.push(frame.prefix)\n }\n\n for (var i = 0; i < len; i++) {\n var edge = edges[i]\n\n stack.push({\n prefix: frame.prefix.concat(edge),\n node: frame.node.edges[edge]\n })\n }\n }\n\n return words\n}\n\n/**\n * Generates a string representation of a TokenSet.\n *\n * This is intended to allow TokenSets to be used as keys\n * in objects, largely to aid the construction and minimisation\n * of a TokenSet. As such it is not designed to be a human\n * friendly representation of the TokenSet.\n *\n * @returns {string}\n */\nlunr.TokenSet.prototype.toString = function () {\n // NOTE: Using Object.keys here as this.edges is very likely\n // to enter 'hash-mode' with many keys being added\n //\n // avoiding a for-in loop here as it leads to the function\n // being de-optimised (at least in V8). From some simple\n // benchmarks the performance is comparable, but allowing\n // V8 to optimize may mean easy performance wins in the future.\n\n if (this._str) {\n return this._str\n }\n\n var str = this.final ? '1' : '0',\n labels = Object.keys(this.edges).sort(),\n len = labels.length\n\n for (var i = 0; i < len; i++) {\n var label = labels[i],\n node = this.edges[label]\n\n str = str + label + node.id\n }\n\n return str\n}\n\n/**\n * Returns a new TokenSet that is the intersection of\n * this TokenSet and the passed TokenSet.\n *\n * This intersection will take into account any wildcards\n * contained within the TokenSet.\n *\n * @param {lunr.TokenSet} b - An other TokenSet to intersect with.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.prototype.intersect = function (b) {\n var output = new lunr.TokenSet,\n frame = undefined\n\n var stack = [{\n qNode: b,\n output: output,\n node: this\n }]\n\n while (stack.length) {\n frame = stack.pop()\n\n // NOTE: As with the #toString method, we are using\n // Object.keys and a for loop instead of a for-in loop\n // as both of these objects enter 'hash' mode, causing\n // the function to be de-optimised in V8\n var qEdges = Object.keys(frame.qNode.edges),\n qLen = qEdges.length,\n nEdges = Object.keys(frame.node.edges),\n nLen = nEdges.length\n\n for (var q = 0; q < qLen; q++) {\n var qEdge = qEdges[q]\n\n for (var n = 0; n < nLen; n++) {\n var nEdge = nEdges[n]\n\n if (nEdge == qEdge || qEdge == '*') {\n var node = frame.node.edges[nEdge],\n qNode = frame.qNode.edges[qEdge],\n final = node.final && qNode.final,\n next = undefined\n\n if (nEdge in frame.output.edges) {\n // an edge already exists for this character\n // no need to create a new node, just set the finality\n // bit unless this node is already final\n next = frame.output.edges[nEdge]\n next.final = next.final || final\n\n } else {\n // no edge exists yet, must create one\n // set the finality bit and insert it\n // into the output\n next = new lunr.TokenSet\n next.final = final\n frame.output.edges[nEdge] = next\n }\n\n stack.push({\n qNode: qNode,\n output: next,\n node: node\n })\n }\n }\n }\n }\n\n return output\n}\nlunr.TokenSet.Builder = function () {\n this.previousWord = \"\"\n this.root = new lunr.TokenSet\n this.uncheckedNodes = []\n this.minimizedNodes = {}\n}\n\nlunr.TokenSet.Builder.prototype.insert = function (word) {\n var node,\n commonPrefix = 0\n\n if (word < this.previousWord) {\n throw new Error (\"Out of order word insertion\")\n }\n\n for (var i = 0; i < word.length && i < this.previousWord.length; i++) {\n if (word[i] != this.previousWord[i]) break\n commonPrefix++\n }\n\n this.minimize(commonPrefix)\n\n if (this.uncheckedNodes.length == 0) {\n node = this.root\n } else {\n node = this.uncheckedNodes[this.uncheckedNodes.length - 1].child\n }\n\n for (var i = commonPrefix; i < word.length; i++) {\n var nextNode = new lunr.TokenSet,\n char = word[i]\n\n node.edges[char] = nextNode\n\n this.uncheckedNodes.push({\n parent: node,\n char: char,\n child: nextNode\n })\n\n node = nextNode\n }\n\n node.final = true\n this.previousWord = word\n}\n\nlunr.TokenSet.Builder.prototype.finish = function () {\n this.minimize(0)\n}\n\nlunr.TokenSet.Builder.prototype.minimize = function (downTo) {\n for (var i = this.uncheckedNodes.length - 1; i >= downTo; i--) {\n var node = this.uncheckedNodes[i],\n childKey = node.child.toString()\n\n if (childKey in this.minimizedNodes) {\n node.parent.edges[node.char] = this.minimizedNodes[childKey]\n } else {\n // Cache the key for this node since\n // we know it can't change anymore\n node.child._str = childKey\n\n this.minimizedNodes[childKey] = node.child\n }\n\n this.uncheckedNodes.pop()\n }\n}\n/*!\n * lunr.Index\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * An index contains the built index of all documents and provides a query interface\n * to the index.\n *\n * Usually instances of lunr.Index will not be created using this constructor, instead\n * lunr.Builder should be used to construct new indexes, or lunr.Index.load should be\n * used to load previously built and serialized indexes.\n *\n * @constructor\n * @param {Object} attrs - The attributes of the built search index.\n * @param {Object} attrs.invertedIndex - An index of term/field to document reference.\n * @param {Object} attrs.fieldVectors - Field vectors\n * @param {lunr.TokenSet} attrs.tokenSet - An set of all corpus tokens.\n * @param {string[]} attrs.fields - The names of indexed document fields.\n * @param {lunr.Pipeline} attrs.pipeline - The pipeline to use for search terms.\n */\nlunr.Index = function (attrs) {\n this.invertedIndex = attrs.invertedIndex\n this.fieldVectors = attrs.fieldVectors\n this.tokenSet = attrs.tokenSet\n this.fields = attrs.fields\n this.pipeline = attrs.pipeline\n}\n\n/**\n * A result contains details of a document matching a search query.\n * @typedef {Object} lunr.Index~Result\n * @property {string} ref - The reference of the document this result represents.\n * @property {number} score - A number between 0 and 1 representing how similar this document is to the query.\n * @property {lunr.MatchData} matchData - Contains metadata about this match including which term(s) caused the match.\n */\n\n/**\n * Although lunr provides the ability to create queries using lunr.Query, it also provides a simple\n * query language which itself is parsed into an instance of lunr.Query.\n *\n * For programmatically building queries it is advised to directly use lunr.Query, the query language\n * is best used for human entered text rather than program generated text.\n *\n * At its simplest queries can just be a single term, e.g. `hello`, multiple terms are also supported\n * and will be combined with OR, e.g `hello world` will match documents that contain either 'hello'\n * or 'world', though those that contain both will rank higher in the results.\n *\n * Wildcards can be included in terms to match one or more unspecified characters, these wildcards can\n * be inserted anywhere within the term, and more than one wildcard can exist in a single term. Adding\n * wildcards will increase the number of documents that will be found but can also have a negative\n * impact on query performance, especially with wildcards at the beginning of a term.\n *\n * Terms can be restricted to specific fields, e.g. `title:hello`, only documents with the term\n * hello in the title field will match this query. Using a field not present in the index will lead\n * to an error being thrown.\n *\n * Modifiers can also be added to terms, lunr supports edit distance and boost modifiers on terms. A term\n * boost will make documents matching that term score higher, e.g. `foo^5`. Edit distance is also supported\n * to provide fuzzy matching, e.g. 'hello~2' will match documents with hello with an edit distance of 2.\n * Avoid large values for edit distance to improve query performance.\n *\n * Each term also supports a presence modifier. By default a term's presence in document is optional, however\n * this can be changed to either required or prohibited. For a term's presence to be required in a document the\n * term should be prefixed with a '+', e.g. `+foo bar` is a search for documents that must contain 'foo' and\n * optionally contain 'bar'. Conversely a leading '-' sets the terms presence to prohibited, i.e. it must not\n * appear in a document, e.g. `-foo bar` is a search for documents that do not contain 'foo' but may contain 'bar'.\n *\n * To escape special characters the backslash character '\\' can be used, this allows searches to include\n * characters that would normally be considered modifiers, e.g. `foo\\~2` will search for a term \"foo~2\" instead\n * of attempting to apply a boost of 2 to the search term \"foo\".\n *\n * @typedef {string} lunr.Index~QueryString\n * @example Simple single term query\n * hello\n * @example Multiple term query\n * hello world\n * @example term scoped to a field\n * title:hello\n * @example term with a boost of 10\n * hello^10\n * @example term with an edit distance of 2\n * hello~2\n * @example terms with presence modifiers\n * -foo +bar baz\n */\n\n/**\n * Performs a search against the index using lunr query syntax.\n *\n * Results will be returned sorted by their score, the most relevant results\n * will be returned first. For details on how the score is calculated, please see\n * the {@link https://lunrjs.com/guides/searching.html#scoring|guide}.\n *\n * For more programmatic querying use lunr.Index#query.\n *\n * @param {lunr.Index~QueryString} queryString - A string containing a lunr query.\n * @throws {lunr.QueryParseError} If the passed query string cannot be parsed.\n * @returns {lunr.Index~Result[]}\n */\nlunr.Index.prototype.search = function (queryString) {\n return this.query(function (query) {\n var parser = new lunr.QueryParser(queryString, query)\n parser.parse()\n })\n}\n\n/**\n * A query builder callback provides a query object to be used to express\n * the query to perform on the index.\n *\n * @callback lunr.Index~queryBuilder\n * @param {lunr.Query} query - The query object to build up.\n * @this lunr.Query\n */\n\n/**\n * Performs a query against the index using the yielded lunr.Query object.\n *\n * If performing programmatic queries against the index, this method is preferred\n * over lunr.Index#search so as to avoid the additional query parsing overhead.\n *\n * A query object is yielded to the supplied function which should be used to\n * express the query to be run against the index.\n *\n * Note that although this function takes a callback parameter it is _not_ an\n * asynchronous operation, the callback is just yielded a query object to be\n * customized.\n *\n * @param {lunr.Index~queryBuilder} fn - A function that is used to build the query.\n * @returns {lunr.Index~Result[]}\n */\nlunr.Index.prototype.query = function (fn) {\n // for each query clause\n // * process terms\n // * expand terms from token set\n // * find matching documents and metadata\n // * get document vectors\n // * score documents\n\n var query = new lunr.Query(this.fields),\n matchingFields = Object.create(null),\n queryVectors = Object.create(null),\n termFieldCache = Object.create(null),\n requiredMatches = Object.create(null),\n prohibitedMatches = Object.create(null)\n\n /*\n * To support field level boosts a query vector is created per\n * field. An empty vector is eagerly created to support negated\n * queries.\n */\n for (var i = 0; i < this.fields.length; i++) {\n queryVectors[this.fields[i]] = new lunr.Vector\n }\n\n fn.call(query, query)\n\n for (var i = 0; i < query.clauses.length; i++) {\n /*\n * Unless the pipeline has been disabled for this term, which is\n * the case for terms with wildcards, we need to pass the clause\n * term through the search pipeline. A pipeline returns an array\n * of processed terms. Pipeline functions may expand the passed\n * term, which means we may end up performing multiple index lookups\n * for a single query term.\n */\n var clause = query.clauses[i],\n terms = null,\n clauseMatches = lunr.Set.empty\n\n if (clause.usePipeline) {\n terms = this.pipeline.runString(clause.term, {\n fields: clause.fields\n })\n } else {\n terms = [clause.term]\n }\n\n for (var m = 0; m < terms.length; m++) {\n var term = terms[m]\n\n /*\n * Each term returned from the pipeline needs to use the same query\n * clause object, e.g. the same boost and or edit distance. The\n * simplest way to do this is to re-use the clause object but mutate\n * its term property.\n */\n clause.term = term\n\n /*\n * From the term in the clause we create a token set which will then\n * be used to intersect the indexes token set to get a list of terms\n * to lookup in the inverted index\n */\n var termTokenSet = lunr.TokenSet.fromClause(clause),\n expandedTerms = this.tokenSet.intersect(termTokenSet).toArray()\n\n /*\n * If a term marked as required does not exist in the tokenSet it is\n * impossible for the search to return any matches. We set all the field\n * scoped required matches set to empty and stop examining any further\n * clauses.\n */\n if (expandedTerms.length === 0 && clause.presence === lunr.Query.presence.REQUIRED) {\n for (var k = 0; k < clause.fields.length; k++) {\n var field = clause.fields[k]\n requiredMatches[field] = lunr.Set.empty\n }\n\n break\n }\n\n for (var j = 0; j < expandedTerms.length; j++) {\n /*\n * For each term get the posting and termIndex, this is required for\n * building the query vector.\n */\n var expandedTerm = expandedTerms[j],\n posting = this.invertedIndex[expandedTerm],\n termIndex = posting._index\n\n for (var k = 0; k < clause.fields.length; k++) {\n /*\n * For each field that this query term is scoped by (by default\n * all fields are in scope) we need to get all the document refs\n * that have this term in that field.\n *\n * The posting is the entry in the invertedIndex for the matching\n * term from above.\n */\n var field = clause.fields[k],\n fieldPosting = posting[field],\n matchingDocumentRefs = Object.keys(fieldPosting),\n termField = expandedTerm + \"/\" + field,\n matchingDocumentsSet = new lunr.Set(matchingDocumentRefs)\n\n /*\n * if the presence of this term is required ensure that the matching\n * documents are added to the set of required matches for this clause.\n *\n */\n if (clause.presence == lunr.Query.presence.REQUIRED) {\n clauseMatches = clauseMatches.union(matchingDocumentsSet)\n\n if (requiredMatches[field] === undefined) {\n requiredMatches[field] = lunr.Set.complete\n }\n }\n\n /*\n * if the presence of this term is prohibited ensure that the matching\n * documents are added to the set of prohibited matches for this field,\n * creating that set if it does not yet exist.\n */\n if (clause.presence == lunr.Query.presence.PROHIBITED) {\n if (prohibitedMatches[field] === undefined) {\n prohibitedMatches[field] = lunr.Set.empty\n }\n\n prohibitedMatches[field] = prohibitedMatches[field].union(matchingDocumentsSet)\n\n /*\n * Prohibited matches should not be part of the query vector used for\n * similarity scoring and no metadata should be extracted so we continue\n * to the next field\n */\n continue\n }\n\n /*\n * The query field vector is populated using the termIndex found for\n * the term and a unit value with the appropriate boost applied.\n * Using upsert because there could already be an entry in the vector\n * for the term we are working with. In that case we just add the scores\n * together.\n */\n queryVectors[field].upsert(termIndex, clause.boost, function (a, b) { return a + b })\n\n /**\n * If we've already seen this term, field combo then we've already collected\n * the matching documents and metadata, no need to go through all that again\n */\n if (termFieldCache[termField]) {\n continue\n }\n\n for (var l = 0; l < matchingDocumentRefs.length; l++) {\n /*\n * All metadata for this term/field/document triple\n * are then extracted and collected into an instance\n * of lunr.MatchData ready to be returned in the query\n * results\n */\n var matchingDocumentRef = matchingDocumentRefs[l],\n matchingFieldRef = new lunr.FieldRef (matchingDocumentRef, field),\n metadata = fieldPosting[matchingDocumentRef],\n fieldMatch\n\n if ((fieldMatch = matchingFields[matchingFieldRef]) === undefined) {\n matchingFields[matchingFieldRef] = new lunr.MatchData (expandedTerm, field, metadata)\n } else {\n fieldMatch.add(expandedTerm, field, metadata)\n }\n\n }\n\n termFieldCache[termField] = true\n }\n }\n }\n\n /**\n * If the presence was required we need to update the requiredMatches field sets.\n * We do this after all fields for the term have collected their matches because\n * the clause terms presence is required in _any_ of the fields not _all_ of the\n * fields.\n */\n if (clause.presence === lunr.Query.presence.REQUIRED) {\n for (var k = 0; k < clause.fields.length; k++) {\n var field = clause.fields[k]\n requiredMatches[field] = requiredMatches[field].intersect(clauseMatches)\n }\n }\n }\n\n /**\n * Need to combine the field scoped required and prohibited\n * matching documents into a global set of required and prohibited\n * matches\n */\n var allRequiredMatches = lunr.Set.complete,\n allProhibitedMatches = lunr.Set.empty\n\n for (var i = 0; i < this.fields.length; i++) {\n var field = this.fields[i]\n\n if (requiredMatches[field]) {\n allRequiredMatches = allRequiredMatches.intersect(requiredMatches[field])\n }\n\n if (prohibitedMatches[field]) {\n allProhibitedMatches = allProhibitedMatches.union(prohibitedMatches[field])\n }\n }\n\n var matchingFieldRefs = Object.keys(matchingFields),\n results = [],\n matches = Object.create(null)\n\n /*\n * If the query is negated (contains only prohibited terms)\n * we need to get _all_ fieldRefs currently existing in the\n * index. This is only done when we know that the query is\n * entirely prohibited terms to avoid any cost of getting all\n * fieldRefs unnecessarily.\n *\n * Additionally, blank MatchData must be created to correctly\n * populate the results.\n */\n if (query.isNegated()) {\n matchingFieldRefs = Object.keys(this.fieldVectors)\n\n for (var i = 0; i < matchingFieldRefs.length; i++) {\n var matchingFieldRef = matchingFieldRefs[i]\n var fieldRef = lunr.FieldRef.fromString(matchingFieldRef)\n matchingFields[matchingFieldRef] = new lunr.MatchData\n }\n }\n\n for (var i = 0; i < matchingFieldRefs.length; i++) {\n /*\n * Currently we have document fields that match the query, but we\n * need to return documents. The matchData and scores are combined\n * from multiple fields belonging to the same document.\n *\n * Scores are calculated by field, using the query vectors created\n * above, and combined into a final document score using addition.\n */\n var fieldRef = lunr.FieldRef.fromString(matchingFieldRefs[i]),\n docRef = fieldRef.docRef\n\n if (!allRequiredMatches.contains(docRef)) {\n continue\n }\n\n if (allProhibitedMatches.contains(docRef)) {\n continue\n }\n\n var fieldVector = this.fieldVectors[fieldRef],\n score = queryVectors[fieldRef.fieldName].similarity(fieldVector),\n docMatch\n\n if ((docMatch = matches[docRef]) !== undefined) {\n docMatch.score += score\n docMatch.matchData.combine(matchingFields[fieldRef])\n } else {\n var match = {\n ref: docRef,\n score: score,\n matchData: matchingFields[fieldRef]\n }\n matches[docRef] = match\n results.push(match)\n }\n }\n\n /*\n * Sort the results objects by score, highest first.\n */\n return results.sort(function (a, b) {\n return b.score - a.score\n })\n}\n\n/**\n * Prepares the index for JSON serialization.\n *\n * The schema for this JSON blob will be described in a\n * separate JSON schema file.\n *\n * @returns {Object}\n */\nlunr.Index.prototype.toJSON = function () {\n var invertedIndex = Object.keys(this.invertedIndex)\n .sort()\n .map(function (term) {\n return [term, this.invertedIndex[term]]\n }, this)\n\n var fieldVectors = Object.keys(this.fieldVectors)\n .map(function (ref) {\n return [ref, this.fieldVectors[ref].toJSON()]\n }, this)\n\n return {\n version: lunr.version,\n fields: this.fields,\n fieldVectors: fieldVectors,\n invertedIndex: invertedIndex,\n pipeline: this.pipeline.toJSON()\n }\n}\n\n/**\n * Loads a previously serialized lunr.Index\n *\n * @param {Object} serializedIndex - A previously serialized lunr.Index\n * @returns {lunr.Index}\n */\nlunr.Index.load = function (serializedIndex) {\n var attrs = {},\n fieldVectors = {},\n serializedVectors = serializedIndex.fieldVectors,\n invertedIndex = Object.create(null),\n serializedInvertedIndex = serializedIndex.invertedIndex,\n tokenSetBuilder = new lunr.TokenSet.Builder,\n pipeline = lunr.Pipeline.load(serializedIndex.pipeline)\n\n if (serializedIndex.version != lunr.version) {\n lunr.utils.warn(\"Version mismatch when loading serialised index. Current version of lunr '\" + lunr.version + \"' does not match serialized index '\" + serializedIndex.version + \"'\")\n }\n\n for (var i = 0; i < serializedVectors.length; i++) {\n var tuple = serializedVectors[i],\n ref = tuple[0],\n elements = tuple[1]\n\n fieldVectors[ref] = new lunr.Vector(elements)\n }\n\n for (var i = 0; i < serializedInvertedIndex.length; i++) {\n var tuple = serializedInvertedIndex[i],\n term = tuple[0],\n posting = tuple[1]\n\n tokenSetBuilder.insert(term)\n invertedIndex[term] = posting\n }\n\n tokenSetBuilder.finish()\n\n attrs.fields = serializedIndex.fields\n\n attrs.fieldVectors = fieldVectors\n attrs.invertedIndex = invertedIndex\n attrs.tokenSet = tokenSetBuilder.root\n attrs.pipeline = pipeline\n\n return new lunr.Index(attrs)\n}\n/*!\n * lunr.Builder\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.Builder performs indexing on a set of documents and\n * returns instances of lunr.Index ready for querying.\n *\n * All configuration of the index is done via the builder, the\n * fields to index, the document reference, the text processing\n * pipeline and document scoring parameters are all set on the\n * builder before indexing.\n *\n * @constructor\n * @property {string} _ref - Internal reference to the document reference field.\n * @property {string[]} _fields - Internal reference to the document fields to index.\n * @property {object} invertedIndex - The inverted index maps terms to document fields.\n * @property {object} documentTermFrequencies - Keeps track of document term frequencies.\n * @property {object} documentLengths - Keeps track of the length of documents added to the index.\n * @property {lunr.tokenizer} tokenizer - Function for splitting strings into tokens for indexing.\n * @property {lunr.Pipeline} pipeline - The pipeline performs text processing on tokens before indexing.\n * @property {lunr.Pipeline} searchPipeline - A pipeline for processing search terms before querying the index.\n * @property {number} documentCount - Keeps track of the total number of documents indexed.\n * @property {number} _b - A parameter to control field length normalization, setting this to 0 disabled normalization, 1 fully normalizes field lengths, the default value is 0.75.\n * @property {number} _k1 - A parameter to control how quickly an increase in term frequency results in term frequency saturation, the default value is 1.2.\n * @property {number} termIndex - A counter incremented for each unique term, used to identify a terms position in the vector space.\n * @property {array} metadataWhitelist - A list of metadata keys that have been whitelisted for entry in the index.\n */\nlunr.Builder = function () {\n this._ref = \"id\"\n this._fields = Object.create(null)\n this._documents = Object.create(null)\n this.invertedIndex = Object.create(null)\n this.fieldTermFrequencies = {}\n this.fieldLengths = {}\n this.tokenizer = lunr.tokenizer\n this.pipeline = new lunr.Pipeline\n this.searchPipeline = new lunr.Pipeline\n this.documentCount = 0\n this._b = 0.75\n this._k1 = 1.2\n this.termIndex = 0\n this.metadataWhitelist = []\n}\n\n/**\n * Sets the document field used as the document reference. Every document must have this field.\n * The type of this field in the document should be a string, if it is not a string it will be\n * coerced into a string by calling toString.\n *\n * The default ref is 'id'.\n *\n * The ref should _not_ be changed during indexing, it should be set before any documents are\n * added to the index. Changing it during indexing can lead to inconsistent results.\n *\n * @param {string} ref - The name of the reference field in the document.\n */\nlunr.Builder.prototype.ref = function (ref) {\n this._ref = ref\n}\n\n/**\n * A function that is used to extract a field from a document.\n *\n * Lunr expects a field to be at the top level of a document, if however the field\n * is deeply nested within a document an extractor function can be used to extract\n * the right field for indexing.\n *\n * @callback fieldExtractor\n * @param {object} doc - The document being added to the index.\n * @returns {?(string|object|object[])} obj - The object that will be indexed for this field.\n * @example Extracting a nested field\n * function (doc) { return doc.nested.field }\n */\n\n/**\n * Adds a field to the list of document fields that will be indexed. Every document being\n * indexed should have this field. Null values for this field in indexed documents will\n * not cause errors but will limit the chance of that document being retrieved by searches.\n *\n * All fields should be added before adding documents to the index. Adding fields after\n * a document has been indexed will have no effect on already indexed documents.\n *\n * Fields can be boosted at build time. This allows terms within that field to have more\n * importance when ranking search results. Use a field boost to specify that matches within\n * one field are more important than other fields.\n *\n * @param {string} fieldName - The name of a field to index in all documents.\n * @param {object} attributes - Optional attributes associated with this field.\n * @param {number} [attributes.boost=1] - Boost applied to all terms within this field.\n * @param {fieldExtractor} [attributes.extractor] - Function to extract a field from a document.\n * @throws {RangeError} fieldName cannot contain unsupported characters '/'\n */\nlunr.Builder.prototype.field = function (fieldName, attributes) {\n if (/\\//.test(fieldName)) {\n throw new RangeError (\"Field '\" + fieldName + \"' contains illegal character '/'\")\n }\n\n this._fields[fieldName] = attributes || {}\n}\n\n/**\n * A parameter to tune the amount of field length normalisation that is applied when\n * calculating relevance scores. A value of 0 will completely disable any normalisation\n * and a value of 1 will fully normalise field lengths. The default is 0.75. Values of b\n * will be clamped to the range 0 - 1.\n *\n * @param {number} number - The value to set for this tuning parameter.\n */\nlunr.Builder.prototype.b = function (number) {\n if (number < 0) {\n this._b = 0\n } else if (number > 1) {\n this._b = 1\n } else {\n this._b = number\n }\n}\n\n/**\n * A parameter that controls the speed at which a rise in term frequency results in term\n * frequency saturation. The default value is 1.2. Setting this to a higher value will give\n * slower saturation levels, a lower value will result in quicker saturation.\n *\n * @param {number} number - The value to set for this tuning parameter.\n */\nlunr.Builder.prototype.k1 = function (number) {\n this._k1 = number\n}\n\n/**\n * Adds a document to the index.\n *\n * Before adding fields to the index the index should have been fully setup, with the document\n * ref and all fields to index already having been specified.\n *\n * The document must have a field name as specified by the ref (by default this is 'id') and\n * it should have all fields defined for indexing, though null or undefined values will not\n * cause errors.\n *\n * Entire documents can be boosted at build time. Applying a boost to a document indicates that\n * this document should rank higher in search results than other documents.\n *\n * @param {object} doc - The document to add to the index.\n * @param {object} attributes - Optional attributes associated with this document.\n * @param {number} [attributes.boost=1] - Boost applied to all terms within this document.\n */\nlunr.Builder.prototype.add = function (doc, attributes) {\n var docRef = doc[this._ref],\n fields = Object.keys(this._fields)\n\n this._documents[docRef] = attributes || {}\n this.documentCount += 1\n\n for (var i = 0; i < fields.length; i++) {\n var fieldName = fields[i],\n extractor = this._fields[fieldName].extractor,\n field = extractor ? extractor(doc) : doc[fieldName],\n tokens = this.tokenizer(field, {\n fields: [fieldName]\n }),\n terms = this.pipeline.run(tokens),\n fieldRef = new lunr.FieldRef (docRef, fieldName),\n fieldTerms = Object.create(null)\n\n this.fieldTermFrequencies[fieldRef] = fieldTerms\n this.fieldLengths[fieldRef] = 0\n\n // store the length of this field for this document\n this.fieldLengths[fieldRef] += terms.length\n\n // calculate term frequencies for this field\n for (var j = 0; j < terms.length; j++) {\n var term = terms[j]\n\n if (fieldTerms[term] == undefined) {\n fieldTerms[term] = 0\n }\n\n fieldTerms[term] += 1\n\n // add to inverted index\n // create an initial posting if one doesn't exist\n if (this.invertedIndex[term] == undefined) {\n var posting = Object.create(null)\n posting[\"_index\"] = this.termIndex\n this.termIndex += 1\n\n for (var k = 0; k < fields.length; k++) {\n posting[fields[k]] = Object.create(null)\n }\n\n this.invertedIndex[term] = posting\n }\n\n // add an entry for this term/fieldName/docRef to the invertedIndex\n if (this.invertedIndex[term][fieldName][docRef] == undefined) {\n this.invertedIndex[term][fieldName][docRef] = Object.create(null)\n }\n\n // store all whitelisted metadata about this token in the\n // inverted index\n for (var l = 0; l < this.metadataWhitelist.length; l++) {\n var metadataKey = this.metadataWhitelist[l],\n metadata = term.metadata[metadataKey]\n\n if (this.invertedIndex[term][fieldName][docRef][metadataKey] == undefined) {\n this.invertedIndex[term][fieldName][docRef][metadataKey] = []\n }\n\n this.invertedIndex[term][fieldName][docRef][metadataKey].push(metadata)\n }\n }\n\n }\n}\n\n/**\n * Calculates the average document length for this index\n *\n * @private\n */\nlunr.Builder.prototype.calculateAverageFieldLengths = function () {\n\n var fieldRefs = Object.keys(this.fieldLengths),\n numberOfFields = fieldRefs.length,\n accumulator = {},\n documentsWithField = {}\n\n for (var i = 0; i < numberOfFields; i++) {\n var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]),\n field = fieldRef.fieldName\n\n documentsWithField[field] || (documentsWithField[field] = 0)\n documentsWithField[field] += 1\n\n accumulator[field] || (accumulator[field] = 0)\n accumulator[field] += this.fieldLengths[fieldRef]\n }\n\n var fields = Object.keys(this._fields)\n\n for (var i = 0; i < fields.length; i++) {\n var fieldName = fields[i]\n accumulator[fieldName] = accumulator[fieldName] / documentsWithField[fieldName]\n }\n\n this.averageFieldLength = accumulator\n}\n\n/**\n * Builds a vector space model of every document using lunr.Vector\n *\n * @private\n */\nlunr.Builder.prototype.createFieldVectors = function () {\n var fieldVectors = {},\n fieldRefs = Object.keys(this.fieldTermFrequencies),\n fieldRefsLength = fieldRefs.length,\n termIdfCache = Object.create(null)\n\n for (var i = 0; i < fieldRefsLength; i++) {\n var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]),\n fieldName = fieldRef.fieldName,\n fieldLength = this.fieldLengths[fieldRef],\n fieldVector = new lunr.Vector,\n termFrequencies = this.fieldTermFrequencies[fieldRef],\n terms = Object.keys(termFrequencies),\n termsLength = terms.length\n\n\n var fieldBoost = this._fields[fieldName].boost || 1,\n docBoost = this._documents[fieldRef.docRef].boost || 1\n\n for (var j = 0; j < termsLength; j++) {\n var term = terms[j],\n tf = termFrequencies[term],\n termIndex = this.invertedIndex[term]._index,\n idf, score, scoreWithPrecision\n\n if (termIdfCache[term] === undefined) {\n idf = lunr.idf(this.invertedIndex[term], this.documentCount)\n termIdfCache[term] = idf\n } else {\n idf = termIdfCache[term]\n }\n\n score = idf * ((this._k1 + 1) * tf) / (this._k1 * (1 - this._b + this._b * (fieldLength / this.averageFieldLength[fieldName])) + tf)\n score *= fieldBoost\n score *= docBoost\n scoreWithPrecision = Math.round(score * 1000) / 1000\n // Converts 1.23456789 to 1.234.\n // Reducing the precision so that the vectors take up less\n // space when serialised. Doing it now so that they behave\n // the same before and after serialisation. Also, this is\n // the fastest approach to reducing a number's precision in\n // JavaScript.\n\n fieldVector.insert(termIndex, scoreWithPrecision)\n }\n\n fieldVectors[fieldRef] = fieldVector\n }\n\n this.fieldVectors = fieldVectors\n}\n\n/**\n * Creates a token set of all tokens in the index using lunr.TokenSet\n *\n * @private\n */\nlunr.Builder.prototype.createTokenSet = function () {\n this.tokenSet = lunr.TokenSet.fromArray(\n Object.keys(this.invertedIndex).sort()\n )\n}\n\n/**\n * Builds the index, creating an instance of lunr.Index.\n *\n * This completes the indexing process and should only be called\n * once all documents have been added to the index.\n *\n * @returns {lunr.Index}\n */\nlunr.Builder.prototype.build = function () {\n this.calculateAverageFieldLengths()\n this.createFieldVectors()\n this.createTokenSet()\n\n return new lunr.Index({\n invertedIndex: this.invertedIndex,\n fieldVectors: this.fieldVectors,\n tokenSet: this.tokenSet,\n fields: Object.keys(this._fields),\n pipeline: this.searchPipeline\n })\n}\n\n/**\n * Applies a plugin to the index builder.\n *\n * A plugin is a function that is called with the index builder as its context.\n * Plugins can be used to customise or extend the behaviour of the index\n * in some way. A plugin is just a function, that encapsulated the custom\n * behaviour that should be applied when building the index.\n *\n * The plugin function will be called with the index builder as its argument, additional\n * arguments can also be passed when calling use. The function will be called\n * with the index builder as its context.\n *\n * @param {Function} plugin The plugin to apply.\n */\nlunr.Builder.prototype.use = function (fn) {\n var args = Array.prototype.slice.call(arguments, 1)\n args.unshift(this)\n fn.apply(this, args)\n}\n/**\n * Contains and collects metadata about a matching document.\n * A single instance of lunr.MatchData is returned as part of every\n * lunr.Index~Result.\n *\n * @constructor\n * @param {string} term - The term this match data is associated with\n * @param {string} field - The field in which the term was found\n * @param {object} metadata - The metadata recorded about this term in this field\n * @property {object} metadata - A cloned collection of metadata associated with this document.\n * @see {@link lunr.Index~Result}\n */\nlunr.MatchData = function (term, field, metadata) {\n var clonedMetadata = Object.create(null),\n metadataKeys = Object.keys(metadata || {})\n\n // Cloning the metadata to prevent the original\n // being mutated during match data combination.\n // Metadata is kept in an array within the inverted\n // index so cloning the data can be done with\n // Array#slice\n for (var i = 0; i < metadataKeys.length; i++) {\n var key = metadataKeys[i]\n clonedMetadata[key] = metadata[key].slice()\n }\n\n this.metadata = Object.create(null)\n\n if (term !== undefined) {\n this.metadata[term] = Object.create(null)\n this.metadata[term][field] = clonedMetadata\n }\n}\n\n/**\n * An instance of lunr.MatchData will be created for every term that matches a\n * document. However only one instance is required in a lunr.Index~Result. This\n * method combines metadata from another instance of lunr.MatchData with this\n * objects metadata.\n *\n * @param {lunr.MatchData} otherMatchData - Another instance of match data to merge with this one.\n * @see {@link lunr.Index~Result}\n */\nlunr.MatchData.prototype.combine = function (otherMatchData) {\n var terms = Object.keys(otherMatchData.metadata)\n\n for (var i = 0; i < terms.length; i++) {\n var term = terms[i],\n fields = Object.keys(otherMatchData.metadata[term])\n\n if (this.metadata[term] == undefined) {\n this.metadata[term] = Object.create(null)\n }\n\n for (var j = 0; j < fields.length; j++) {\n var field = fields[j],\n keys = Object.keys(otherMatchData.metadata[term][field])\n\n if (this.metadata[term][field] == undefined) {\n this.metadata[term][field] = Object.create(null)\n }\n\n for (var k = 0; k < keys.length; k++) {\n var key = keys[k]\n\n if (this.metadata[term][field][key] == undefined) {\n this.metadata[term][field][key] = otherMatchData.metadata[term][field][key]\n } else {\n this.metadata[term][field][key] = this.metadata[term][field][key].concat(otherMatchData.metadata[term][field][key])\n }\n\n }\n }\n }\n}\n\n/**\n * Add metadata for a term/field pair to this instance of match data.\n *\n * @param {string} term - The term this match data is associated with\n * @param {string} field - The field in which the term was found\n * @param {object} metadata - The metadata recorded about this term in this field\n */\nlunr.MatchData.prototype.add = function (term, field, metadata) {\n if (!(term in this.metadata)) {\n this.metadata[term] = Object.create(null)\n this.metadata[term][field] = metadata\n return\n }\n\n if (!(field in this.metadata[term])) {\n this.metadata[term][field] = metadata\n return\n }\n\n var metadataKeys = Object.keys(metadata)\n\n for (var i = 0; i < metadataKeys.length; i++) {\n var key = metadataKeys[i]\n\n if (key in this.metadata[term][field]) {\n this.metadata[term][field][key] = this.metadata[term][field][key].concat(metadata[key])\n } else {\n this.metadata[term][field][key] = metadata[key]\n }\n }\n}\n/**\n * A lunr.Query provides a programmatic way of defining queries to be performed\n * against a {@link lunr.Index}.\n *\n * Prefer constructing a lunr.Query using the {@link lunr.Index#query} method\n * so the query object is pre-initialized with the right index fields.\n *\n * @constructor\n * @property {lunr.Query~Clause[]} clauses - An array of query clauses.\n * @property {string[]} allFields - An array of all available fields in a lunr.Index.\n */\nlunr.Query = function (allFields) {\n this.clauses = []\n this.allFields = allFields\n}\n\n/**\n * Constants for indicating what kind of automatic wildcard insertion will be used when constructing a query clause.\n *\n * This allows wildcards to be added to the beginning and end of a term without having to manually do any string\n * concatenation.\n *\n * The wildcard constants can be bitwise combined to select both leading and trailing wildcards.\n *\n * @constant\n * @default\n * @property {number} wildcard.NONE - The term will have no wildcards inserted, this is the default behaviour\n * @property {number} wildcard.LEADING - Prepend the term with a wildcard, unless a leading wildcard already exists\n * @property {number} wildcard.TRAILING - Append a wildcard to the term, unless a trailing wildcard already exists\n * @see lunr.Query~Clause\n * @see lunr.Query#clause\n * @see lunr.Query#term\n * @example query term with trailing wildcard\n * query.term('foo', { wildcard: lunr.Query.wildcard.TRAILING })\n * @example query term with leading and trailing wildcard\n * query.term('foo', {\n * wildcard: lunr.Query.wildcard.LEADING | lunr.Query.wildcard.TRAILING\n * })\n */\n\nlunr.Query.wildcard = new String (\"*\")\nlunr.Query.wildcard.NONE = 0\nlunr.Query.wildcard.LEADING = 1\nlunr.Query.wildcard.TRAILING = 2\n\n/**\n * Constants for indicating what kind of presence a term must have in matching documents.\n *\n * @constant\n * @enum {number}\n * @see lunr.Query~Clause\n * @see lunr.Query#clause\n * @see lunr.Query#term\n * @example query term with required presence\n * query.term('foo', { presence: lunr.Query.presence.REQUIRED })\n */\nlunr.Query.presence = {\n /**\n * Term's presence in a document is optional, this is the default value.\n */\n OPTIONAL: 1,\n\n /**\n * Term's presence in a document is required, documents that do not contain\n * this term will not be returned.\n */\n REQUIRED: 2,\n\n /**\n * Term's presence in a document is prohibited, documents that do contain\n * this term will not be returned.\n */\n PROHIBITED: 3\n}\n\n/**\n * A single clause in a {@link lunr.Query} contains a term and details on how to\n * match that term against a {@link lunr.Index}.\n *\n * @typedef {Object} lunr.Query~Clause\n * @property {string[]} fields - The fields in an index this clause should be matched against.\n * @property {number} [boost=1] - Any boost that should be applied when matching this clause.\n * @property {number} [editDistance] - Whether the term should have fuzzy matching applied, and how fuzzy the match should be.\n * @property {boolean} [usePipeline] - Whether the term should be passed through the search pipeline.\n * @property {number} [wildcard=lunr.Query.wildcard.NONE] - Whether the term should have wildcards appended or prepended.\n * @property {number} [presence=lunr.Query.presence.OPTIONAL] - The terms presence in any matching documents.\n */\n\n/**\n * Adds a {@link lunr.Query~Clause} to this query.\n *\n * Unless the clause contains the fields to be matched all fields will be matched. In addition\n * a default boost of 1 is applied to the clause.\n *\n * @param {lunr.Query~Clause} clause - The clause to add to this query.\n * @see lunr.Query~Clause\n * @returns {lunr.Query}\n */\nlunr.Query.prototype.clause = function (clause) {\n if (!('fields' in clause)) {\n clause.fields = this.allFields\n }\n\n if (!('boost' in clause)) {\n clause.boost = 1\n }\n\n if (!('usePipeline' in clause)) {\n clause.usePipeline = true\n }\n\n if (!('wildcard' in clause)) {\n clause.wildcard = lunr.Query.wildcard.NONE\n }\n\n if ((clause.wildcard & lunr.Query.wildcard.LEADING) && (clause.term.charAt(0) != lunr.Query.wildcard)) {\n clause.term = \"*\" + clause.term\n }\n\n if ((clause.wildcard & lunr.Query.wildcard.TRAILING) && (clause.term.slice(-1) != lunr.Query.wildcard)) {\n clause.term = \"\" + clause.term + \"*\"\n }\n\n if (!('presence' in clause)) {\n clause.presence = lunr.Query.presence.OPTIONAL\n }\n\n this.clauses.push(clause)\n\n return this\n}\n\n/**\n * A negated query is one in which every clause has a presence of\n * prohibited. These queries require some special processing to return\n * the expected results.\n *\n * @returns boolean\n */\nlunr.Query.prototype.isNegated = function () {\n for (var i = 0; i < this.clauses.length; i++) {\n if (this.clauses[i].presence != lunr.Query.presence.PROHIBITED) {\n return false\n }\n }\n\n return true\n}\n\n/**\n * Adds a term to the current query, under the covers this will create a {@link lunr.Query~Clause}\n * to the list of clauses that make up this query.\n *\n * The term is used as is, i.e. no tokenization will be performed by this method. Instead conversion\n * to a token or token-like string should be done before calling this method.\n *\n * The term will be converted to a string by calling `toString`. Multiple terms can be passed as an\n * array, each term in the array will share the same options.\n *\n * @param {object|object[]} term - The term(s) to add to the query.\n * @param {object} [options] - Any additional properties to add to the query clause.\n * @returns {lunr.Query}\n * @see lunr.Query#clause\n * @see lunr.Query~Clause\n * @example adding a single term to a query\n * query.term(\"foo\")\n * @example adding a single term to a query and specifying search fields, term boost and automatic trailing wildcard\n * query.term(\"foo\", {\n * fields: [\"title\"],\n * boost: 10,\n * wildcard: lunr.Query.wildcard.TRAILING\n * })\n * @example using lunr.tokenizer to convert a string to tokens before using them as terms\n * query.term(lunr.tokenizer(\"foo bar\"))\n */\nlunr.Query.prototype.term = function (term, options) {\n if (Array.isArray(term)) {\n term.forEach(function (t) { this.term(t, lunr.utils.clone(options)) }, this)\n return this\n }\n\n var clause = options || {}\n clause.term = term.toString()\n\n this.clause(clause)\n\n return this\n}\nlunr.QueryParseError = function (message, start, end) {\n this.name = \"QueryParseError\"\n this.message = message\n this.start = start\n this.end = end\n}\n\nlunr.QueryParseError.prototype = new Error\nlunr.QueryLexer = function (str) {\n this.lexemes = []\n this.str = str\n this.length = str.length\n this.pos = 0\n this.start = 0\n this.escapeCharPositions = []\n}\n\nlunr.QueryLexer.prototype.run = function () {\n var state = lunr.QueryLexer.lexText\n\n while (state) {\n state = state(this)\n }\n}\n\nlunr.QueryLexer.prototype.sliceString = function () {\n var subSlices = [],\n sliceStart = this.start,\n sliceEnd = this.pos\n\n for (var i = 0; i < this.escapeCharPositions.length; i++) {\n sliceEnd = this.escapeCharPositions[i]\n subSlices.push(this.str.slice(sliceStart, sliceEnd))\n sliceStart = sliceEnd + 1\n }\n\n subSlices.push(this.str.slice(sliceStart, this.pos))\n this.escapeCharPositions.length = 0\n\n return subSlices.join('')\n}\n\nlunr.QueryLexer.prototype.emit = function (type) {\n this.lexemes.push({\n type: type,\n str: this.sliceString(),\n start: this.start,\n end: this.pos\n })\n\n this.start = this.pos\n}\n\nlunr.QueryLexer.prototype.escapeCharacter = function () {\n this.escapeCharPositions.push(this.pos - 1)\n this.pos += 1\n}\n\nlunr.QueryLexer.prototype.next = function () {\n if (this.pos >= this.length) {\n return lunr.QueryLexer.EOS\n }\n\n var char = this.str.charAt(this.pos)\n this.pos += 1\n return char\n}\n\nlunr.QueryLexer.prototype.width = function () {\n return this.pos - this.start\n}\n\nlunr.QueryLexer.prototype.ignore = function () {\n if (this.start == this.pos) {\n this.pos += 1\n }\n\n this.start = this.pos\n}\n\nlunr.QueryLexer.prototype.backup = function () {\n this.pos -= 1\n}\n\nlunr.QueryLexer.prototype.acceptDigitRun = function () {\n var char, charCode\n\n do {\n char = this.next()\n charCode = char.charCodeAt(0)\n } while (charCode > 47 && charCode < 58)\n\n if (char != lunr.QueryLexer.EOS) {\n this.backup()\n }\n}\n\nlunr.QueryLexer.prototype.more = function () {\n return this.pos < this.length\n}\n\nlunr.QueryLexer.EOS = 'EOS'\nlunr.QueryLexer.FIELD = 'FIELD'\nlunr.QueryLexer.TERM = 'TERM'\nlunr.QueryLexer.EDIT_DISTANCE = 'EDIT_DISTANCE'\nlunr.QueryLexer.BOOST = 'BOOST'\nlunr.QueryLexer.PRESENCE = 'PRESENCE'\n\nlunr.QueryLexer.lexField = function (lexer) {\n lexer.backup()\n lexer.emit(lunr.QueryLexer.FIELD)\n lexer.ignore()\n return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexTerm = function (lexer) {\n if (lexer.width() > 1) {\n lexer.backup()\n lexer.emit(lunr.QueryLexer.TERM)\n }\n\n lexer.ignore()\n\n if (lexer.more()) {\n return lunr.QueryLexer.lexText\n }\n}\n\nlunr.QueryLexer.lexEditDistance = function (lexer) {\n lexer.ignore()\n lexer.acceptDigitRun()\n lexer.emit(lunr.QueryLexer.EDIT_DISTANCE)\n return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexBoost = function (lexer) {\n lexer.ignore()\n lexer.acceptDigitRun()\n lexer.emit(lunr.QueryLexer.BOOST)\n return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexEOS = function (lexer) {\n if (lexer.width() > 0) {\n lexer.emit(lunr.QueryLexer.TERM)\n }\n}\n\n// This matches the separator used when tokenising fields\n// within a document. These should match otherwise it is\n// not possible to search for some tokens within a document.\n//\n// It is possible for the user to change the separator on the\n// tokenizer so it _might_ clash with any other of the special\n// characters already used within the search string, e.g. :.\n//\n// This means that it is possible to change the separator in\n// such a way that makes some words unsearchable using a search\n// string.\nlunr.QueryLexer.termSeparator = lunr.tokenizer.separator\n\nlunr.QueryLexer.lexText = function (lexer) {\n while (true) {\n var char = lexer.next()\n\n if (char == lunr.QueryLexer.EOS) {\n return lunr.QueryLexer.lexEOS\n }\n\n // Escape character is '\\'\n if (char.charCodeAt(0) == 92) {\n lexer.escapeCharacter()\n continue\n }\n\n if (char == \":\") {\n return lunr.QueryLexer.lexField\n }\n\n if (char == \"~\") {\n lexer.backup()\n if (lexer.width() > 0) {\n lexer.emit(lunr.QueryLexer.TERM)\n }\n return lunr.QueryLexer.lexEditDistance\n }\n\n if (char == \"^\") {\n lexer.backup()\n if (lexer.width() > 0) {\n lexer.emit(lunr.QueryLexer.TERM)\n }\n return lunr.QueryLexer.lexBoost\n }\n\n // \"+\" indicates term presence is required\n // checking for length to ensure that only\n // leading \"+\" are considered\n if (char == \"+\" && lexer.width() === 1) {\n lexer.emit(lunr.QueryLexer.PRESENCE)\n return lunr.QueryLexer.lexText\n }\n\n // \"-\" indicates term presence is prohibited\n // checking for length to ensure that only\n // leading \"-\" are considered\n if (char == \"-\" && lexer.width() === 1) {\n lexer.emit(lunr.QueryLexer.PRESENCE)\n return lunr.QueryLexer.lexText\n }\n\n if (char.match(lunr.QueryLexer.termSeparator)) {\n return lunr.QueryLexer.lexTerm\n }\n }\n}\n\nlunr.QueryParser = function (str, query) {\n this.lexer = new lunr.QueryLexer (str)\n this.query = query\n this.currentClause = {}\n this.lexemeIdx = 0\n}\n\nlunr.QueryParser.prototype.parse = function () {\n this.lexer.run()\n this.lexemes = this.lexer.lexemes\n\n var state = lunr.QueryParser.parseClause\n\n while (state) {\n state = state(this)\n }\n\n return this.query\n}\n\nlunr.QueryParser.prototype.peekLexeme = function () {\n return this.lexemes[this.lexemeIdx]\n}\n\nlunr.QueryParser.prototype.consumeLexeme = function () {\n var lexeme = this.peekLexeme()\n this.lexemeIdx += 1\n return lexeme\n}\n\nlunr.QueryParser.prototype.nextClause = function () {\n var completedClause = this.currentClause\n this.query.clause(completedClause)\n this.currentClause = {}\n}\n\nlunr.QueryParser.parseClause = function (parser) {\n var lexeme = parser.peekLexeme()\n\n if (lexeme == undefined) {\n return\n }\n\n switch (lexeme.type) {\n case lunr.QueryLexer.PRESENCE:\n return lunr.QueryParser.parsePresence\n case lunr.QueryLexer.FIELD:\n return lunr.QueryParser.parseField\n case lunr.QueryLexer.TERM:\n return lunr.QueryParser.parseTerm\n default:\n var errorMessage = \"expected either a field or a term, found \" + lexeme.type\n\n if (lexeme.str.length >= 1) {\n errorMessage += \" with value '\" + lexeme.str + \"'\"\n }\n\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n}\n\nlunr.QueryParser.parsePresence = function (parser) {\n var lexeme = parser.consumeLexeme()\n\n if (lexeme == undefined) {\n return\n }\n\n switch (lexeme.str) {\n case \"-\":\n parser.currentClause.presence = lunr.Query.presence.PROHIBITED\n break\n case \"+\":\n parser.currentClause.presence = lunr.Query.presence.REQUIRED\n break\n default:\n var errorMessage = \"unrecognised presence operator'\" + lexeme.str + \"'\"\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n\n var nextLexeme = parser.peekLexeme()\n\n if (nextLexeme == undefined) {\n var errorMessage = \"expecting term or field, found nothing\"\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n\n switch (nextLexeme.type) {\n case lunr.QueryLexer.FIELD:\n return lunr.QueryParser.parseField\n case lunr.QueryLexer.TERM:\n return lunr.QueryParser.parseTerm\n default:\n var errorMessage = \"expecting term or field, found '\" + nextLexeme.type + \"'\"\n throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n }\n}\n\nlunr.QueryParser.parseField = function (parser) {\n var lexeme = parser.consumeLexeme()\n\n if (lexeme == undefined) {\n return\n }\n\n if (parser.query.allFields.indexOf(lexeme.str) == -1) {\n var possibleFields = parser.query.allFields.map(function (f) { return \"'\" + f + \"'\" }).join(', '),\n errorMessage = \"unrecognised field '\" + lexeme.str + \"', possible fields: \" + possibleFields\n\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n\n parser.currentClause.fields = [lexeme.str]\n\n var nextLexeme = parser.peekLexeme()\n\n if (nextLexeme == undefined) {\n var errorMessage = \"expecting term, found nothing\"\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n\n switch (nextLexeme.type) {\n case lunr.QueryLexer.TERM:\n return lunr.QueryParser.parseTerm\n default:\n var errorMessage = \"expecting term, found '\" + nextLexeme.type + \"'\"\n throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n }\n}\n\nlunr.QueryParser.parseTerm = function (parser) {\n var lexeme = parser.consumeLexeme()\n\n if (lexeme == undefined) {\n return\n }\n\n parser.currentClause.term = lexeme.str.toLowerCase()\n\n if (lexeme.str.indexOf(\"*\") != -1) {\n parser.currentClause.usePipeline = false\n }\n\n var nextLexeme = parser.peekLexeme()\n\n if (nextLexeme == undefined) {\n parser.nextClause()\n return\n }\n\n switch (nextLexeme.type) {\n case lunr.QueryLexer.TERM:\n parser.nextClause()\n return lunr.QueryParser.parseTerm\n case lunr.QueryLexer.FIELD:\n parser.nextClause()\n return lunr.QueryParser.parseField\n case lunr.QueryLexer.EDIT_DISTANCE:\n return lunr.QueryParser.parseEditDistance\n case lunr.QueryLexer.BOOST:\n return lunr.QueryParser.parseBoost\n case lunr.QueryLexer.PRESENCE:\n parser.nextClause()\n return lunr.QueryParser.parsePresence\n default:\n var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n }\n}\n\nlunr.QueryParser.parseEditDistance = function (parser) {\n var lexeme = parser.consumeLexeme()\n\n if (lexeme == undefined) {\n return\n }\n\n var editDistance = parseInt(lexeme.str, 10)\n\n if (isNaN(editDistance)) {\n var errorMessage = \"edit distance must be numeric\"\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n\n parser.currentClause.editDistance = editDistance\n\n var nextLexeme = parser.peekLexeme()\n\n if (nextLexeme == undefined) {\n parser.nextClause()\n return\n }\n\n switch (nextLexeme.type) {\n case lunr.QueryLexer.TERM:\n parser.nextClause()\n return lunr.QueryParser.parseTerm\n case lunr.QueryLexer.FIELD:\n parser.nextClause()\n return lunr.QueryParser.parseField\n case lunr.QueryLexer.EDIT_DISTANCE:\n return lunr.QueryParser.parseEditDistance\n case lunr.QueryLexer.BOOST:\n return lunr.QueryParser.parseBoost\n case lunr.QueryLexer.PRESENCE:\n parser.nextClause()\n return lunr.QueryParser.parsePresence\n default:\n var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n }\n}\n\nlunr.QueryParser.parseBoost = function (parser) {\n var lexeme = parser.consumeLexeme()\n\n if (lexeme == undefined) {\n return\n }\n\n var boost = parseInt(lexeme.str, 10)\n\n if (isNaN(boost)) {\n var errorMessage = \"boost must be numeric\"\n throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n }\n\n parser.currentClause.boost = boost\n\n var nextLexeme = parser.peekLexeme()\n\n if (nextLexeme == undefined) {\n parser.nextClause()\n return\n }\n\n switch (nextLexeme.type) {\n case lunr.QueryLexer.TERM:\n parser.nextClause()\n return lunr.QueryParser.parseTerm\n case lunr.QueryLexer.FIELD:\n parser.nextClause()\n return lunr.QueryParser.parseField\n case lunr.QueryLexer.EDIT_DISTANCE:\n return lunr.QueryParser.parseEditDistance\n case lunr.QueryLexer.BOOST:\n return lunr.QueryParser.parseBoost\n case lunr.QueryLexer.PRESENCE:\n parser.nextClause()\n return lunr.QueryParser.parsePresence\n default:\n var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n }\n}\n\n /**\n * export the module via AMD, CommonJS or as a browser global\n * Export code from https://github.com/umdjs/umd/blob/master/returnExports.js\n */\n ;(function (root, factory) {\n if (typeof define === 'function' && define.amd) {\n // AMD. Register as an anonymous module.\n define(factory)\n } else if (typeof exports === 'object') {\n /**\n * Node. Does not work with strict CommonJS, but\n * only CommonJS-like enviroments that support module.exports,\n * like Node.\n */\n module.exports = factory()\n } else {\n // Browser globals (root is window)\n root.lunr = factory()\n }\n }(this, function () {\n /**\n * Just return a value to define the module export.\n * This example returns an object, but the module\n * can return a function as the exported value.\n */\n return lunr\n }))\n})();\n", "/*!\n * escape-html\n * Copyright(c) 2012-2013 TJ Holowaychuk\n * Copyright(c) 2015 Andreas Lubbe\n * Copyright(c) 2015 Tiancheng \"Timothy\" Gu\n * MIT Licensed\n */\n\n'use strict';\n\n/**\n * Module variables.\n * @private\n */\n\nvar matchHtmlRegExp = /[\"'&<>]/;\n\n/**\n * Module exports.\n * @public\n */\n\nmodule.exports = escapeHtml;\n\n/**\n * Escape special characters in the given string of html.\n *\n * @param {string} string The string to escape for inserting into HTML\n * @return {string}\n * @public\n */\n\nfunction escapeHtml(string) {\n var str = '' + string;\n var match = matchHtmlRegExp.exec(str);\n\n if (!match) {\n return str;\n }\n\n var escape;\n var html = '';\n var index = 0;\n var lastIndex = 0;\n\n for (index = match.index; index < str.length; index++) {\n switch (str.charCodeAt(index)) {\n case 34: // \"\n escape = '"';\n break;\n case 38: // &\n escape = '&';\n break;\n case 39: // '\n escape = ''';\n break;\n case 60: // <\n escape = '<';\n break;\n case 62: // >\n escape = '>';\n break;\n default:\n continue;\n }\n\n if (lastIndex !== index) {\n html += str.substring(lastIndex, index);\n }\n\n lastIndex = index + 1;\n html += escape;\n }\n\n return lastIndex !== index\n ? html + str.substring(lastIndex, index)\n : html;\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport lunr from \"lunr\"\n\nimport \"~/polyfills\"\n\nimport { Search, SearchIndexConfig } from \"../../_\"\nimport {\n SearchMessage,\n SearchMessageType\n} from \"../message\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Add support for usage with `iframe-worker` polyfill\n *\n * While `importScripts` is synchronous when executed inside of a web worker,\n * it's not possible to provide a synchronous polyfilled implementation. The\n * cool thing is that awaiting a non-Promise is a noop, so extending the type\n * definition to return a `Promise` shouldn't break anything.\n *\n * @see https://bit.ly/2PjDnXi - GitHub comment\n */\ndeclare global {\n function importScripts(...urls: string[]): Promise | void\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nlet index: Search\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch (= import) multi-language support through `lunr-languages`\n *\n * This function automatically imports the stemmers necessary to process the\n * languages, which are defined through the search index configuration.\n *\n * If the worker runs inside of an `iframe` (when using `iframe-worker` as\n * a shim), the base URL for the stemmers to be loaded must be determined by\n * searching for the first `script` element with a `src` attribute, which will\n * contain the contents of this script.\n *\n * @param config - Search index configuration\n *\n * @returns Promise resolving with no result\n */\nasync function setupSearchLanguages(\n config: SearchIndexConfig\n): Promise {\n let base = \"../lunr\"\n\n /* Detect `iframe-worker` and fix base URL */\n if (typeof parent !== \"undefined\" && \"IFrameWorker\" in parent) {\n const worker = document.querySelector(\"script[src]\")!\n const [path] = worker.src.split(\"/worker\")\n\n /* Prefix base with path */\n base = base.replace(\"..\", path)\n }\n\n /* Add scripts for languages */\n const scripts = []\n for (const lang of config.lang) {\n switch (lang) {\n\n /* Add segmenter for Japanese */\n case \"ja\":\n scripts.push(`${base}/tinyseg.js`)\n break\n\n /* Add segmenter for Hindi and Thai */\n case \"hi\":\n case \"th\":\n scripts.push(`${base}/wordcut.js`)\n break\n }\n\n /* Add language support */\n if (lang !== \"en\")\n scripts.push(`${base}/min/lunr.${lang}.min.js`)\n }\n\n /* Add multi-language support */\n if (config.lang.length > 1)\n scripts.push(`${base}/min/lunr.multi.min.js`)\n\n /* Load scripts synchronously */\n if (scripts.length)\n await importScripts(\n `${base}/min/lunr.stemmer.support.min.js`,\n ...scripts\n )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Message handler\n *\n * @param message - Source message\n *\n * @returns Target message\n */\nexport async function handler(\n message: SearchMessage\n): Promise {\n switch (message.type) {\n\n /* Search setup message */\n case SearchMessageType.SETUP:\n await setupSearchLanguages(message.data.config)\n index = new Search(message.data)\n return {\n type: SearchMessageType.READY\n }\n\n /* Search query message */\n case SearchMessageType.QUERY:\n return {\n type: SearchMessageType.RESULT,\n data: index ? index.search(message.data) : { items: [] }\n }\n\n /* All other messages */\n default:\n throw new TypeError(\"Invalid message type\")\n }\n}\n\n/* ----------------------------------------------------------------------------\n * Worker\n * ------------------------------------------------------------------------- */\n\n/* @ts-expect-error - expose Lunr.js in global scope, or stemmers won't work */\nself.lunr = lunr\n\n/* Handle messages */\naddEventListener(\"message\", async ev => {\n postMessage(await handler(ev.data))\n})\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Polyfills\n * ------------------------------------------------------------------------- */\n\n/* Polyfill `Object.entries` */\nif (!Object.entries)\n Object.entries = function (obj: object) {\n const data: [string, string][] = []\n for (const key of Object.keys(obj))\n // @ts-expect-error - ignore property access warning\n data.push([key, obj[key]])\n\n /* Return entries */\n return data\n }\n\n/* Polyfill `Object.values` */\nif (!Object.values)\n Object.values = function (obj: object) {\n const data: string[] = []\n for (const key of Object.keys(obj))\n // @ts-expect-error - ignore property access warning\n data.push(obj[key])\n\n /* Return values */\n return data\n }\n\n/* ------------------------------------------------------------------------- */\n\n/* Polyfills for `Element` */\nif (typeof Element !== \"undefined\") {\n\n /* Polyfill `Element.scrollTo` */\n if (!Element.prototype.scrollTo)\n Element.prototype.scrollTo = function (\n x?: ScrollToOptions | number, y?: number\n ): void {\n if (typeof x === \"object\") {\n this.scrollLeft = x.left!\n this.scrollTop = x.top!\n } else {\n this.scrollLeft = x!\n this.scrollTop = y!\n }\n }\n\n /* Polyfill `Element.replaceWith` */\n if (!Element.prototype.replaceWith)\n Element.prototype.replaceWith = function (\n ...nodes: Array\n ): void {\n const parent = this.parentNode\n if (parent) {\n if (nodes.length === 0)\n parent.removeChild(this)\n\n /* Replace children and create text nodes */\n for (let i = nodes.length - 1; i >= 0; i--) {\n let node = nodes[i]\n if (typeof node !== \"object\")\n node = document.createTextNode(node)\n else if (node.parentNode)\n node.parentNode.removeChild(node)\n\n /* Replace child or insert before previous sibling */\n if (!i)\n parent.replaceChild(node, this)\n else\n parent.insertBefore(this.previousSibling!, node)\n }\n }\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport escapeHTML from \"escape-html\"\n\nimport { SearchIndexDocument } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search document\n */\nexport interface SearchDocument extends SearchIndexDocument {\n parent?: SearchIndexDocument /* Parent article */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search document mapping\n */\nexport type SearchDocumentMap = Map\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search document mapping\n *\n * @param docs - Search index documents\n *\n * @returns Search document map\n */\nexport function setupSearchDocumentMap(\n docs: SearchIndexDocument[]\n): SearchDocumentMap {\n const documents = new Map()\n const parents = new Set()\n for (const doc of docs) {\n const [path, hash] = doc.location.split(\"#\")\n\n /* Extract location, title and tags */\n const location = doc.location\n const title = doc.title\n const tags = doc.tags\n\n /* Escape and cleanup text */\n const text = escapeHTML(doc.text)\n .replace(/\\s+(?=[,.:;!?])/g, \"\")\n .replace(/\\s+/g, \" \")\n\n /* Handle section */\n if (hash) {\n const parent = documents.get(path)!\n\n /* Ignore first section, override article */\n if (!parents.has(parent)) {\n parent.title = doc.title\n parent.text = text\n\n /* Remember that we processed the article */\n parents.add(parent)\n\n /* Add subsequent section */\n } else {\n documents.set(location, {\n location,\n title,\n text,\n parent\n })\n }\n\n /* Add article */\n } else {\n documents.set(location, {\n location,\n title,\n text,\n ...tags && { tags }\n })\n }\n }\n return documents\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport escapeHTML from \"escape-html\"\n\nimport { SearchIndexConfig } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlight function\n *\n * @param value - Value\n *\n * @returns Highlighted value\n */\nexport type SearchHighlightFn = (value: string) => string\n\n/**\n * Search highlight factory function\n *\n * @param query - Query value\n *\n * @returns Search highlight function\n */\nexport type SearchHighlightFactoryFn = (query: string) => SearchHighlightFn\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search highlighter\n *\n * @param config - Search index configuration\n * @param escape - Whether to escape HTML\n *\n * @returns Search highlight factory function\n */\nexport function setupSearchHighlighter(\n config: SearchIndexConfig, escape: boolean\n): SearchHighlightFactoryFn {\n const separator = new RegExp(config.separator, \"img\")\n const highlight = (_: unknown, data: string, term: string) => {\n return `${data}${term}`\n }\n\n /* Return factory function */\n return (query: string) => {\n query = query\n .replace(/[\\s*+\\-:~^]+/g, \" \")\n .trim()\n\n /* Create search term match expression */\n const match = new RegExp(`(^|${config.separator})(${\n query\n .replace(/[|\\\\{}()[\\]^$+*?.-]/g, \"\\\\$&\")\n .replace(separator, \"|\")\n })`, \"img\")\n\n /* Highlight string value */\n return value => (\n escape\n ? escapeHTML(value)\n : value\n )\n .replace(match, highlight)\n .replace(/<\\/mark>(\\s+)]*>/img, \"$1\")\n }\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search query clause\n */\nexport interface SearchQueryClause {\n presence: lunr.Query.presence /* Clause presence */\n term: string /* Clause term */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search query terms\n */\nexport type SearchQueryTerms = Record\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Parse a search query for analysis\n *\n * @param value - Query value\n *\n * @returns Search query clauses\n */\nexport function parseSearchQuery(\n value: string\n): SearchQueryClause[] {\n const query = new (lunr as any).Query([\"title\", \"text\"])\n const parser = new (lunr as any).QueryParser(value, query)\n\n /* Parse and return query clauses */\n parser.parse()\n return query.clauses\n}\n\n/**\n * Analyze the search query clauses in regard to the search terms found\n *\n * @param query - Search query clauses\n * @param terms - Search terms\n *\n * @returns Search query terms\n */\nexport function getSearchQueryTerms(\n query: SearchQueryClause[], terms: string[]\n): SearchQueryTerms {\n const clauses = new Set(query)\n\n /* Match query clauses against terms */\n const result: SearchQueryTerms = {}\n for (let t = 0; t < terms.length; t++)\n for (const clause of clauses)\n if (terms[t].startsWith(clause.term)) {\n result[clause.term] = true\n clauses.delete(clause)\n }\n\n /* Annotate unmatched non-stopword query clauses */\n for (const clause of clauses)\n if (lunr.stopWordFilter?.(clause.term as any))\n result[clause.term] = false\n\n /* Return query terms */\n return result\n}\n", "/*\n * Copyright (c) 2016-2022 Martin Donath \n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n SearchDocument,\n SearchDocumentMap,\n setupSearchDocumentMap\n} from \"../document\"\nimport {\n SearchHighlightFactoryFn,\n setupSearchHighlighter\n} from \"../highlighter\"\nimport { SearchOptions } from \"../options\"\nimport {\n SearchQueryTerms,\n getSearchQueryTerms,\n parseSearchQuery\n} from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index configuration\n */\nexport interface SearchIndexConfig {\n lang: string[] /* Search languages */\n separator: string /* Search separator */\n}\n\n/**\n * Search index document\n */\nexport interface SearchIndexDocument {\n location: string /* Document location */\n title: string /* Document title */\n text: string /* Document text */\n tags?: string[] /* Document tags */\n boost?: number /* Document boost */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search index\n *\n * This interfaces describes the format of the `search_index.json` file which\n * is automatically built by the MkDocs search plugin.\n */\nexport interface SearchIndex {\n config: SearchIndexConfig /* Search index configuration */\n docs: SearchIndexDocument[] /* Search index documents */\n options: SearchOptions /* Search options */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search metadata\n */\nexport interface SearchMetadata {\n score: number /* Score (relevance) */\n terms: SearchQueryTerms /* Search query terms */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search result document\n */\nexport type SearchResultDocument = SearchDocument & SearchMetadata\n\n/**\n * Search result item\n */\nexport type SearchResultItem = SearchResultDocument[]\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search result\n */\nexport interface SearchResult {\n items: SearchResultItem[] /* Search result items */\n suggestions?: string[] /* Search suggestions */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Compute the difference of two lists of strings\n *\n * @param a - 1st list of strings\n * @param b - 2nd list of strings\n *\n * @returns Difference\n */\nfunction difference(a: string[], b: string[]): string[] {\n const [x, y] = [new Set(a), new Set(b)]\n return [\n ...new Set([...x].filter(value => !y.has(value)))\n ]\n}\n\n/* ----------------------------------------------------------------------------\n * Class\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nexport class Search {\n\n /**\n * Search document mapping\n *\n * A mapping of URLs (including hash fragments) to the actual articles and\n * sections of the documentation. The search document mapping must be created\n * regardless of whether the index was prebuilt or not, as Lunr.js itself\n * only stores the actual index.\n */\n protected documents: SearchDocumentMap\n\n /**\n * Search highlight factory function\n */\n protected highlight: SearchHighlightFactoryFn\n\n /**\n * The underlying Lunr.js search index\n */\n protected index: lunr.Index\n\n /**\n * Search options\n */\n protected options: SearchOptions\n\n /**\n * Create the search integration\n *\n * @param data - Search index\n */\n public constructor({ config, docs, options }: SearchIndex) {\n this.options = options\n\n /* Set up document map and highlighter factory */\n this.documents = setupSearchDocumentMap(docs)\n this.highlight = setupSearchHighlighter(config, false)\n\n /* Set separator for tokenizer */\n lunr.tokenizer.separator = new RegExp(config.separator)\n\n /* Create search index */\n this.index = lunr(function () {\n\n /* Set up multi-language support */\n if (config.lang.length === 1 && config.lang[0] !== \"en\") {\n this.use((lunr as any)[config.lang[0]])\n } else if (config.lang.length > 1) {\n this.use((lunr as any).multiLanguage(...config.lang))\n }\n\n /* Compute functions to be removed from the pipeline */\n const fns = difference([\n \"trimmer\", \"stopWordFilter\", \"stemmer\"\n ], options.pipeline)\n\n /* Remove functions from the pipeline for registered languages */\n for (const lang of config.lang.map(language => (\n language === \"en\" ? lunr : (lunr as any)[language]\n ))) {\n for (const fn of fns) {\n this.pipeline.remove(lang[fn])\n this.searchPipeline.remove(lang[fn])\n }\n }\n\n /* Set up reference */\n this.ref(\"location\")\n\n /* Set up fields */\n this.field(\"title\", { boost: 1e3 })\n this.field(\"text\")\n this.field(\"tags\", { boost: 1e6, extractor: doc => {\n const { tags = [] } = doc as SearchDocument\n return tags.reduce((list, tag) => [\n ...list,\n ...lunr.tokenizer(tag)\n ], [] as lunr.Token[])\n } })\n\n /* Index documents */\n for (const doc of docs)\n this.add(doc, { boost: doc.boost })\n })\n }\n\n /**\n * Search for matching documents\n *\n * The search index which MkDocs provides is divided up into articles, which\n * contain the whole content of the individual pages, and sections, which only\n * contain the contents of the subsections obtained by breaking the individual\n * pages up at `h1` ... `h6`. As there may be many sections on different pages\n * with identical titles (for example within this very project, e.g. \"Usage\"\n * or \"Installation\"), they need to be put into the context of the containing\n * page. For this reason, section results are grouped within their respective\n * articles which are the top-level results that are returned.\n *\n * @param query - Query value\n *\n * @returns Search results\n */\n public search(query: string): SearchResult {\n if (query) {\n try {\n const highlight = this.highlight(query)\n\n /* Parse query to extract clauses for analysis */\n const clauses = parseSearchQuery(query)\n .filter(clause => (\n clause.presence !== lunr.Query.presence.PROHIBITED\n ))\n\n /* Perform search and post-process results */\n const groups = this.index.search(`${query}*`)\n\n /* Apply post-query boosts based on title and search query terms */\n .reduce((item, { ref, score, matchData }) => {\n const document = this.documents.get(ref)\n if (typeof document !== \"undefined\") {\n const { location, title, text, tags, parent } = document\n\n /* Compute and analyze search query terms */\n const terms = getSearchQueryTerms(\n clauses,\n Object.keys(matchData.metadata)\n )\n\n /* Highlight title and text and apply post-query boosts */\n const boost = +!parent + +Object.values(terms).every(t => t)\n item.push({\n location,\n title: highlight(title),\n text: highlight(text),\n ...tags && { tags: tags.map(highlight) },\n score: score * (1 + boost),\n terms\n })\n }\n return item\n }, [])\n\n /* Sort search results again after applying boosts */\n .sort((a, b) => b.score - a.score)\n\n /* Group search results by page */\n .reduce((items, result) => {\n const document = this.documents.get(result.location)\n if (typeof document !== \"undefined\") {\n const ref = \"parent\" in document\n ? document.parent!.location\n : document.location\n items.set(ref, [...items.get(ref) || [], result])\n }\n return items\n }, new Map())\n\n /* Generate search suggestions, if desired */\n let suggestions: string[] | undefined\n if (this.options.suggestions) {\n const titles = this.index.query(builder => {\n for (const clause of clauses)\n builder.term(clause.term, {\n fields: [\"title\"],\n presence: lunr.Query.presence.REQUIRED,\n wildcard: lunr.Query.wildcard.TRAILING\n })\n })\n\n /* Retrieve suggestions for best match */\n suggestions = titles.length\n ? 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    Glossary

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    Construct

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    • To construct a workflow in dewret is to pull the connected steps into a single structure.
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    Sub Workflow

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      A subworkflow is a nested or hierarchical workflow. It is a workflow defined within another workflow, allowing for the encapsulation and reuse of complex operations as a single, higher-level step in the parent workflow.

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      Specific type of task designed to encapsulate multiple tasks. Nested tasks are the culmination (or result) of multiple tasks represented as a single task in a dewret workflow.

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    Render

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    • To render a workflow is to generate an executable workflow in a specific workflow language such as CWL and Snakemake.
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    Step

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    • A step in a dewret workflow represents a single unit of work. It contains a single task and the arguments for that task. Corresponds to a CWL Step or a Snakemake Rule
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    • A task is the function scheduled to be executed later. Corresponds to a CWL Process
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    Workflow

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    • A workflow is designed to define, manage, and execute a series of tasks that make use of both local and global parameters.
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    Quickstart

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    Introduction

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    Description

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    Dewret is a tool designed for creating complex workflows, written in a dynamic style, to be rendered to a static representation. Dewret provides a programmatic python interface to multiple declarative workflow engines, where workflows are often written in a yaml-like syntax. It makes it easier for users to define tasks and organize them into workflows. Currently, Dewret supports two renderers: Snakemake and CWL, which generate yamls in the corresponding workflow languages.

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    What are Workflows?

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    Workflows are a collection of tasks or steps designed to automate complex processes. These processes are common in fields like data science, scientific computing and software development, where you can ensure automation. Traditionally, managing workflows can be challenging due to the diversity of backend systems and the complexity of configurations involved.

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    What Makes Dewret Unique? Why should I use Dewret?

    +

    Dewret stands out by providing a unified and simplified interface for workflow management, making it accessible to users with varying levels of experience. Here are some key features that make Dewret unique: +- Consistency: offers a consistent interface for defining tasks and workflows. +- Optimization: creating a declarative workflow opens up possibilities for static analysis and refactoring before execution. +- Customization: dewret offers the ability to create custom renderers for workflows in desired languages. This includes default support for CWL and Snakemake workflow languages. The capability to render a single workflow into multiple declarative languages enables users to experiment with different workflow engines. +- Git-versionable workflows: while code can be versioned, changes in a dynamic workflow may not clearly correspond to changes in the executed workflow. By defining a static workflow that is rendered from the dynamic or programmatic workflow, we maintain a precise and trackable history. +- Default Renderers: Snakemake and CWL. +- Debugging: a number of classes of workflow planning bugs will not appear until late in a simulation run that might take days or weeks. Having a declarative and static workflow definition document post-render provides enhanced possibilities for static analysis, helping to catch these issues before startup. +- Continuous Integration and Testing: complex dynamic workflows can be rapidly sense-checked in CI without needing all the hardware and internal algorithms present to run them.

    +

    Installation for pure users

    +

    If you simply want to use Dewret to run workflows, you can install it from PyPI or Conda.

    +

    From PyPI:

    +
    pip install dewret
+
    +

    From Conda:

    +
    conda install conda-forge::dewret
+
    +

    Installation for developers

    +

    From a cloned repository:

    +
    pip install -e .
+
    +

    Usage

    +

    You can render a simple Common Workflow Language CWL workflow from a graph composed of one or more tasks as follows:

    +
    # workflow.py
+
+from dewret.tasks import task
+
+@task()
+def increment(num: int) -> int:
+    return num + 1
+
    +
    $ python -m dewret --pretty workflow.py increment num:3
+
    +
    class: Workflow
+cwlVersion: 1.2
+outputs:
+  out:
+    outputSource: increment-e138626779553199eb2bd678356b640f-num
+    type: int
+steps:
+  increment-e138626779553199eb2bd678356b640f-num
+    in:
+      num:
+        default: 3
+    out:
+    - out
+    run: increment
+
    +

    By default dewret uses a dask backend so that dewret.task wraps a dask.delayed, and renders a CWL workflow.

    +

    Programmatic Usage

    +

    Building and rendering may be done programmatically, +which provides the opportunity to use custom renderers +and backends, as well as bespoke serialization or formatting.

    +
    >>> import sys
+>>> import yaml
+>>> from dewret.tasks import task, construct
+>>> from dewret.renderers.cwl import render
+>>> 
+>>> @task()
+... def increment(num: int) -> int:
+...     return num + 1
+>>>
+>>> result = increment(num=3)
+>>> workflow = construct(result, simplify_ids=True)
+>>> cwl = render(workflow)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  increment-1-num:
+    default: 3
+    label: num
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: increment-1/out
+    type: int
+steps:
+  increment-1:
+    in:
+      num:
+        source: increment-1-num
+    out:
+    - out
+    run: increment
+
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/docs/renderer_tutorial/index.html b/docs/renderer_tutorial/index.html new file mode 100644 index 00000000..7bacd0bd --- /dev/null +++ b/docs/renderer_tutorial/index.html @@ -0,0 +1,1502 @@ + + + + + + + + + + + + + + + + Renderer Tutorial - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
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    + + + +
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    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Step-by-Step Guide to Writing a Custom Renderer

    +

    1. Understand the Target Workflow Language

    +

    Before writing any code, it is essential to fully understand the target workflow language. This includes syntax, structure, and specific requirements. By breaking down each key dewret.workflow task into smaller components, you can better map your workflow definitions to the target language.

    +

    Example:

    +

    In Snakemake, a workflow task is generally created by: +1. Defining the task. - rule process_data +2. Defining the input required for the rule to run(dependencies). - input: "data/raw_data.txt" +3. Defining the output required for the rule to be considered finished. - output: "data/processed_data.txt" +4. Defining the actual work that the task will do. - in this case:shell: ...

    +
    rule process_data: # Example Snakemake rule/task
+    input:
+        "data/raw_data.txt"
+    output:
+        output_file="data/processed_data.txt"
+    run:
+        with open(output.output_file, "w") as f:
+            f.write("data")
+
+        return output_file
+
    +

    2. Create WorkflowDefinition.

    +

    The WorkflowDefinition class is responsible for transforming each step from a constructed dewret workflow into an executable step in the target workflow language (e.g. a Snakemake rule). This class should encapsulate workflow-level information, such as the list of steps to be executed, and any workflow-scope input/ouput. It should also contain a class method that initializes the WorkflowDefinition from an dewret Workflow (such as from_workflow below), and a method that renders the workflow as a Python dict (as in the render method below).

    +

    Example:

    +
    @define
+class WorkflowDefinition:
+    steps: list[StepDefinition]
+
+    # Returns a WorkflowDefinition instanace.
+    # Steps contains all of the tasks you want to convert to the target WL tasks.
+    @classmethod
+    def from_workflow(cls, workflow: Workflow) -> "WorkflowDefinition":
+        return cls(steps=[StepDefinition.from_step(step) for step in workflow.steps])
+
+    # Returns each task as a Snakemake executable rule.
+    def render(self) -> dict[str, RawType]:
+        return {
+            f"rule {step.name.replace("-", "_")}": step.render() for step in self.steps
+        }
+
    +

    3. Create a StepDefinition.

    +

    Create a StepsDefinition class create each of the code blocks needed for a rule(step) to be executable in Snakemake. +When you have defined each block in your target workflow language task from step 1, +you can go ahead and create, for each of the code blocks required to run a Snakemake rule, a BlockDefinition to handle the rendering of each block.

    +

    Example:

    +

    In the Snakemake example, we have created: +1. InputDefinition - Handles the input block, which contains what is required for a rule to be executed. It also handles the params block since the code for extracting the input and params blocks is similar. +2. RunDefinition - Handles the run block which contains the instructions needed for this specific task. +3. OutputDefinition - Handles the output block which is required for the rule to be considered successfully finished.

    +
    @define
+class StepDefinition:
+    """Represents a Snakemake-renderable step definition in a dewret workflow.
+
+    Attributes:
+        name (str): The name of the step.
+        run (str): The run block definition for the step.
+        params (List[str]): The parameter definitions for the step.
+        output (list[str]: The output definition for the step.
+        input (List[str]): The input definitions for the step.
+
+    Methods:
+        from_step(cls, step: Step) -> "StepDefinition": Constructs a StepDefinition
+            object from a Step object, extracting step information and components
+            from the step and converting them to Snakemake format.
+        render(self) -> dict[str, MainTypes]: Renders the step definition as a dictionary
+            suitable for use in Snakemake workflows.
+    """
+
+    # You can consider each step as a separate rule.
+    # Each field in this class represents a separate block in the rule definition
+    name: str # name of the rule
+    input: list[str] # Input block
+    params: list[str] # Params block
+    output: list[str] # Output block
+    run: list[str] # Run block - where the instructions for the task are
+
+    @classmethod
+    def from_step(cls, step: Step) -> "StepDefinition":
+        """Constructs a StepDefinition object from a Step.
+
+        Args:
+            step (Step): The Step object from which step information and components
+                are extracted.
+
+        Returns:
+            StepDefinition: A StepDefinition object containing the converted step
+                information and components.
+        """
+        input_block = InputDefinition.from_step(step).render()
+        run_block = RunDefinition.from_task(step.task).render()
+        output_block = OutputDefinition.from_step(step).render()
+        return cls(
+            name=step.name, 
+            run=run_block,
+            params=input_block["params"],
+            input=input_block["inputs"],
+            output=output_block,
+        )
+
+    def render(self) -> dict[str, MainTypes]:
+        """Renders the step definition as a dictionary.
+
+        Returns:
+            dict[str, MainTypes]: A dictionary containing the components of the step
+                definition, for use in Snakemake workflows.
+        """
+        return {
+            "run": self.run,
+            "input": self.input,
+            "params": self.params,
+            "output": self.output,
+        }
+
    +

    4. Create the Separate block definitions.

    +

    In this step, you'll define classes to handle the rendering of each code block required for a rule (step) to be executable in the target workflow language. Each of these classes will encapsulate the logic for converting parts of a workflow step into the target language format.

    +

    Example:

    +

    For the Snakemake workflow language, we will define:

    +
      +
    1. InputDefinition: Handles the input block and parameter block.
      2. +
    2. RunDefinition: Handles the run block.
      3. +
    3. OutputDefinition: Handles the output block.
      4. +
    +

    InputDefinition:

    +

    The InputDefinition class is responsible for rendering the inputs and parameters required for a Snakemake rule.

    +
    @define
+class InputDefinition:
+    """Represents input and parameter definitions block for a Snakemake-renderable workflow step.
+
+    Attributes:
+        inputs (List[str]): A list of input definitions.
+        params (List[str]): A list of parameter definitions.
+
+    Methods:
+        from_step(cls, step: Step) -> "InputDefinition": Constructs an InputDefinition
+            object from a Step object, extracting inputs and parameters and converting
+            them to Snakemake-compatible format.
+        render(self) -> dict[str, str]: Renders the input and parameter definitions
+            as a dictionary for use in Snakemake Input and Params blocks.
+    """
+
+    # As we already mention input and params block have similar generation
+    # So it made sence to encapsulate them into one Definition
+    inputs: list[str]
+    params: list[str]
+
+    @classmethod
+    def from_step(cls, step: Step) -> "InputDefinition":
+        """Constructs an InputDefinition object from a Step.
+
+        Args:
+            step (Step): The Step object from which input and parameter block definitions are
+                extracted.
+
+        Returns:
+            InputDefinition: An InputDefinition object.
+        """
+        params = []
+        inputs = []
+        # The keys represent the names of the arguments of the @tasks in our snakemake_workflow.py.
+        # The params represent the values.
+        for key, param in step.arguments.items():
+            # We check if the param is a reference.
+            # If it is then it's an input requirement for the rule to run, so we put it in the input block
+            if isinstance(param, Reference):
+                ref = ReferenceDefinition.from_reference(param).render().replace("-","_").replace("/out", ".output")
+                input = f"{key}=rules.{ref}.output_file"
+                inputs.append(input)
+                params.append(input + ",")
+            # If it's not - we put it in the params block for use in the RunDefinition
+            elif isinstance(param, Raw):
+                customized = f"{key}={to_snakemake_type(param)},"
+                params.append(customized)
+
+        # Since the params must be comma separated except the last one - we remove the last comma
+        if params:
+            params[len(params) - 1] = params[len(params) - 1].replace(",", "")
+
+        return cls(inputs=inputs, params=params)
+
+    def render(self) -> dict[str, list[str]]:
+        """Renders the input and parameter definitions as a dictionary.
+
+        Returns:
+            dict[str, list[MainTypes]]: A dictionary containing the input and parameter definitions,
+                for use in Snakemake Input and Params blocks.
+        """
+        return {"inputs": self.inputs, "params": self.params}
+
    +

    RunDefinition:

    +

    The RunDefinition class is responsible for rendering the run block, which contains the actual instructions for the task.

    +
    @define
+class RunDefinition:
+    # This is where we handle the execution of the task itself.
+    """Represents a Snakemake-renderable run block for a dewret workflow step.
+
+    Attributes:
+        method_name (str): The name of the method to be executed in the snakefile run block.
+        rel_import (str): The relative import path of the method.
+        args (List[str]): The arguments to be passed to the method.
+
+    Methods:
+        from_task(cls, task: Task) -> "RunDefinition": Constructs a RunDefinition
+            object from a Task object, extracting method information and arguments
+            from the task and converting them to Snakemake-compatible format.
+
+        render(self) -> list[str]: A list containing the import statement and the method
+            call statement, for use in Snakemake run block.
+    """
+
+    method_name: str
+    rel_import: str
+    args: list[str]
+
+    @classmethod
+    def from_task(cls, task: Task) -> "RunDefinition":
+        """Constructs a RunDefinition object from a Task.
+
+        Args:
+            task (Task): The Task object from which method information and arguments
+                are extracted.
+
+        Returns:
+            RunDefinition: A RunDefinition object containing the converted method
+                information and arguments.
+        """
+        # Since we can import our snakemake_workflow.py @tasks we need the relative path
+        relative_path = get_method_rel_path(task.target)
+        # If we need to make any customization to the import
+        rel_import = f"{relative_path}"
+
+        args = get_method_args(task.target)
+        signature = [
+            f"{param_name}=params.{param_name}"
+            for param_name in args.parameters.keys()
+        ]
+
+        return cls(method_name=task.name, rel_import=rel_import, args=signature)
+
+    def render(self) -> list[str]:
+        """Renders the run block as a list of strings.
+
+        Returns:
+            list[str]: A list containing the import statement and the method
+                call statement, for use in Snakemake run block.
+        """
+        signature = ", ".join(f"{arg}" for arg in self.args)
+        # The comma after the last element is mandatory for the structure of rule onces it's used in yaml.dump
+        return [
+            f"import {self.rel_import}\n",
+            f"{self.rel_import}.{self.method_name}({signature})\n",
+        ]
+
    +

    OutputDefinition:

    +

    The OutputDefinition class is responsible for rendering the output block, which specifies the output files or results that indicate the rule has successfully completed.

    +
    @define
+class OutputDefinition:
+    """Represents the output definition block for a Snakemake-renderable workflow step.
+
+    Attributes:
+        output_file (str): The output file definition.
+
+    Methods:
+        from_step(cls, step: Step) -> "OutputDefinition": Constructs an OutputDefinition
+            object from a Step object, extracting and converting the output file definition
+            to Snakemake-compatible format.
+
+        render(self) -> list[str]: Renders the output definition as a list
+            suitable for use in Snakemake Output block.
+    """
+
+    output_file: str
+
+    @classmethod
+    def from_step(cls, step: Step) -> "OutputDefinition":
+        """Constructs an OutputDefinition object from a Step.
+
+        Args:
+            step (Step): The Step object from which the output file definition is extracted.
+
+        Returns:
+            OutputDefinition: An OutputDefinition object, for use in Snakemake Output block.
+        """
+        # Since snakemake commonly communicates using files.
+        # Output file must always be called - `output_file`
+        # Further code could be added to handled if it's a reference in case we want take care of multiple tasks writing to the same output file.
+        output_file = step.arguments["output_file"]
+        if isinstance(output_file, Raw):
+            args = to_snakemake_type(output_file)
+
+        return cls(output_file=args)
+
+    def render(self) -> list[str]:
+        """Renders the output definition as a list.
+
+        Returns:
+            list[str]: A list containing the output file definition, for use in a Snakemake Output block.
+        """
+        # The comma after the last element is mandatory for the structure of rule onces it's used in yaml.dump
+        # It adds the new line to the output block
+        return [
+            f"output_file={self.output_file}",
+        ]
+
    +

    Integrate these block definitions into the StepDefinition class as demonstrated in Step 3. Each StepDefinition will use these block definitions to render the complete step in the target workflow language.

    +

    5. Helper methods.

    +

    In this step, you'll define helper methods that will assist you in converting workflow components into the target workflow language format. In our case these methods will handle type conversion, extracting method arguments, and computing relative paths.

    +

    Example:

    +

    We'll define the following helper methods for our Snakemake renderer:

    +
      +
    1. to_snakemake_type(param: Raw) -> str: Converts a raw type to a Snakemake-compatible Python type.
      2. +
    2. get_method_args(func: Lazy) -> inspect.Signature: Retrieves the argument names and types of a lazy-evaluatable function.
      3. +
    3. get_method_rel_path(func: Lazy) -> str: Computes the relative path of the module containing the given function.
      4. +
    +

    Type Conversion Helper:

    +
    # Basic types returned from dewret will look like this "str|valueOfParam".
+# We'll need to convert them.
+def to_Snakemake_type(param: Raw) -> str:
+    typ = str(param)
+    if typ.__contains__("str"):
+        return f'"{typ.replace("str|", "")}"'
+    elif typ.__contains__("bool"):
+        return typ.replace("bool|", "")
+    elif typ.__contains__("dict"):
+        return typ.replace("dict|", "")
+    elif typ.__contains__("list"):
+        return typ.replace("list|", "")
+    elif typ.__contains__("float"):
+        return typ.replace("float|", "")
+    elif typ.__contains__("int"):
+        return typ.replace("int|", "")
+    else:
+        raise TypeError(f"Cannot render complex type ({typ})")
+
    +

    Argument Extraction Helper:

    +
    # We need to get the signature of the method. 
+def get_method_args(func: Lazy) -> inspect.Signature:
+    args = inspect.signature(func)
+    return args
+
    +

    Relative Path Computation Helper:

    +
    # Computes the relative path
+def get_method_rel_path(func: Lazy) -> str:
+    source_file = inspect.getsourcefile(func)
+    if source_file:
+        relative_path = os.path.relpath(source_file, start=os.getcwd())
+        module_name = os.path.splitext(relative_path)[0].replace(os.path.sep, ".")
+
+    return module_name
+
    +

    Imports and custom types required in the SMK example:

    +
    import os
+import yaml
+import inspect
+import typing
+
+from attrs import define
+from dewret.utils import Raw, BasicType
+from dewret.workflow import Lazy
+from dewret.workflow import Reference, Workflow, Step, Task
+
+RawType = typing.Union[BasicType, list[str], list["RawType"], dict[str, "RawType"]]
+
    +

    To run this example:

    +
      +
    1. Import the snakemake renderer into your @tasks file
      2. +
    2. There's an example in snakemake_tasks.py
      3. +
    3. Run it: +
      python snakemake_tasks.py
+
      4. +
    + +

    A: Get details on how that happens and probably yes.

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    Renderers

    +

    Renderers are a function that takes a task, which can +be assumed to have a __workflow__ member of type Workflow, and return +a YAML-serializable nested dict structure.

    +

    CWL

    +

    The default renderer is for the Common Workflow Language. It implements a very small subset +of functionality, and is not yet strictly standards compliant. It assumes that all run +names can be interpreted in the context of the workflow module's global scope.

    +

    Custom

    +

    ...

    + + + + + + +
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    + + + + + + + + + \ No newline at end of file diff --git a/docs/workflows/index.html b/docs/workflows/index.html new file mode 100644 index 00000000..9deba860 --- /dev/null +++ b/docs/workflows/index.html @@ -0,0 +1,1639 @@ + + + + + + + + + + + + + + + + Workflows - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    Workflows

    +

    Description

    +

    A dewret workflow is composed of one or more steps that may make use of both local and global parameters. Each step is defined by a dewret task that is created by using the @task() decorator, and each task may be used by multiple steps.

    +

    Setup

    +

    We can pull in dewret tools to produce CWL with a small number of imports.

    +
    >>> import sys
+>>> import yaml
+>>> from dewret.tasks import task, construct
+>>> from dewret.workflow import param
+>>> from dewret.renderers.cwl import render
+
    +

    Dependencies

    +

    Specifying step interdependencies is possible by combining lazy-evaluated function +calls. The output series of steps is not guaranteed to be in order of execution.

    +

    Dewret hashes the parameters to identify and unify steps. This lets you do, for example:

    +
    graph TD
+    A[increment] --> B[double]
+    A[increment] --> C[mod10]
+    B[double] --> D[sum]
+    C[mod10] --> D[sum]
    +

    In code, this would be:

    +
    >>> import sys
+>>> import yaml
+>>> from dewret.tasks import task, construct
+>>> from dewret.renderers.cwl import render
+>>> @task()
+... def increment(num: int) -> int:
+...     """Increment an integer."""
+...     return num + 1
+>>> 
+>>> @task()
+... def double(num: int) -> int:
+...     """Double an integer."""
+...     return 2 * num
+>>> 
+>>> @task()
+... def mod10(num: int) -> int:
+...     """Take num mod 10."""
+...     return num % 10
+>>> 
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...     """Add two integers."""
+...     return left + right
+>>>
+>>> result = sum(
+...     left=double(num=increment(num=23)),
+...     right=mod10(num=increment(num=23))
+... )
+>>> wkflw = construct(result, simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  increment-1-num:
+    default: 23
+    label: num
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: sum-1/out
+    type: int
+steps:
+  double-1:
+    in:
+      num:
+        source: increment-1/out
+    out:
+    - out
+    run: double
+  increment-1:
+    in:
+      num:
+        source: increment-1-num
+    out:
+    - out
+    run: increment
+  mod10-1:
+    in:
+      num:
+        source: increment-1/out
+    out:
+    - out
+    run: mod10
+  sum-1:
+    in:
+      left:
+        source: double-1/out
+      right:
+        source: mod10-1/out
+    out:
+    - out
+    run: sum
+
    +

    Notice that the increment tasks appears twice in the CWL workflow definition, being referenced twice in the python code above. +This duplication can be avoided by explicitly indicating that the parameters are the same, with the param function.

    +
    >>> import sys
+>>> import yaml
+>>> from dewret.workflow import param
+>>> from dewret.tasks import task, construct
+>>> from dewret.renderers.cwl import render
+>>> @task()
+... def increment(num: int) -> int:
+...     """Increment an integer."""
+...     return num + 1
+>>> 
+>>> @task()
+... def double(num: int) -> int:
+...     """Double an integer."""
+...     return 2 * num
+>>> 
+>>> @task()
+... def mod10(num: int) -> int:
+...     """Take num mod 10."""
+...     return num % 10
+>>> 
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...     """Add two integers."""
+...     return left + right
+>>>
+>>> num = param("num", default=3)
+>>> result = sum(
+...     left=double(num=increment(num=num)),
+...     right=mod10(num=increment(num=num))
+... )
+>>> wkflw = construct(result, simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  num:
+    default: 3
+    label: num
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: sum-1/out
+    type: int
+steps:
+  double-1:
+    in:
+      num:
+        source: increment-1/out
+    out:
+    - out
+    run: double
+  increment-1:
+    in:
+      num:
+        source: num
+    out:
+    - out
+    run: increment
+  mod10-1:
+    in:
+      num:
+        source: increment-1/out
+    out:
+    - out
+    run: mod10
+  sum-1:
+    in:
+      left:
+        source: double-1/out
+      right:
+        source: mod10-1/out
+    out:
+    - out
+    run: sum
+
    +

    Parameters

    +

    The tool will spot global variables that you have used when building your tasks, +and treat them as parameters. It will try to get the type from the typehint, or +the value that you have set it to. This only works for basic types (and dict/lists of +those).

    +

    While global variables are implicit input to the Python function note that:

    +
      +
    1. in CWL, they will be rendered as explicit global input to a step
      2. +
    2. as input, they are read-only, and must not be updated
      3. +
    +

    For example: +

    >>> import sys
+>>> import yaml
+>>> from dewret.workflow import param
+>>> from dewret.tasks import task, construct
+>>> from dewret.renderers.cwl import render
+>>> INPUT_NUM = 3
+>>> @task()
+... def rotate(num: int) -> int:
+...    """Rotate an integer."""
+...    return (num + INPUT_NUM) % INPUT_NUM
+>>>
+>>> result = rotate(num=5)
+>>> wkflw = construct(result, simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  INPUT_NUM:
+    default: 3
+    label: INPUT_NUM
+    type: int
+  rotate-1-num:
+    default: 5
+    label: num
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: rotate-1/out
+    type: int
+steps:
+  rotate-1:
+    in:
+      INPUT_NUM:
+        source: INPUT_NUM
+      num:
+        source: rotate-1-num
+    out:
+    - out
+    run: rotate
+

    +

    Nested tasks

    +

    When you wish to combine tasks together programmatically, +you can use nested tasks. These are run at render time, not +execution time. In other words, they do not appear in the +final graph, and so must only combine other tasks.

    +

    For example:

    +
    graph TD
+    A[rotate] --> B[rotate]
+    B[rotate] --> C[double_rotate]
    +

    As code:

    +

    >>> import sys
+>>> import yaml
+>>> from dewret.core import set_configuration
+>>> from dewret.tasks import task, construct, workflow
+>>> from dewret.renderers.cwl import render
+>>> INPUT_NUM = 3
+>>> @task()
+... def rotate(num: int) -> int:
+...     """Rotate an integer."""
+...     return (num + INPUT_NUM) % INPUT_NUM
+>>>
+>>> @workflow()
+... def double_rotate(num: int) -> int:
+...     """Rotate an integer twice."""
+...     return rotate(num=rotate(num=num))
+>>>
+>>> with set_configuration(flatten_all_nested=True):
+...     result = double_rotate(num=3)
+...     wkflw = construct(result, simplify_ids=True)
+...     cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  INPUT_NUM:
+    default: 3
+    label: INPUT_NUM
+    type: int
+  num:
+    default: 3
+    label: num
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: rotate-1/out
+    type: int
+steps:
+  rotate-1:
+    in:
+      INPUT_NUM:
+        source: INPUT_NUM
+      num:
+        source: rotate-2/out
+    out:
+    - out
+    run: rotate
+  rotate-2:
+    in:
+      INPUT_NUM:
+        source: INPUT_NUM
+      num:
+        source: num
+    out:
+    - out
+    run: rotate
+
    +Note that, as with all dewret calculations, only the steps +necessary to achieve the ultimate output are included in the final +graph. Therefore, nested tasks must return a step execution +(task that is being called) that forces any other calculations +you wish to happen. In other words, if a task in a +nested task does not have an impact on the return value, +it will disappear. +For example, the following code renders the same workflow as in the previous example:

    +
    @workflow()
+def double_rotate(num: int) -> int:
+   """Rotate an integer twice."""
+   unused_var = increment(num=num)
+   return rotate(num=rotate(num=num))
+
    +

    Step Output Fields

    +

    Each step, by default, is treated as having +a single result. However, we allow a mechanism +for specifying multiple fields, using attrs or dataclasses.

    +

    Where needed, fields can be accessed outside of tasks +by dot notation and dewret will map that access to a +specific output field in CWL.

    +

    Note that in the example below, shuffle is still +only seen once in the graph:

    +
    graph TD
+    A[shuffle] --> B[hearts]
+    A[shuffle] --> C[diamonds]
+    B[hearts] --> D[sum]
+    C[diamonds] --> D[sum]
    +

    As code:

    +
    >>> import sys
+>>> import yaml
+>>> from attrs import define
+>>> from numpy import random
+>>> from dewret.tasks import task, construct
+>>> from dewret.renderers.cwl import render
+>>> @define
+... class PackResult:
+...     hearts: int
+...     clubs: int
+...     spades: int
+...     diamonds: int
+>>>
+>>> @task()
+... def shuffle(max_cards_per_suit: int) -> PackResult:
+...    """Fill a random pile from a card deck, suit by suit."""
+...    return PackResult(
+...        hearts=random.randint(max_cards_per_suit),
+...        clubs=random.randint(max_cards_per_suit),
+...        spades=random.randint(max_cards_per_suit),
+...        diamonds=random.randint(max_cards_per_suit)
+...    )
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...    return left + right
+>>> red_total = sum(
+...     left=shuffle(max_cards_per_suit=13).hearts,
+...     right=shuffle(max_cards_per_suit=13).diamonds
+... )
+>>> wkflw = construct(red_total, simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  shuffle-1-max_cards_per_suit:
+    default: 13
+    label: max_cards_per_suit
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: sum-1/out
+    type: int
+steps:
+  shuffle-1:
+    in:
+      max_cards_per_suit:
+        source: shuffle-1-max_cards_per_suit
+    out:
+      clubs:
+        label: clubs
+        type: int
+      diamonds:
+        label: diamonds
+        type: int
+      hearts:
+        label: hearts
+        type: int
+      spades:
+        label: spades
+        type: int
+    run: shuffle
+  sum-1:
+    in:
+      left:
+        source: shuffle-1/hearts
+      right:
+        source: shuffle-1/diamonds
+    out:
+    - out
+    run: sum
+
    +

    Here, we show the same example with dataclasses.

    +
    >>> import sys
+>>> import yaml
+>>> from dataclasses import dataclass
+>>> from numpy import random
+>>> from dewret.tasks import task, construct
+>>> from dewret.renderers.cwl import render
+>>> @dataclass
+... class PackResult:
+...     hearts: int
+...     clubs: int
+...     spades: int
+...     diamonds: int
+>>>
+>>> @task()
+... def shuffle(max_cards_per_suit: int) -> PackResult:
+...    """Fill a random pile from a card deck, suit by suit."""
+...    return PackResult(
+...        hearts=random.randint(max_cards_per_suit),
+...        clubs=random.randint(max_cards_per_suit),
+...        spades=random.randint(max_cards_per_suit),
+...        diamonds=random.randint(max_cards_per_suit)
+...    )
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...    return left + right
+>>>
+>>> red_total = sum(
+...     left=shuffle(max_cards_per_suit=13).hearts,
+...     right=shuffle(max_cards_per_suit=13).diamonds
+... )
+>>> wkflw = construct(red_total, simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  shuffle-1-max_cards_per_suit:
+    default: 13
+    label: max_cards_per_suit
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: sum-1/out
+    type: int
+steps:
+  shuffle-1:
+    in:
+      max_cards_per_suit:
+        source: shuffle-1-max_cards_per_suit
+    out:
+      clubs:
+        label: clubs
+        type: int
+      diamonds:
+        label: diamonds
+        type: int
+      hearts:
+        label: hearts
+        type: int
+      spades:
+        label: spades
+        type: int
+    run: shuffle
+  sum-1:
+    in:
+      left:
+        source: shuffle-1/hearts
+      right:
+        source: shuffle-1/diamonds
+    out:
+    - out
+    run: sum
+
    +

    Subworkflow

    +

    A special form of nested task is available to help divide up +more complex workflows: the subworkflow. By wrapping logic in subflows, +dewret will produce multiple output workflows that reference each other.

    +
    >>> import sys
+>>> import yaml
+>>> from attrs import define
+>>> from numpy import random
+>>> from dewret.tasks import task, construct, workflow
+>>> from dewret.renderers.cwl import render
+>>> @define
+... class PackResult:
+...     hearts: int
+...     clubs: int
+...     spades: int
+...     diamonds: int
+>>>
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...    return left + right
+>>>
+>>> @task()
+... def shuffle(max_cards_per_suit: int) -> PackResult:
+...    """Fill a random pile from a card deck, suit by suit."""
+...    return PackResult(
+...        hearts=random.randint(max_cards_per_suit),
+...        clubs=random.randint(max_cards_per_suit),
+...        spades=random.randint(max_cards_per_suit),
+...        diamonds=random.randint(max_cards_per_suit)
+...    )
+>>> @workflow()
+... def red_total() -> int:
+...     return sum(
+...         left=shuffle(max_cards_per_suit=13).hearts,
+...         right=shuffle(max_cards_per_suit=13).diamonds
+...     )
+>>> @workflow()
+... def black_total() -> int:
+...     return sum(
+...         left=shuffle(max_cards_per_suit=13).spades,
+...         right=shuffle(max_cards_per_suit=13).clubs
+...     )
+>>> total = sum(left=red_total(), right=black_total())
+>>> wkflw = construct(total, simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs: {}
+outputs:
+  out:
+    label: out
+    outputSource: sum-1/out
+    type: int
+steps:
+  black_total-1:
+    in: {}
+    out:
+    - out
+    run: black_total
+  red_total-1:
+    in: {}
+    out:
+    - out
+    run: red_total
+  sum-1:
+    in:
+      left:
+        source: red_total-1/out
+      right:
+        source: black_total-1/out
+    out:
+    - out
+    run: sum
+
    +

    As we have used subworkflow to wrap the colour totals, the outer workflow +contains references to them only. The subworkflows are now returned by render +as a second term.

    +
    >>> import sys
+>>> import yaml
+>>> from attrs import define
+>>> from numpy import random
+>>> from dewret.tasks import task, construct, workflow
+>>> from dewret.renderers.cwl import render
+>>> @define
+... class PackResult:
+...     hearts: int
+...     clubs: int
+...     spades: int
+...     diamonds: int
+>>>
+>>> @task()
+... def shuffle(max_cards_per_suit: int) -> PackResult:
+...    """Fill a random pile from a card deck, suit by suit."""
+...    return PackResult(
+...        hearts=random.randint(max_cards_per_suit),
+...        clubs=random.randint(max_cards_per_suit),
+...        spades=random.randint(max_cards_per_suit),
+...        diamonds=random.randint(max_cards_per_suit)
+...    )
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...    return left + right
+>>>
+>>> @workflow()
+... def red_total() -> int:
+...     return sum(
+...         left=shuffle(max_cards_per_suit=13).hearts,
+...         right=shuffle(max_cards_per_suit=13).diamonds
+...     )
+>>> @workflow()
+... def black_total() -> int:
+...     return sum(
+...         left=shuffle(max_cards_per_suit=13).spades,
+...         right=shuffle(max_cards_per_suit=13).clubs
+...     )
+>>> total = sum(left=red_total(), right=black_total())
+>>> wkflw = construct(total, simplify_ids=True)
+>>> cwl = render(wkflw)
+>>> yaml.dump(cwl["red_total-1"], sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs: {}
+outputs:
+  out:
+    label: out
+    outputSource: sum-1-1/out
+    type: int
+steps:
+  shuffle-1-1:
+    in:
+      max_cards_per_suit:
+        default: 13
+    out:
+      clubs:
+        label: clubs
+        type: int
+      diamonds:
+        label: diamonds
+        type: int
+      hearts:
+        label: hearts
+        type: int
+      spades:
+        label: spades
+        type: int
+    run: shuffle
+  sum-1-1:
+    in:
+      left:
+        source: shuffle-1-1/hearts
+      right:
+        source: shuffle-1-1/diamonds
+    out:
+    - out
+    run: sum
+
    +

    Input Factories

    +

    Sometimes we want to take complex Python input, not just raw types. +Not all serialization support this, but the factory function lets us +wrap a simple call, usually a constructor, that takes only raw arguments. +This can then rendered as either a step or a parameter depending on whether +the chosen renderer has the capability.

    +

    Below is the default output, treating Pack as a task.

    +
    >>> import sys
+>>> import yaml
+>>> from dewret.tasks import workflow, factory, workflow, construct, task
+>>> from attrs import define
+>>> from dewret.renderers.cwl import render
+>>> @define
+... class PackResult:
+...     hearts: int
+...     clubs: int
+...     spades: int
+...     diamonds: int
+>>>
+>>> Pack = factory(PackResult)
+>>>
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...    return left + right
+>>>
+>>> @workflow()
+... def black_total(pack: PackResult) -> int:
+...     return sum(
+...         left=pack.spades,
+...         right=pack.clubs
+...     )
+>>> pack = Pack(hearts=13, spades=13, diamonds=13, clubs=13)
+>>> wkflw = construct(black_total(pack=pack), simplify_ids=True)
+>>> cwl = render(wkflw)["__root__"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  PackResult-1-clubs:
+    default: 13
+    label: clubs
+    type: int
+  PackResult-1-diamonds:
+    default: 13
+    label: diamonds
+    type: int
+  PackResult-1-hearts:
+    default: 13
+    label: hearts
+    type: int
+  PackResult-1-spades:
+    default: 13
+    label: spades
+    type: int
+outputs:
+  out:
+    label: out
+    outputSource: black_total-1/out
+    type: int
+steps:
+  PackResult-1:
+    in:
+      clubs:
+        source: PackResult-1-clubs
+      diamonds:
+        source: PackResult-1-diamonds
+      hearts:
+        source: PackResult-1-hearts
+      spades:
+        source: PackResult-1-spades
+    out:
+      clubs:
+        label: clubs
+        type: int
+      diamonds:
+        label: diamonds
+        type: int
+      hearts:
+        label: hearts
+        type: int
+      spades:
+        label: spades
+        type: int
+    run: PackResult
+  black_total-1:
+    in:
+      pack:
+        source: PackResult-1/out
+    out:
+    - out
+    run: black_total
+
    +

    The CWL renderer is also able to treat pack as a parameter, if complex +types are allowed.

    +
    >>> import sys
+>>> import yaml
+>>> from dewret.tasks import task, factory, workflow, construct
+>>> from attrs import define
+>>> from dewret.renderers.cwl import render
+>>> @define
+... class PackResult:
+...     hearts: int
+...     clubs: int
+...     spades: int
+...     diamonds: int
+>>>
+>>> Pack = factory(PackResult)
+>>> @task()
+... def sum(left: int, right: int) -> int:
+...    return left + right
+>>>
+>>> @workflow()
+... def black_total(pack: PackResult) -> int:
+...     return sum(
+...         left=pack.spades,
+...         right=pack.clubs
+...     )
+>>> pack = Pack(hearts=13, spades=13, diamonds=13, clubs=13)
+>>> wkflw = construct(black_total(pack=pack), simplify_ids=True)
+>>> cwl = render(wkflw, allow_complex_types=True, factories_as_params=True)["black_total-1"]
+>>> yaml.dump(cwl, sys.stdout, indent=2)
+class: Workflow
+cwlVersion: 1.2
+inputs:
+  pack:
+    label: pack
+    type: record
+outputs:
+  out:
+    label: out
+    outputSource: sum-1-1/out
+    type: int
+steps:
+  sum-1-1:
+    in:
+      left:
+        source: pack/spades
+      right:
+        source: pack/clubs
+    out:
+    - out
+    run: sum
+
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/index.html b/index.html new file mode 100644 index 00000000..e2abbbeb --- /dev/null +++ b/index.html @@ -0,0 +1,790 @@ + + + + + + + + + + + + + + + + dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    dewret

    +

    DEclarative Workflow REndering Tool

    +

    Pron: durr-it, like "durable"

    +

    Introduction

    +

    Dewret allows certain workflows written in a dynamic +style to be rendered to a static representation.

    +

    Advantages of doing so include:

    +
      +
    • git-versionable workflows: while code can be versioned, the changes of a dynamic workflow + do not necessarily clearly correspond to changes in the executed workflow. This maintains + a precise trackable history.
      • +
    • plan and play: the workflow can be rapidly iterated, analysed and optimized before it + is sent for real execution on expensive or restricted HPC hardware.
      • +
    • optimization: creating the workflow explicitly opens up possibilities for static analysis + and refactoring before real execution.
      • +
    • debugging: a number of classes of workflow planning bugs will not appear until late + in a simulation run that might take days or weeks. This catches them before startup.
      • +
    • continuous integration and testing: complex dynamic workflows can be rapidly sense-checked + in CI without needing all the hardware and internal algorithms present to run them.
      • +
    +

    Documentation

    +

    For further information, see the documentation.

    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/annotations/index.html b/reference/dewret/annotations/index.html new file mode 100644 index 00000000..32d6b97c --- /dev/null +++ b/reference/dewret/annotations/index.html @@ -0,0 +1,1106 @@ + + + + + + + + + + + + + + + + Annotations - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    +
    + + + + + + + + + +

    Module dewret.annotations

    +

    Tooling for managing annotations.

    +

    Provides FunctionAnalyser, a toolkit that takes a Callable and can interrogate it +for annotations, with some intelligent searching beyond the obvious location.

    +

    Variables

    +
    AtRender
+
    +
    Fixed
+
    +
    T
+
    +

    Classes

    +

    FunctionAnalyser

    +
    class FunctionAnalyser(
+    fn: Callable[..., Any]
+)
+
    +

    Convenience class for analysing a function with reduced duplication of effort.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    _fnNonethe wrapped callableNone
    _annotationsNonestored annotations for the function.None
    +

    Instance variables

    +
    free_vars
+
    +

    Get the free variables for this Callable.

    +
    globals
+
    +

    Get the globals for this Callable.

    +
    return_type
+
    +

    Return type of the callable.

    +

    Returns: expected type of the return value.

    +

    Methods

    +

    argument_has

    +
    def argument_has(
+    self,
+    arg: str,
+    annotation: type,
+    exhaustive: bool = False
+) -> bool
+
    +

    Check if the named argument has the given annotation.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    argNoneargument to retrieve.None
    annotationNoneAnnotated to search for.None
    exhaustiveNonewhether to check the globals and other modules.None
    +

    get_all_imported_names

    +
    def get_all_imported_names(
+    self
+) -> dict[str, tuple[module, str]]
+
    +

    Find all of the annotations that were imported into this module.

    +

    get_all_module_names

    +
    def get_all_module_names(
+    self
+) -> dict[str, typing.Any]
+
    +

    Find all of the annotations within this module.

    +

    get_argument_annotation

    +
    def get_argument_annotation(
+    self,
+    arg: str,
+    exhaustive: bool = False
+) -> Any
+
    +

    Retrieve the annotations for this argument.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    argNonename of the argument.None
    exhaustiveNoneTrue if we should search outside the function itself, into the module globals, and into imported modules.None
    +

    is_at_construct_arg

    +
    def is_at_construct_arg(
+    self,
+    arg: str,
+    exhaustive: bool = False
+) -> bool
+
    +

    Convience function to check for AtConstruct, wrapping FunctionAnalyser.argument_has.

    +

    with_new_globals

    +
    def with_new_globals(
+    self,
+    new_globals: dict[str, typing.Any]
+) -> Callable[..., Any]
+
    +

    Create a Callable that will run the current Callable with new globals.

    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/backends/backend_dask/index.html b/reference/dewret/backends/backend_dask/index.html new file mode 100644 index 00000000..559a3a38 --- /dev/null +++ b/reference/dewret/backends/backend_dask/index.html @@ -0,0 +1,1220 @@ + + + + + + + + + + + + + + + + Backend Dask - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.backends.backend_dask

    +

    Dask backend.

    +

    Lazy-evaluation via dask.delayed.

    +

    Variables

    +
    config
+
    +

    Functions

    +

    is_lazy

    +
    def is_lazy(
+    task: Any
+) -> bool
+
    +

    Checks if a task is really a lazy-evaluated function for this backend.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonesuspected lazy-evaluated function.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneTrue if so, False otherwise.
    +

    lazy

    +
    def lazy(
+    obj='__no__default__',
+    name=None,
+    pure=None,
+    nout=None,
+    traverse=True
+)
+
    +

    Wraps a function or object to produce a Delayed.

    +

    Delayed objects act as proxies for the object they wrap, but all +operations on them are done lazily by building up a dask graph internally.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    objobjectThe function or object to wrapNone
    nameDask keyThe key to use in the underlying graph for the wrapped object. Defaults
    to hashing content. Note that this only affects the name of the object
    wrapped by this call to delayed, and not the output of delayed
    function calls - for that use dask_key_name= as described below.

    .. note::

    Because this name is used as the key in task graphs, you should
    ensure that it uniquely identifies obj. If you'd like to provide
    a descriptive name that is still unique, combine the descriptive name
    with :func:dask.base.tokenize of the array_like. See
    :ref:graphs for more.    		s
    pureboolIndicates whether calling the resulting Delayed object is a pure
    operation. If True, arguments to the call are hashed to produce
    deterministic keys. If not provided, the default is to check the global
    delayed_pure setting, and fallback to False if unset.    		to
    noutintThe number of outputs returned from calling the resulting Delayed
    object. If provided, the Delayed output of the call can be iterated
    into nout objects, allowing for unpacking of results. By default
    iteration over Delayed objects will error. Note, that nout=1
    expects obj to return a tuple of length 1, and consequently for
    nout=0, obj should return an empty tuple.    		iteration
    traverseboolBy default dask traverses builtin python collections looking for dask
    objects passed to delayed. For large collections this can be
    expensive. If obj doesn't contain any dask objects, set
    traverse=False to avoid doing this traversal.    		dask
    +

    run

    +
    def run(
+    workflow: dewret.workflow.Workflow | None,
+    task: dewret.workflow.Lazy | list[dewret.workflow.Lazy] | tuple[dewret.workflow.Lazy],
+    thread_pool: concurrent.futures.thread.ThreadPoolExecutor | None = None,
+    **kwargs: Any
+) -> Any
+
    +

    Execute a task as the output of a workflow.

    +

    Runs a task with dask.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneWorkflow in which to record the execution.None
    taskNonedask.delayed function, wrapped by dewret, that we wish to compute.None
    thread_poolNonethread pool for executing the workflows, to allow initialization of configuration contextvars.None
    **kwargsNoneany configuration arguments for this backend.None
    +

    unwrap

    +
    def unwrap(
+    task: dewret.workflow.Lazy
+) -> collections.abc.Callable[..., typing.Any]
+
    +

    Unwraps a lazy-evaluated function to get the function.

    +

    In recent dask (>=2024.3) this works with inspect.wraps, but earlier +versions do not have the __wrapped__ property.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonetask to be unwrapped.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneOriginal target.
    +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    RuntimeErrorif the task is not a wrapped function.
    +

    Classes

    +

    Delayed

    +
    class Delayed(
+    *args,
+    **kwargs
+)
+
    +

    Description of a dask delayed.

    +

    Since dask.delayed does not have a hintable type, this +stands in its place, making sure that all the features of a +dask.delayed are available.

    +

    More info: https://github.com/dask/dask/issues/7779

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Methods

    +

    compute

    +
    def compute(
+    self,
+    __workflow__: dewret.workflow.Workflow | None
+) -> dewret.workflow.StepReference[typing.Any]
+
    +

    Evaluate this dask.delayed.

    +

    Evaluate a delayed (dask lazy-evaluated) function. dewret +will have replaced it with a wrapper that expects a Workflow +and all arguments will already be known to the wrapped delayed +so the signature here is simple.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneWorkflow that this is tied to, if applicable.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReference to the final output step.
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/backends/index.html b/reference/dewret/backends/index.html new file mode 100644 index 00000000..ff943767 --- /dev/null +++ b/reference/dewret/backends/index.html @@ -0,0 +1,783 @@ + + + + + + + + + + + + + + + + Index - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.backends

    +

    Backends for supplying lazy-evaluation.

    +

    By default, we use dask, but it is possible to extend this functionality +as the behaviour is encapsulated within these modules.

    +

    Sub-modules

    + + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/core/index.html b/reference/dewret/core/index.html new file mode 100644 index 00000000..688595d6 --- /dev/null +++ b/reference/dewret/core/index.html @@ -0,0 +1,8678 @@ + + + + + + + + + + + + + + + + Core - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.core

    +

    Base classes that need to be available everywhere.

    +

    Mainly tooling around configuration, protocols and superclasses for References +and WorkflowComponents, that are concretized elsewhere.

    +

    Variables

    +
    BasicType
+
    +
    CONFIGURATION
+
    +
    ExprType
+
    +
    FirmType
+
    +
    RawType
+
    +
    RenderConfiguration
+
    +
    T
+
    +
    U
+
    +

    Functions

    +

    default_construct_config

    +
    def default_construct_config(
+    
+) -> dewret.core.ConstructConfiguration
+
    +

    Gets the default construct-time configuration.

    +

    This is the primary mechanism for configuring dewret internals, so these defaults +should be carefully chosen and, if they change, that likely has an impact on backwards compatibility +from a SemVer perspective.

    +

    Returns: configuration dictionary with default construct values.

    +

    default_renderer_config

    +
    def default_renderer_config(
+    
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Gets the default renderer configuration.

    +

    This may be called frequently, but is cached so note that any changes to the +wrapped config function will not be reflected during the process.

    +

    It is a light wrapper for default_config in the supplier renderer module.

    +

    Returns: the default configuration dict for the chosen renderer.

    +

    get_configuration

    +
    def get_configuration(
+    key: str
+) -> Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]
+
    +

    Retrieve the configuration or (silently) return the default.

    +

    Helps avoid a proliferation of set_configuration calls by not erroring if it has not been called. +However, the cost is that the user may accidentally put configuration-affected logic outside a +set_configuration call and be surprised that the behaviour is inexplicibly not as expected.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    keyNoneconfiguration key to retrieve.None
    +

    get_render_configuration

    +
    def get_render_configuration(
+    key: str
+) -> Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]
+
    +

    Retrieve configuration for the active renderer.

    +

    Finds the current user-set configuration, defaulting back to the chosen renderer module's declared +defaults.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    keyNoneconfiguration key to retrieve.None
    +

    set_configuration

    +
    def set_configuration(
+    **kwargs: *<class 'dewret.core.ConstructConfigurationTypedDict'>
+) -> Iterator[_contextvars.ContextVar[dewret.core.GlobalConfiguration]]
+
    +

    Sets the construct-time configuration.

    +

    This is a context manager, so that a setting can be temporarily overridden and automatically restored.

    +

    set_render_configuration

    +
    def set_render_configuration(
+    kwargs: dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+) -> Iterator[_contextvars.ContextVar[dewret.core.GlobalConfiguration]]
+
    +

    Sets the render-time configuration.

    +

    This is a context manager, so that a setting can be temporarily overridden and automatically restored.

    +

    Returns: the yielded global configuration ContextVar.

    +

    strip_annotations

    +
    def strip_annotations(
+    parent_type: type
+) -> tuple[type, tuple[str]]
+
    +

    Discovers and removes annotations from a parent type.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    parent_typeNonea type, possibly Annotated.None
    +

    Classes

    +

    BaseRenderModule

    +
    class BaseRenderModule(
+    *args,
+    **kwargs
+)
+
    +

    Common routines for all renderer modules.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Descendants

    +
      +
    • dewret.core.RawRenderModule
      • +
    • dewret.core.StructuredRenderModule
      • +
    +

    Static methods

    +

    default_config

    +
    def default_config(
+    
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Retrieve default settings.

    +

    These will not change during execution, but can be overridden by dewret.core.set_render_configuration.

    +

    Returns: a static, serializable dict.

    +

    ConstructConfiguration

    +
    class ConstructConfiguration(
+    flatten_all_nested: bool = False,
+    allow_positional_args: bool = False,
+    allow_plain_dict_fields: bool = False,
+    field_separator: str = '/',
+    field_index_types: str = 'int',
+    simplify_ids: bool = False
+)
+
    +

    Basic configuration of the construction process.

    +

    Holds configuration that may be relevant to construst(...) calls or, realistically, +anything prior to rendering. It should hold generic configuration that is renderer-independent.

    +

    Instance variables

    +
    allow_plain_dict_fields
+
    +
    allow_positional_args
+
    +
    field_index_types
+
    +
    field_separator
+
    +
    flatten_all_nested
+
    +
    simplify_ids
+
    +

    ConstructConfigurationTypedDict

    +
    class ConstructConfigurationTypedDict(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Basic configuration of the construction process.

    +

    Holds configuration that may be relevant to construst(...) calls or, realistically, +anything prior to rendering. It should hold generic configuration that is renderer-independent.

    +

    THIS MUST BE KEPT IDENTICAL TO ConstructConfiguration.

    +

    Ancestors (in MRO)

    +
      +
    • builtins.dict
      • +
    +

    Methods

    +

    clear

    +
    def clear(
+    ...
+)
+
    +

    D.clear() -> None. Remove all items from D.

    +

    copy

    +
    def copy(
+    ...
+)
+
    +

    D.copy() -> a shallow copy of D

    +

    fromkeys

    +
    def fromkeys(
+    iterable,
+    value=None,
+    /
+)
+
    +

    Create a new dictionary with keys from iterable and values set to value.

    +

    get

    +
    def get(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Return the value for key if key is in the dictionary, else default.

    +

    items

    +
    def items(
+    ...
+)
+
    +

    D.items() -> a set-like object providing a view on D's items

    +

    keys

    +
    def keys(
+    ...
+)
+
    +

    D.keys() -> a set-like object providing a view on D's keys

    +

    pop

    +
    def pop(
+    ...
+)
+
    +

    D.pop(k[,d]) -> v, remove specified key and return the corresponding value.

    +

    If the key is not found, return the default if given; otherwise, +raise a KeyError.

    +

    popitem

    +
    def popitem(
+    self,
+    /
+)
+
    +

    Remove and return a (key, value) pair as a 2-tuple.

    +

    Pairs are returned in LIFO (last-in, first-out) order. +Raises KeyError if the dict is empty.

    +

    setdefault

    +
    def setdefault(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Insert key with a value of default if key is not in the dictionary.

    +

    Return the value for key if key is in the dictionary, else default.

    +

    update

    +
    def update(
+    ...
+)
+
    +

    D.update([E, ]**F) -> None. Update D from dict/iterable E and F.

    +

    If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] +If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v +In either case, this is followed by: for k in F: D[k] = F[k]

    +

    values

    +
    def values(
+    ...
+)
+
    +

    D.values() -> an object providing a view on D's values

    +

    GlobalConfiguration

    +
    class GlobalConfiguration(
+    construct: dewret.core.ConstructConfiguration,
+    render: dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+)
+
    +

    Overall configuration structure.

    +

    Having a single configuration dict allows us to manage only one ContextVar.

    +

    Instance variables

    +
    construct
+
    +
    render
+
    +

    IterableMixin

    +
    class IterableMixin(
+    typ: type[~U] | None = None,
+    **kwargs: Any
+)
+
    +

    Functionality for iterating over references to give new references.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.core.Reference
      • +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Descendants

    +
      +
    • dewret.workflow.IterableParameterReference
      • +
    • dewret.workflow.IterableStepReference
      • +
    +

    Class variables

    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Printable name of the reference.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    IteratedGenerator

    +
    class IteratedGenerator(
+    to_wrap: dewret.core.Reference[~U]
+)
+
    +

    Sentinel value for capturing that an iteration has occured without performing it.

    +

    Allows us to lazily evaluate a loop, for instance, in the renderer. This may be relevant +if the renderer wishes to postpone iteration to runtime, and simply record it is required, +rather than evaluating the iterator.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Generic
      • +
    +

    Raw

    +
    class Raw(
+    value: Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]
+)
+
    +

    Value object for any raw types.

    +

    This is able to hash raw types consistently and provides +a single type for validating type-consistency.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    valueNonethe real value, e.g. a str, int, ...None
    +

    RawRenderModule

    +
    class RawRenderModule(
+    *args,
+    **kwargs
+)
+
    +

    Render module that returns raw text.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.core.BaseRenderModule
      • +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Static methods

    +

    default_config

    +
    def default_config(
+    
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Retrieve default settings.

    +

    These will not change during execution, but can be overridden by dewret.core.set_render_configuration.

    +

    Returns: a static, serializable dict.

    +

    Methods

    +

    render_raw

    +
    def render_raw(
+    self,
+    workflow: dewret.core.WorkflowProtocol,
+    **kwargs: dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+) -> dict[str, str]
+
    +

    Turn a workflow into flat strings.

    +

    Returns: one or more subworkflows with a __root__ key representing the outermost workflow, at least.

    +

    Reference

    +
    class Reference(
+    *args: Any,
+    typ: type[~U] | None = None,
+    **kwargs: Any
+)
+
    +

    Superclass for all symbolic references to values.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Descendants

    +
      +
    • dewret.core.IterableMixin
      • +
    • dewret.workflow.ParameterReference
      • +
    • dewret.workflow.StepReference
      • +
    +

    Class variables

    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Printable name of the reference.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    RenderCall

    +
    class RenderCall(
+    *args,
+    **kwargs
+)
+
    +

    Callable that will render out workflow(s).

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    StructuredRenderModule

    +
    class StructuredRenderModule(
+    *args,
+    **kwargs
+)
+
    +

    Render module that returns JSON/YAML-serializable structures.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.core.BaseRenderModule
      • +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Static methods

    +

    default_config

    +
    def default_config(
+    
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Retrieve default settings.

    +

    These will not change during execution, but can be overridden by dewret.core.set_render_configuration.

    +

    Returns: a static, serializable dict.

    +

    Methods

    +

    render

    +
    def render(
+    self,
+    workflow: dewret.core.WorkflowProtocol,
+    **kwargs: dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+) -> dict[str, dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]]
+
    +

    Turn a workflow into a serializable structure.

    +

    Returns: one or more subworkflows with a __root__ key representing the outermost workflow, at least.

    +

    UnevaluatableError

    +
    class UnevaluatableError(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Signposts that a user has tried to treat a reference as the real (runtime) value.

    +

    For example, by comparing to a concrete integer or value, etc.

    +

    Ancestors (in MRO)

    +
      +
    • builtins.Exception
      • +
    • builtins.BaseException
      • +
    +

    Class variables

    +
    args
+
    +

    Methods

    +

    add_note

    +
    def add_note(
+    ...
+)
+
    +

    Exception.add_note(note) --

    +

    add a note to the exception

    +

    with_traceback

    +
    def with_traceback(
+    ...
+)
+
    +

    Exception.with_traceback(tb) --

    +

    set self.traceback to tb and return self.

    +

    WorkflowComponent

    +
    class WorkflowComponent(
+    *args: Any,
+    workflow: dewret.core.WorkflowProtocol,
+    **kwargs: Any
+)
+
    +

    Base class for anything directly tied to an individual Workflow.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNonethe Workflow that this is tied to.None
    +

    Descendants

    +
      +
    • dewret.core.Reference
      • +
    • dewret.workflow.BaseStep
      • +
    • dewret.workflow.ParameterReference
      • +
    +

    WorkflowProtocol

    +
    class WorkflowProtocol(
+    *args,
+    **kwargs
+)
+
    +

    Expected structure for a workflow.

    +

    We do not expect various workflow implementations, but this allows us to define the +interface expected by the core classes.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Methods

    +

    remap

    +
    def remap(
+    self,
+    name: str
+) -> str
+
    +

    Perform any name-changing for steps, etc. in the workflow.

    +

    This enables, for example, simplifying all the IDs to an integer sequence.

    +

    Returns: remapped name.

    +

    set_result

    +
    def set_result(
+    self,
+    result: sympy.core.basic.Basic | list[sympy.core.basic.Basic] | tuple[sympy.core.basic.Basic]
+) -> None
+
    +

    Set the step that should produce a result for the overall workflow.

    +

    simplify_ids

    +
    def simplify_ids(
+    self,
+    infix: list[str] | None = None
+) -> None
+
    +

    Drop the non-human-readable IDs if possible, in favour of integer sequences.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    infixNoneany inherited intermediary identifiers, to allow nesting, or None.None
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/index.html b/reference/dewret/index.html new file mode 100644 index 00000000..7b5d6ebc --- /dev/null +++ b/reference/dewret/index.html @@ -0,0 +1,788 @@ + + + + + + + + + + + + + + + + Index - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
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    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/render/index.html b/reference/dewret/render/index.html new file mode 100644 index 00000000..bf805f93 --- /dev/null +++ b/reference/dewret/render/index.html @@ -0,0 +1,960 @@ + + + + + + + + + + + + + + + + Render - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
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    +
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    + + + +
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    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.render

    +

    Utilities for building renderers.

    +

    Provides the routines for calling varied renderers in a standard way, and for +renderers to reuse to build up their own functionality.

    +

    Variables

    +
    T
+
    +

    Functions

    +

    base_render

    +
    def base_render(
+    workflow: dewret.workflow.Workflow,
+    build_cb: Callable[[dewret.workflow.Workflow], ~T]
+) -> dict[str, ~T]
+
    +

    Render to a dict-like structure.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneworkflow to evaluate result.None
    build_cbNonea callback to call for each workflow found.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    +

    get_render_method

    +
    def get_render_method(
+    renderer: pathlib.Path | dewret.core.RawRenderModule | dewret.core.StructuredRenderModule,
+    pretty: bool = False
+) -> dewret.core.RenderCall
+
    +

    Create a ready-made callable to render the workflow that is appropriate for the renderer module.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    rendererNonea module or path to a module.None
    prettyNonewhether the renderer should attempt to YAML-format the output (if relevant).None
    +

    structured_to_raw

    +
    def structured_to_raw(
+    rendered: Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']],
+    pretty: bool = False
+) -> str
+
    +

    Serialize a serializable structure to a string.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    renderedNonea possibly-nested, static basic Python structure.None
    prettyNonewhether to attempt YAML dumping with an indent of 2.None
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/renderers/cwl/index.html b/reference/dewret/renderers/cwl/index.html new file mode 100644 index 00000000..0b6c3b9c --- /dev/null +++ b/reference/dewret/renderers/cwl/index.html @@ -0,0 +1,3282 @@ + + + + + + + + + + + + + + + + Cwl - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
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    +
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    + + + +
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    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.renderers.cwl

    +

    CWL Renderer.

    +

    Outputs a Common Workflow Language representation of the +current workflow.

    +

    Variables

    +
    InputSchemaType
+
    +

    Functions

    +

    cwl_type_from_value

    +
    def cwl_type_from_value(
+    label: str,
+    val: Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType'], dewret.utils.Unset]
+) -> dewret.renderers.cwl.CommandInputSchema
+
    +

    Find a CWL type for a given (possibly Unset) value.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    labelNonethe label for the variable being checked to prefill the input def and improve debugging info.None
    valNonea raw Python variable or an unset variable.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneInput schema type.
    +

    default_renderer_config

    +
    def default_renderer_config(
+    
+) -> dewret.renderers.cwl.CWLRendererConfiguration
+
    +

    Default configuration for this renderer.

    +

    This is a hook-like call to give a configuration dict that this renderer +will respect, and sets any necessary default values.

    +

    Returns: a dict with (preferably) raw type structures to enable easy setting + from YAML/JSON.

    +

    raw_to_command_input_schema

    +
    def raw_to_command_input_schema(
+    label: str,
+    value: Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType'], dewret.utils.Unset]
+) -> Union[str, dewret.renderers.cwl.CommandInputSchema, list[str], list['InputSchemaType'], dict[str, 'str | InputSchemaType']]
+
    +

    Infer the CWL input structure for this value.

    +

    Inspects the value, to work out an appropriate structure +describing it in CWL.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    labelNonename of the variable.None
    valueNonebasic-typed variable from which to build structure.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneStructure used to define (possibly compound) basic types for input.
    +

    render

    +
    def render(
+    workflow: dewret.workflow.Workflow,
+    **kwargs: *<class 'dewret.renderers.cwl.CWLRendererConfiguration'>
+) -> dict[str, dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]]
+
    +

    Render to a dict-like structure.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneworkflow to evaluate result.None
    **kwargsNoneadditional configuration arguments - these should match CWLRendererConfiguration.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    +

    render_expression

    +
    def render_expression(
+    ref: Any
+) -> 'ReferenceDefinition'
+
    +

    Turn a rich (sympy) expression into a CWL JS expression.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    refNonea structure whose elements are all string-renderable or sympy Basic.None
    +

    to_cwl_type

    +
    def to_cwl_type(
+    label: str,
+    typ: type
+) -> dewret.renderers.cwl.CommandInputSchema
+
    +

    Map Python types to CWL types.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    labelNonethe label for the variable being checked to prefill the input def and improve debugging info.None
    typNonea Python basic type.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneCWL specification type dict.
    +

    to_name

    +
    def to_name(
+    result: dewret.core.Reference[typing.Any]
+) -> str
+
    +

    Take a reference and get a name representing it.

    +

    The primary purpose of this method is to deal with the case where a reference is to the +whole result, as we always put this into an imagined out field for CWL consistency.

    +

    Returns: the name of the reference, including any field portion, appending an "out" fieldname if none.

    +

    to_output_schema

    +
    def to_output_schema(
+    label: str,
+    typ: type[typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType'], attr.AttrsInstance, dewret.utils.DataclassProtocol]],
+    output_source: str | None = None
+) -> dewret.renderers.cwl.CommandOutputSchema
+
    +

    Turn a step's output into an output schema.

    +

    Takes a source, type and label and provides a description for CWL.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    labelNonename of this field.None
    typNoneeither a basic type, compound of basic types, or a TypedDict representing a pre-defined result structure.None
    output_sourceNoneif provided, a CWL step result reference to input here.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneCWL CommandOutputSchema-like structure for embedding into an outputs block
    +

    with_field

    +
    def with_field(
+    result: Any
+) -> str
+
    +

    Get a string representing any 'field' suffix of a value.

    +

    This only makes sense in the context of a Reference, which can represent +a deep reference with a known variable (parameter or step result, say) using +its __field__ attribute. Defaults to "out" as this produces compliant CWL +where every output has a "fieldname".

    +

    Returns: a string representation of the field portion of the passed value or "out".

    +

    with_type

    +
    def with_type(
+    result: Any
+) -> type | typing.Any
+
    +

    Get a Python type from a value.

    +

    Does so either by using its __type__ field (for example, for References) +or if unavailable, using type().

    +

    Returns: a Python type.

    +

    Classes

    +

    CWLRendererConfiguration

    +
    class CWLRendererConfiguration(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Configuration for the renderer.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    allow_complex_typesNonecan input/output types be other than raw?None
    factories_as_paramsNoneshould factories be treated as input or steps?None
    +

    Ancestors (in MRO)

    +
      +
    • builtins.dict
      • +
    +

    Methods

    +

    clear

    +
    def clear(
+    ...
+)
+
    +

    D.clear() -> None. Remove all items from D.

    +

    copy

    +
    def copy(
+    ...
+)
+
    +

    D.copy() -> a shallow copy of D

    +

    fromkeys

    +
    def fromkeys(
+    iterable,
+    value=None,
+    /
+)
+
    +

    Create a new dictionary with keys from iterable and values set to value.

    +

    get

    +
    def get(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Return the value for key if key is in the dictionary, else default.

    +

    items

    +
    def items(
+    ...
+)
+
    +

    D.items() -> a set-like object providing a view on D's items

    +

    keys

    +
    def keys(
+    ...
+)
+
    +

    D.keys() -> a set-like object providing a view on D's keys

    +

    pop

    +
    def pop(
+    ...
+)
+
    +

    D.pop(k[,d]) -> v, remove specified key and return the corresponding value.

    +

    If the key is not found, return the default if given; otherwise, +raise a KeyError.

    +

    popitem

    +
    def popitem(
+    self,
+    /
+)
+
    +

    Remove and return a (key, value) pair as a 2-tuple.

    +

    Pairs are returned in LIFO (last-in, first-out) order. +Raises KeyError if the dict is empty.

    +

    setdefault

    +
    def setdefault(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Insert key with a value of default if key is not in the dictionary.

    +

    Return the value for key if key is in the dictionary, else default.

    +

    update

    +
    def update(
+    ...
+)
+
    +

    D.update([E, ]**F) -> None. Update D from dict/iterable E and F.

    +

    If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] +If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v +In either case, this is followed by: for k in F: D[k] = F[k]

    +

    values

    +
    def values(
+    ...
+)
+
    +

    D.values() -> an object providing a view on D's values

    +

    CommandInputSchema

    +
    class CommandInputSchema(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Structure for referring to a raw type in CWL.

    +

    Encompasses several CWL types. In future, it may be best to +use cwltool or another library for these basic structures.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    typeNoneCWL type of this input.None
    labelNonename to show for this input.None
    fieldsNone(for record) individual fields in a dict-like structure.None
    itemsNone(for array) type that each field will have.None
    +

    Ancestors (in MRO)

    +
      +
    • builtins.dict
      • +
    +

    Methods

    +

    clear

    +
    def clear(
+    ...
+)
+
    +

    D.clear() -> None. Remove all items from D.

    +

    copy

    +
    def copy(
+    ...
+)
+
    +

    D.copy() -> a shallow copy of D

    +

    fromkeys

    +
    def fromkeys(
+    iterable,
+    value=None,
+    /
+)
+
    +

    Create a new dictionary with keys from iterable and values set to value.

    +

    get

    +
    def get(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Return the value for key if key is in the dictionary, else default.

    +

    items

    +
    def items(
+    ...
+)
+
    +

    D.items() -> a set-like object providing a view on D's items

    +

    keys

    +
    def keys(
+    ...
+)
+
    +

    D.keys() -> a set-like object providing a view on D's keys

    +

    pop

    +
    def pop(
+    ...
+)
+
    +

    D.pop(k[,d]) -> v, remove specified key and return the corresponding value.

    +

    If the key is not found, return the default if given; otherwise, +raise a KeyError.

    +

    popitem

    +
    def popitem(
+    self,
+    /
+)
+
    +

    Remove and return a (key, value) pair as a 2-tuple.

    +

    Pairs are returned in LIFO (last-in, first-out) order. +Raises KeyError if the dict is empty.

    +

    setdefault

    +
    def setdefault(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Insert key with a value of default if key is not in the dictionary.

    +

    Return the value for key if key is in the dictionary, else default.

    +

    update

    +
    def update(
+    ...
+)
+
    +

    D.update([E, ]**F) -> None. Update D from dict/iterable E and F.

    +

    If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] +If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v +In either case, this is followed by: for k in F: D[k] = F[k]

    +

    values

    +
    def values(
+    ...
+)
+
    +

    D.values() -> an object providing a view on D's values

    +

    CommandOutputSchema

    +
    class CommandOutputSchema(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Structure for referring to an output in CWL.

    +

    As a simplification, this is an input schema with an extra +outputSource field.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    outputSourceNonestep result to use for this output.None
    +

    Ancestors (in MRO)

    +
      +
    • builtins.dict
      • +
    +

    Methods

    +

    clear

    +
    def clear(
+    ...
+)
+
    +

    D.clear() -> None. Remove all items from D.

    +

    copy

    +
    def copy(
+    ...
+)
+
    +

    D.copy() -> a shallow copy of D

    +

    fromkeys

    +
    def fromkeys(
+    iterable,
+    value=None,
+    /
+)
+
    +

    Create a new dictionary with keys from iterable and values set to value.

    +

    get

    +
    def get(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Return the value for key if key is in the dictionary, else default.

    +

    items

    +
    def items(
+    ...
+)
+
    +

    D.items() -> a set-like object providing a view on D's items

    +

    keys

    +
    def keys(
+    ...
+)
+
    +

    D.keys() -> a set-like object providing a view on D's keys

    +

    pop

    +
    def pop(
+    ...
+)
+
    +

    D.pop(k[,d]) -> v, remove specified key and return the corresponding value.

    +

    If the key is not found, return the default if given; otherwise, +raise a KeyError.

    +

    popitem

    +
    def popitem(
+    self,
+    /
+)
+
    +

    Remove and return a (key, value) pair as a 2-tuple.

    +

    Pairs are returned in LIFO (last-in, first-out) order. +Raises KeyError if the dict is empty.

    +

    setdefault

    +
    def setdefault(
+    self,
+    key,
+    default=None,
+    /
+)
+
    +

    Insert key with a value of default if key is not in the dictionary.

    +

    Return the value for key if key is in the dictionary, else default.

    +

    update

    +
    def update(
+    ...
+)
+
    +

    D.update([E, ]**F) -> None. Update D from dict/iterable E and F.

    +

    If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] +If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v +In either case, this is followed by: for k in F: D[k] = F[k]

    +

    values

    +
    def values(
+    ...
+)
+
    +

    D.values() -> an object providing a view on D's values

    +

    InputsDefinition

    +
    class InputsDefinition(
+    inputs: dict[str, 'CommandInputParameter']
+)
+
    +

    CWL-renderable representation of an input parameter block.

    +

    Turns dewret results into a CWL input block.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    inputNonesequence of results from a workflow.None
    +

    Class variables

    +
    CommandInputParameter
+
    +

    Static methods

    +

    from_parameters

    +
    def from_parameters(
+    parameters: list[typing.Union[dewret.workflow.ParameterReference[typing.Any], dewret.workflow.FactoryCall]]
+) -> 'InputsDefinition'
+
    +

    Takes a list of parameters into a CWL structure.

    +

    Uses the parameters to fill out the necessary input fields.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneCWL-like structure representing all workflow outputs.
    +

    Instance variables

    +
    inputs
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Render to a dict-like structure.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    +

    OutputsDefinition

    +
    class OutputsDefinition(
+    outputs: dict[str, 'CommandOutputSchema'] | list['CommandOutputSchema'] | dewret.renderers.cwl.CommandOutputSchema
+)
+
    +

    CWL-renderable set of workflow outputs.

    +

    Turns dewret results into a CWL output block.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    outputsNonesequence of results from a workflow.None
    +

    Static methods

    +

    from_results

    +
    def from_results(
+    results: dict[str, dewret.workflow.StepReference[typing.Any]] | list[dewret.workflow.StepReference[typing.Any]] | tuple[dewret.workflow.StepReference[typing.Any], ...]
+) -> 'OutputsDefinition'
+
    +

    Takes a mapping of results into a CWL structure.

    +

    Pulls the result type from the signature, ultimately, if possible.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneCWL-like structure representing all workflow outputs.
    +

    Instance variables

    +
    outputs
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]] | list[typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Render to a dict-like structure.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    +

    ReferenceDefinition

    +
    class ReferenceDefinition(
+    source: str | None,
+    value_from: str | None
+)
+
    +

    CWL-renderable internal reference.

    +

    Normally points to a value or a step.

    +

    Static methods

    +

    from_reference

    +
    def from_reference(
+    ref: dewret.core.Reference[typing.Any]
+) -> 'ReferenceDefinition'
+
    +

    Build from a Reference.

    +

    Converts a dewret.workflow.Reference into a CWL-rendering object.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    refNonereference to convert.None
    +

    Instance variables

    +
    source
+
    +
    value_from
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Render to a dict-like structure.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    +

    StepDefinition

    +
    class StepDefinition(
+    name: str,
+    run: str,
+    out: dict[str, 'CommandInputSchema'] | list[str],
+    in_: collections.abc.Mapping[str, dewret.renderers.cwl.ReferenceDefinition | dewret.core.Raw]
+)
+
    +

    CWL-renderable step.

    +

    Coerces the dewret structure of a step into that +needed for valid CWL.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    nameNoneidentifier to call this step by.None
    runNonetask to execute for this step.None
    in_Noneinputs from values or other steps.None
    +

    Static methods

    +

    from_step

    +
    def from_step(
+    step: dewret.workflow.BaseStep
+) -> 'StepDefinition'
+
    +

    Build from a BaseStep.

    +

    Converts a dewret.workflow.Step into a CWL-rendering object.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepNonestep to convert.None
    +

    Instance variables

    +
    in_
+
    +
    name
+
    +
    out
+
    +
    run
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Render to a dict-like structure.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    +

    WorkflowDefinition

    +
    class WorkflowDefinition(
+    steps: list[dewret.renderers.cwl.StepDefinition],
+    inputs: dewret.renderers.cwl.InputsDefinition,
+    outputs: dewret.renderers.cwl.OutputsDefinition,
+    name: None | str
+)
+
    +

    CWL-renderable workflow.

    +

    Coerces the dewret structure of a workflow into that +needed for valid CWL.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepsNonesequence of steps in the workflow.None
    +

    Static methods

    +

    from_workflow

    +
    def from_workflow(
+    workflow: dewret.workflow.Workflow,
+    name: None | str = None
+) -> 'WorkflowDefinition'
+
    +

    Build from a Workflow.

    +

    Converts a dewret.workflow.Workflow into a CWL-rendering object.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneworkflow to convert.None
    nameNonename of this workflow, if it should have one.None
    +

    Instance variables

    +
    inputs
+
    +
    name
+
    +
    outputs
+
    +
    steps
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]]
+
    +

    Render to a dict-like structure.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReduced form as a native Python dict structure for
    serialization.
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/renderers/index.html b/reference/dewret/renderers/index.html new file mode 100644 index 00000000..2df0ee86 --- /dev/null +++ b/reference/dewret/renderers/index.html @@ -0,0 +1,784 @@ + + + + + + + + + + + + + + + + Index - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    + +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/renderers/snakemake/index.html b/reference/dewret/renderers/snakemake/index.html new file mode 100644 index 00000000..e0b89489 --- /dev/null +++ b/reference/dewret/renderers/snakemake/index.html @@ -0,0 +1,2366 @@ + + + + + + + + + + + + + + + + Snakemake - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.renderers.snakemake

    +

    Snakemake Renderer.

    +

    Outputs a Snakemake representation of the +current workflow.

    +

    Variables

    +
    MainTypes
+
    +

    Functions

    +

    get_method_args

    +
    def get_method_args(
+    func: dewret.workflow.Lazy
+) -> inspect.Signature
+
    +

    Retrieve the argument names and types of a lazy-evaluatable function.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    funcNoneA function that adheres to the Lazy protocol.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneAn ItemsView object containing the argument names and their corresponding
    inspect.Parameter objects.
    +

    get_method_rel_path

    +
    def get_method_rel_path(
+    func: dewret.workflow.Lazy
+) -> Any
+
    +

    Get the relative path of the module containing the given function.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    funcLazyThe function for which the relative path is to be determined.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    anyThe relative path of the module containing the function.
    +

    raw_render

    +
    def raw_render(
+    workflow: dewret.workflow.Workflow
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list[str], list['MainTypes'], dict[str, 'MainTypes']]]
+
    +

    Render the workflow as a Snakemake (SMK) string.

    +

    This function converts a Workflow object into a object containing snakemake rules.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowWorkflowThe workflow to be rendered.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    dict[str, MainTypes]A dictionary containing the components of the Workflow
    definition, for use in Snakemake workflows.
    +

    render

    +
    def render(
+    workflow: dewret.workflow.Workflow
+) -> dict[str, typing.Any]
+
    +

    Render the workflow as a Snakemake (SMK) string.

    +

    This function converts a Workflow object into a Snakemake-compatible yaml.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowWorkflowThe workflow to be rendered.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    strA Snakemake-compatible yaml representation of the workflow.
    +

    to_snakemake_type

    +
    def to_snakemake_type(
+    param: dewret.core.Raw
+) -> str
+
    +

    Convert a raw type to a corresponding Snakemake-compatible Python type.

    +

    This function maps raw types to their corresponding Python types as +used in Snakemake. Snakemake primarily uses Python types for its parameters, +and this function ensures that the provided type is appropriately converted.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    paramNoneThe raw type to be converted, which can be of any type.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA string representing the corresponding Python type for Snakemake.
    +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    TypeErrorIf the parameter's type cannot be mapped to a known Python type.
    +

    Classes

    +

    InputDefinition

    +
    class InputDefinition(
+    inputs: list[str],
+    params: list[str]
+)
+
    +

    Represents input and parameter definitions block for a Snakemake-renderable workflow step.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    inputsList[str]A list of input definitions.None
    paramsList[str]A list of parameter definitions.None
    +

    Static methods

    +

    from_step

    +
    def from_step(
+    step: dewret.workflow.BaseStep
+) -> 'InputDefinition'
+
    +

    Constructs an InputDefinition object from a Step.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepStepThe Step object from which input and parameter block definitions are
    extracted.    		None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    InputDefinitionAn InputDefinition object.
    +

    Instance variables

    +
    inputs
+
    +
    params
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, list[str]]
+
    +

    Renders the input and parameter definitions as a dictionary.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    dict[str, list[MainTypes]]A dictionary containing the input and parameter definitions,
    for use in Snakemake Input and Params blocks.
    +

    OutputDefinition

    +
    class OutputDefinition(
+    output_file: str
+)
+
    +

    Represents the output definition block for a Snakemake-renderable workflow step.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    output_filestrThe output file definition.None
    +

    Static methods

    +

    from_step

    +
    def from_step(
+    step: dewret.workflow.BaseStep
+) -> 'OutputDefinition'
+
    +

    Constructs an OutputDefinition object from a Step.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepStepThe Step object from which the output file definition is extracted.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    OutputDefinitionAn OutputDefinition object, for use in Snakemake Output block.
    +

    Instance variables

    +
    output_file
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> list[str]
+
    +

    Renders the output definition as a list.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    list[str]A list containing the output file definition, for use in a Snakemake Output block.
    +

    ReferenceDefinition

    +
    class ReferenceDefinition(
+    source: str
+)
+
    +

    Represents a Snakemake-renderable internal reference.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    sourcestrThe source of the internal reference.None
    +

    Static methods

    +

    from_reference

    +
    def from_reference(
+    ref: dewret.core.Reference[typing.Any]
+) -> 'ReferenceDefinition'
+
    +

    Build from a Reference.

    +

    Converts a dewret.workflow.Reference into a Snakemake-rendering object.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    refNonereference to convert.None
    +

    Instance variables

    +
    source
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> str
+
    +

    Render the internal reference definition as a string.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    strinternal reference.
    +

    RunDefinition

    +
    class RunDefinition(
+    method_name: str,
+    rel_import: str,
+    args: list[str]
+)
+
    +

    Represents a Snakemake-renderable run block for a dewret workflow step.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    method_namestrThe name of the method to be executed in the snakefile run block.None
    rel_importstrThe relative import path of the method.None
    argsList[str]The arguments to be passed to the method.None
    +

    Static methods

    +

    from_task

    +
    def from_task(
+    task: dewret.workflow.Task | dewret.workflow.Workflow
+) -> 'RunDefinition'
+
    +

    Constructs a RunDefinition object from a Task.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskTaskThe Task object from which method information and arguments
    are extracted.    		None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    RunDefinitionA RunDefinition object containing the converted method
    information and arguments.
    +

    Instance variables

    +
    args
+
    +
    method_name
+
    +
    rel_import
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> list[str]
+
    +

    Renders the run block as a list of strings.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    list[str]A list containing the import statement and the method
    call statement, for use in Snakemake run block.
    +

    StepDefinition

    +
    class StepDefinition(
+    name: str,
+    run: list[str],
+    params: list[str],
+    output: list[str],
+    input: list[str]
+)
+
    +

    Represents a Snakemake-renderable step definition in a dewret workflow.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    namestrThe name of the step.None
    runstrThe run block definition for the step.None
    paramsList[str]The parameter definitions for the step.None
    output (list[str]NoneThe output definition for the step.None
    inputList[str]The input definitions for the step.None
    +

    Static methods

    +

    from_step

    +
    def from_step(
+    step: dewret.workflow.BaseStep
+) -> 'StepDefinition'
+
    +

    Constructs a StepDefinition object from a Step.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepStepThe Step object from which step information and components
    are extracted.    		None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    StepDefinitionA StepDefinition object containing the converted step
    information and components.
    +

    Instance variables

    +
    input
+
    +
    name
+
    +
    output
+
    +
    params
+
    +
    run
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list[str], list['MainTypes'], dict[str, 'MainTypes']]]
+
    +

    Renders the step definition as a dictionary.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    dict[str, MainTypes]A dictionary containing the components of the step
    definition, for use in Snakemake workflows.
    +

    WorkflowDefinition

    +
    class WorkflowDefinition(
+    steps: list[dewret.renderers.snakemake.StepDefinition]
+)
+
    +

    Represents a Snakemake-renderable workflow definition from a dewret workflow.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepsList[StepDefinition]A list of StepDefinition objects representing the steps
    in the workflow.    		None
    +

    Static methods

    +

    from_workflow

    +
    def from_workflow(
+    workflow: dewret.workflow.Workflow
+) -> 'WorkflowDefinition'
+
    +

    Creates a WorkflowDefinition object from a Workflow object.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowWorkflowThe workflow to be rendered.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    strWorkflowDefinition with definited steps.
    +

    Instance variables

    +
    steps
+
    +

    Methods

    +

    render

    +
    def render(
+    self
+) -> dict[str, typing.Union[str, float, bool, bytes, int, NoneType, list[str], list['MainTypes'], dict[str, 'MainTypes']]]
+
    +

    Render the WorkflowDefinition.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    dict[str, MainTypes]A dictionary containing the components of the Workflow
    definition, for use in Snakemake workflows.
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/tasks/index.html b/reference/dewret/tasks/index.html new file mode 100644 index 00000000..32308045 --- /dev/null +++ b/reference/dewret/tasks/index.html @@ -0,0 +1,1935 @@ + + + + + + + + + + + + + + + + Tasks - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.tasks

    +

    Abstraction layer for task operations.

    +

    Access dask, or other, backends consistently using this module. It provides +decorators and execution calls that manage tasks. Note that the task +decorator should be called with no arguments, and will return the appropriate +decorator for the current backend.

    +

    Typical usage example:

    +
    >>> @task()
+... def increment(num: int) -> int:
+...     return num + 1
+
    +

    Variables

    +
    DEFAULT_BACKEND
+
    +
    Param
+
    +
    RetType
+
    +
    construct
+
    +

    An alias pointing to an instance of the TaskManager class.

    +

    Used for constructing a set of tasks into a dewret workflow instance.

    +

    Functions

    +

    ensure_lazy

    +
    def ensure_lazy(
+    task: Any
+) -> dewret.workflow.Lazy | None
+
    +

    Evaluate a single task for a known workflow.

    +

    As we mask our lazy-evaluable functions to appear as their original +types to the type system (see dewret.tasks.task), we must cast them +back, to allow the type-checker to comb the remainder of the code.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonethe suspected task to check.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneOriginal task, cast to a Lazy, or None.
    +

    evaluate

    +
    def evaluate(
+    task: dewret.workflow.Lazy | list[dewret.workflow.Lazy] | tuple[dewret.workflow.Lazy],
+    __workflow__: dewret.workflow.Workflow,
+    thread_pool: concurrent.futures.thread.ThreadPoolExecutor | None = None,
+    **kwargs: Any
+) -> Any
+
    +

    Evaluate a single task for a known workflow.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonethe task to evaluate.None
    workflowNoneworkflow within which this exists.None
    thread_poolNoneexisting pool of threads to run this in, or None.None
    **kwargsNoneany arguments to pass to the task.None
    +

    factory

    +
    def factory(
+    fn: collections.abc.Callable[..., ~RetType]
+) -> collections.abc.Callable[..., ~RetType]
+
    +

    Create a factory, that can be treated as complex input to a workflow.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fnNonea callable to create the entity.None
    +

    in_nested_task

    +
    def in_nested_task(
+    
+) -> Generator[NoneType, NoneType, NoneType]
+
    +

    Informs the builder that we are within a nested task.

    +

    This is only really relevant in the subworkflow context.

    +

    TODO: check impact of ContextVar being thread-sensitive on build.

    +

    is_in_nested_task

    +
    def is_in_nested_task(
+    
+) -> bool
+
    +

    Check if we are within a nested task.

    +

    Used, for example, to see if discovered parameters should be +treated as "local" (i.e. should take a default to the step) or +global (i.e. should be turned into a workflow parameter) if we +are inside or outside a subworkflow, respectively.

    +

    lazy

    +
    def lazy(
+    
+) -> collections.abc.Callable[[collections.abc.Callable[..., typing.Any]], dewret.workflow.Lazy]
+
    +

    Get the lazy decorator for this backend.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReal decorator for this backend.
    +

    set_backend

    +
    def set_backend(
+    backend: dewret.tasks.Backend
+) -> None
+
    +

    Choose a backend.

    +

    Will raise an error if a backend is already chosen.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    backendNonechosen backend to use from here-on in.None
    +

    task

    +
    def task(
+    nested: bool = False,
+    flatten_nested: bool = True,
+    is_factory: bool = False
+) -> collections.abc.Callable[[collections.abc.Callable[~Param, ~RetType]], collections.abc.Callable[~Param, ~RetType]]
+
    +

    Decorator factory abstracting backend's own task decorator.

    +

    For example:

    +
    >>> @task()
+... def increment(num: int) -> int:
+...     return num + 1
+
    +

    If the backend is dask (the default), it is will evaluate this +as a dask.delayed. Note that, with any backend, dewret will +hijack the decorator to record the attempted evalution rather than +actually evaluating the lazy function. Nonetheless, this hijacking +will still be executed with the backend's lazy executor, so +dask.delayed will still be called, for example, in the dask case.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    nestedNonewhether this should be executed to find other tasks.None
    flatten_nestedNone(only relevant to nested tasks) should this nested task
    be considered a distinct subworkflow, or is it just organizational
    for the outer workflow.    		None
    is_factoryNonewhether this task should be marked as a 'factory', rather than
    a normal step.    		None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneDecorator for the current backend to mark lazy-executable tasks.
    +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    TypeErrorif arguments are missing or incorrect, in line with usual
    Python behaviour.
    +

    unwrap

    +
    def unwrap(
+    task: dewret.workflow.Lazy
+) -> collections.abc.Callable[..., typing.Any]
+
    +

    Unwraps a lazy-evaluated function to get the function.

    +

    Ideally, we could use the __wrapped__ property but not all +workflow engines support this, and most importantly, dask has +only done so as of 2024.03.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonetask to be unwrapped.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneOriginal target.
    +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    RuntimeErrorif the task is not a wrapped function.
    +

    workflow

    +
    def workflow(
+    
+) -> collections.abc.Callable[[collections.abc.Callable[~Param, ~RetType]], collections.abc.Callable[~Param, ~RetType]]
+
    +

    Shortcut for marking a task as nested.

    +

    A nested task is one which calls other tasks and does not +do anything else important. It will not actually get called +at runtime, but should map entirely into the graph. As such, +arithmetic operations on results, etc. will cause errors at +render-time. Combining tasks is acceptable, and intended. The +effect of the nested task will be considered equivalent to whatever +reaching whatever step reference is returned at the end.

    +
    >>> @task()
+... def increment(num: int) -> int:
+...     return num + 1
+
+>>> @workflow()
+... def double_increment(num: int) -> int:
+...     return increment(increment(num=num))
+
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneTask that runs at render, not execution, time.
    +

    Classes

    +

    Backend

    +
    class Backend(
+    *args,
+    **kwds
+)
+
    +

    Stringy enum representing available backends.

    +

    Ancestors (in MRO)

    +
      +
    • enum.Enum
      • +
    +

    Class variables

    +
    DASK
+
    +
    name
+
    +
    value
+
    +

    TaskException

    +
    class TaskException(
+    task: dewret.workflow.Task | collections.abc.Callable[..., typing.Any],
+    dec_tb: traceback | None,
+    tb: traceback | None,
+    message: str,
+    *args: Any,
+    **kwargs: Any
+)
+
    +

    Exception tied to a specific task.

    +

    Primarily aimed at parsing issues, but this will ensure that +a message is shown with useful debug information for the +workflow writer.

    +

    Ancestors (in MRO)

    +
      +
    • builtins.Exception
      • +
    • builtins.BaseException
      • +
    +

    Class variables

    +
    args
+
    +

    Methods

    +

    add_note

    +
    def add_note(
+    ...
+)
+
    +

    Exception.add_note(note) --

    +

    add a note to the exception

    +

    with_traceback

    +
    def with_traceback(
+    ...
+)
+
    +

    Exception.with_traceback(tb) --

    +

    set self.traceback to tb and return self.

    +

    TaskManager

    +
    class TaskManager(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Overarching backend-agnostic task manager.

    +

    Gatekeeps the specific backend implementation. This can be +instantiated without choosing a backend, but the first call to +any of its methods will concretize that choice - either as +the default, or the backend set via TaskManager.set_backend. +It cannot be changed after this point.

    +

    Methods

    +

    backend

    +
    def backend(
+    ...
+)
+
    +

    Import backend module.

    +

    Cached property to load the backend module, if it has not been already.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneBackend module for the specific choice of backend.
    +

    ensure_lazy

    +
    def ensure_lazy(
+    self,
+    task: Any
+) -> dewret.workflow.Lazy | None
+
    +

    Evaluate a single task for a known workflow.

    +

    As we mask our lazy-evaluable functions to appear as their original +types to the type system (see dewret.tasks.task), we must cast them +back, to allow the type-checker to comb the remainder of the code.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonethe suspected task to check.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneOriginal task, cast to a Lazy, or None.
    +

    evaluate

    +
    def evaluate(
+    self,
+    task: dewret.workflow.Lazy | list[dewret.workflow.Lazy] | tuple[dewret.workflow.Lazy],
+    __workflow__: dewret.workflow.Workflow,
+    thread_pool: concurrent.futures.thread.ThreadPoolExecutor | None = None,
+    **kwargs: Any
+) -> Any
+
    +

    Evaluate a single task for a known workflow.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonethe task to evaluate.None
    workflowNoneworkflow within which this exists.None
    thread_poolNoneexisting pool of threads to run this in, or None.None
    **kwargsNoneany arguments to pass to the task.None
    +

    make_lazy

    +
    def make_lazy(
+    self
+) -> collections.abc.Callable[[collections.abc.Callable[..., typing.Any]], dewret.workflow.Lazy]
+
    +

    Get the lazy decorator for this backend.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneReal decorator for this backend.
    +

    set_backend

    +
    def set_backend(
+    self,
+    backend: dewret.tasks.Backend
+) -> dewret.tasks.Backend
+
    +

    Choose a backend.

    +

    Sets the backend, provided it has not already been loaded.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    backendNonechosen backend, to override the default.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneBackend that was set.
    +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    RuntimeErrorwhen a backend has already been loaded.
    +

    unwrap

    +
    def unwrap(
+    self,
+    task: dewret.workflow.Lazy
+) -> collections.abc.Callable[..., typing.Any]
+
    +

    Unwraps a lazy-evaluated function to get the function.

    +

    Ideally, we could use the __wrapped__ property but not all +workflow engines support this, and most importantly, dask has +only done so as of 2024.03.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonetask to be unwrapped.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneOriginal target.
    +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    RuntimeErrorif the task is not a wrapped function.
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/utils/index.html b/reference/dewret/utils/index.html new file mode 100644 index 00000000..5aa62b61 --- /dev/null +++ b/reference/dewret/utils/index.html @@ -0,0 +1,1357 @@ + + + + + + + + + + + + + + + + Utils - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.utils

    +

    Utility module.

    +

    General types and functions to centralize common logic.

    +

    Functions

    +

    crawl_raw

    +
    def crawl_raw(
+    value: Any,
+    action: Optional[Callable[[Any], Any]] = None
+) -> Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]
+
    +

    Takes a Raw-like structure and makes it RawType.

    +

    Particularly useful for squashing any TypedDicts.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    valueNonevalue to squashNone
    actionNonean callback to apply to each found entry, or None.None
    +

    ensure_raw

    +
    def ensure_raw(
+    value: Any,
+    cast_tuple: bool = False
+) -> Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]
+
    +

    Check if a variable counts as "raw".

    +

    This works around a checking issue that isinstance of a union of types +assigned to a variable, such as RawType, may throw errors even though Python 3.11+ +does not. Instead, we explicitly make the full union in the statement below.

    +

    firm_to_raw

    +
    def firm_to_raw(
+    value: Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType'], list['FirmType'], dict[str, 'FirmType'], tuple['FirmType', ...]]
+) -> Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType']]
+
    +

    Convenience wrapper for firm structures.

    +

    Turns structures that would be raw, except for tuples, into raw structures +by mapping any tuples to lists.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    valueNonea firm structure (contains raw/tuple values).None
    +

    flatten_if_set

    +
    def flatten_if_set(
+    value: Any
+) -> Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType'], dewret.utils.Unset]
+
    +

    Takes a Raw-like structure and makes it RawType or Unset.

    +

    Flattens if the value is set, but otherwise returns the unset +sentinel value as-is.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    valueNonevalue to squashNone
    +

    hasher

    +
    def hasher(
+    construct: Union[str, float, bool, bytes, int, NoneType, list['RawType'], dict[str, 'RawType'], list['FirmType'], dict[str, 'FirmType'], tuple['FirmType', ...]]
+) -> str
+
    +

    Consistently hash a RawType or tuple structure.

    +

    Turns a possibly nested structure of basic types, dicts, lists and tuples +into a consistent hash.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    constructNonestructure to hash.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneHash string that should be unique to the construct. The limits of this uniqueness
    have not yet been explicitly calculated.
    +

    is_expr

    +
    def is_expr(
+    value: Any,
+    permitted_references: type = <class 'dewret.core.Reference'>
+) -> bool
+
    +

    Confirms whether a structure has only raw or expression types.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    valueNonea crawlable structure.None
    permitted_referencesNonea class representing the allowed types of References.None
    +

    is_firm

    +
    def is_firm(
+    value: Any,
+    check: Optional[Callable[[Any], bool]] = None
+) -> bool
+
    +

    Confirms whether a function is firm.

    +

    That is, whether its contents are raw or tuples.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    valueNonevalue to check.None
    checkNoneany additional check to apply.None
    +

    is_raw

    +
    def is_raw(
+    value: Any,
+    check: Optional[Callable[[Any], bool]] = None
+) -> bool
+
    +

    Check if a variable counts as "raw".

    +

    This works around a checking issue that isinstance of a union of types +assigned to a variable, such as RawType, may throw errors even though Python 3.11+ +does not. Instead, we explicitly make the full union in the statement below.

    +

    is_raw_type

    +
    def is_raw_type(
+    typ: type
+) -> bool
+
    +

    Check if a type counts as "raw".

    +

    load_module_or_package

    +
    def load_module_or_package(
+    target_name: str,
+    path: pathlib.Path
+) -> module
+
    +

    Convenience loader for modules.

    +

    If an __init__.py is found in the same location as the target, it will try to load the renderer module +as if it is contained in a package and, if it cannot, will fall back to loading the single file.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    target_nameNonemodule name that should appear in sys.modules.None
    pathNonelocation of the module.None
    +

    make_traceback

    +
    def make_traceback(
+    skip: int = 2
+) -> traceback | None
+
    +

    Creates a traceback for the current frame.

    +

    Necessary to allow tracebacks to be prepped for +potential errors in lazy-evaluated functions.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    skipNonenumber of frames to skip before starting traceback.None
    +

    Classes

    +

    DataclassProtocol

    +
    class DataclassProtocol(
+    *args,
+    **kwargs
+)
+
    +

    Format of a dataclass.

    +

    Since dataclasses do not expose a proper type, we use this to +represent them.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Unset

    +
    class Unset(
+    /,
+    *args,
+    **kwargs
+)
+
    +

    Unset variable, with no default value.

    +

    Descendants

    +
      +
    • dewret.workflow.UnsetType
      • +
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/reference/dewret/workflow/index.html b/reference/dewret/workflow/index.html new file mode 100644 index 00000000..2783a6a1 --- /dev/null +++ b/reference/dewret/workflow/index.html @@ -0,0 +1,19741 @@ + + + + + + + + + + + + + + + + Workflow - dewret + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +
    + + + + +
    + + +
    + +
    + + + + + + +
    +
    + + + +
    +
    +
    + + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    +
    + + + +
    +
    + + + + + + + + + +

    Module dewret.workflow

    +

    Overarching workflow concepts.

    +

    Basic constructs for describing a workflow.

    +

    Variables

    +
    AVAILABLE_TYPES
+
    +
    CHECK_IDS
+
    +
    LazyFactory
+
    +
    RetType
+
    +
    StepExecution
+
    +
    T
+
    +
    TYPE_CHECKING
+
    +
    Target
+
    +
    U
+
    +
    UNSET
+
    +
    logger
+
    +

    Functions

    +

    expr_to_references

    +
    def expr_to_references(
+    expression: 'Any',
+    remap: 'Callable[[Any], Any] | None' = None
+) -> 'tuple[ExprType, list[Reference[Any] | Parameter[Any]]]'
+
    +

    Pull out any references, or other free symbols, from an expression.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    expressionNonenormally a reference that can be immediately returned, but may be a sympy expression or a dict/tuple/list/etc. of such.None
    remapNonea callable to project certain values down before extracting symbols, or None.None
    +

    is_task

    +
    def is_task(
+    task: 'Lazy'
+) -> 'bool'
+
    +

    Decide whether this is a task.

    +

    Checks whether the wrapped function has the magic +attribute __step_expression__ set to True, which is +done within task creation.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonelazy-evaluated value, suspected to be a task.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneTrue if task is indeed a task.
    +

    param

    +
    def param(
+    name: 'str',
+    default: 'T | UnsetType[T] | Unset' = <dewret.utils.Unset object at 0x7f1e70dddb90>,
+    tethered: 'Literal[False] | None | Step | Workflow' = False,
+    typ: 'type[T] | Unset' = <dewret.utils.Unset object at 0x7f1e70dddb90>,
+    autoname: 'bool' = False
+) -> 'T'
+
    +

    Create a parameter.

    +

    Will cast so it looks like the original type.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneParameter class cast to the type of the supplied default.
    +

    unify_workflows

    +
    def unify_workflows(
+    expression: 'Any',
+    base_workflow: 'Workflow | None',
+    set_only: 'bool' = False
+) -> 'tuple[Basic | None, Workflow | None]'
+
    +

    Takes an expression and ensures all of its references exist in the same workflow.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    expressionNoneany valid argument to dewret.workflow.expr_to_references.None
    base_workflowNonethe desired workflow to align on, or None.None
    set_onlyNonewhether to bother assimilating all the workflows (False), or to assume that has been done (False).None
    +

    Classes

    +

    BaseStep

    +
    class BaseStep(
+    workflow: 'Workflow',
+    task: 'Task | Workflow',
+    arguments: 'Mapping[str, Reference[Any] | Raw]',
+    raw_as_parameter: 'bool' = False
+)
+
    +

    Lazy-evaluated function call.

    +

    Individual function call to a lazy-evaluatable function, tracked +for building up the Workflow.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    taskNonethe Task being called in this step.None
    argumentsNonekey-value pairs of arguments to this step.None
    +

    Ancestors (in MRO)

    +
      +
    • dewret.core.WorkflowComponent
      • +
    +

    Descendants

    +
      +
    • dewret.workflow.NestedStep
      • +
    • dewret.workflow.Step
      • +
    +

    Class variables

    +
    positional_args
+
    +

    Instance variables

    +
    id
+
    +

    Consistent ID based on the value.

    +
    name
+
    +

    Name for this step.

    +

    May be remapped by the workflow to something nicer +than the ID.

    +
    return_type
+
    +

    Take the type of the wrapped function from the target.

    +

    Unwraps and inspects the signature, meaning that the original +wrapped function must have a typehint for the return value.

    +

    Methods

    +

    make_reference

    +
    def make_reference(
+    self,
+    **kwargs: 'Any'
+) -> "'StepReference[T]'"
+
    +

    Create a reference to this step.

    +

    Builds a reference to the (result of) this step, which will be iterable if appropriate.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    **kwargsNonearguments for reference constructor, which will be supplemented appropriately.None
    +

    set_workflow

    +
    def set_workflow(
+    self,
+    workflow: 'Workflow',
+    with_arguments: 'bool' = True
+) -> 'None'
+
    +

    Move the step reference to another workflow.

    +

    This method is primarily intended to be called by a step, allowing it to +switch to a new workflow. It also updates the workflow reference for any +arguments that are steps themselves, if specified.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneThe new target workflow to which the step should be moved.None
    with_argumentsNoneIf True, also update the workflow reference for the step's arguments.None
    +

    FactoryCall

    +
    class FactoryCall(
+    workflow: 'Workflow',
+    task: 'Task | Workflow',
+    arguments: 'Mapping[str, Reference[Any] | Raw]',
+    raw_as_parameter: 'bool' = False
+)
+
    +

    Call to a factory function.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.workflow.Step
      • +
    • dewret.workflow.BaseStep
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Class variables

    +
    positional_args
+
    +

    Instance variables

    +
    id
+
    +

    Consistent ID based on the value.

    +
    name
+
    +

    Name for this step.

    +

    May be remapped by the workflow to something nicer +than the ID.

    +
    return_type
+
    +

    Take the type of the wrapped function from the target.

    +

    Unwraps and inspects the signature, meaning that the original +wrapped function must have a typehint for the return value.

    +

    Methods

    +

    make_reference

    +
    def make_reference(
+    self,
+    **kwargs: 'Any'
+) -> "'StepReference[T]'"
+
    +

    Create a reference to this step.

    +

    Builds a reference to the (result of) this step, which will be iterable if appropriate.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    **kwargsNonearguments for reference constructor, which will be supplemented appropriately.None
    +

    set_workflow

    +
    def set_workflow(
+    self,
+    workflow: 'Workflow',
+    with_arguments: 'bool' = True
+) -> 'None'
+
    +

    Move the step reference to another workflow.

    +

    This method is primarily intended to be called by a step, allowing it to +switch to a new workflow. It also updates the workflow reference for any +arguments that are steps themselves, if specified.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneThe new target workflow to which the step should be moved.None
    with_argumentsNoneIf True, also update the workflow reference for the step's arguments.None
    +

    FieldableMixin

    +
    class FieldableMixin(
+    self: 'FieldableProtocol',
+    *args: 'Any',
+    field: 'str | int | tuple[str | int, ...] | None' = None,
+    **kwargs: 'Any'
+)
+
    +

    Tooling for enhancing a type with referenceable fields.

    +

    Descendants

    +
      +
    • dewret.workflow.ParameterReference
      • +
    • dewret.workflow.StepReference
      • +
    +

    Methods

    +

    find_field

    +
    def find_field(
+    self: 'FieldableProtocol',
+    field: 'str | int',
+    fallback_type: 'type | None' = None,
+    **init_kwargs: 'Any'
+) -> 'Reference[Any]'
+
    +

    Field within the reference, if possible.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fieldNonethe field to search for.None
    fallback_typeNonethe type to use if we do not know a more specific one.None
    **init_kwargsNonearguments to use for constructing a new reference (via __make_reference__).None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA field-specific version of this reference.
    +

    FieldableProtocol

    +
    class FieldableProtocol(
+    *args: 'Any',
+    field: 'str | None' = None,
+    **kwargs: 'Any'
+)
+
    +

    Expected interfaces for a type that can take fields.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fieldNonetuple representing the named fields, either strings or integers.None
    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Instance variables

    +
    name
+
    +

    The name for the target, accounting for the field.

    +

    IterableParameterReference

    +
    class IterableParameterReference(
+    typ: type[~U] | None = None,
+    **kwargs: Any
+)
+
    +

    Iterable form of parameter references.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.core.IterableMixin
      • +
    • dewret.workflow.ParameterReference
      • +
    • dewret.workflow.FieldableMixin
      • +
    • dewret.core.Reference
      • +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Class variables

    +
    ParameterReferenceMetadata
+
    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Printable name of the reference.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    find_field

    +
    def find_field(
+    self: 'FieldableProtocol',
+    field: 'str | int',
+    fallback_type: 'type | None' = None,
+    **init_kwargs: 'Any'
+) -> 'Reference[Any]'
+
    +

    Field within the reference, if possible.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fieldNonethe field to search for.None
    fallback_typeNonethe type to use if we do not know a more specific one.None
    **init_kwargsNonearguments to use for constructing a new reference (via __make_reference__).None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA field-specific version of this reference.
    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    IterableStepReference

    +
    class IterableStepReference(
+    typ: type[~U] | None = None,
+    **kwargs: Any
+)
+
    +

    Iterable form of a step reference.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.core.IterableMixin
      • +
    • dewret.workflow.StepReference
      • +
    • dewret.workflow.FieldableMixin
      • +
    • dewret.core.Reference
      • +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Class variables

    +
    StepReferenceMetadata
+
    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Printable name of the reference.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    find_field

    +
    def find_field(
+    self: 'FieldableProtocol',
+    field: 'str | int',
+    fallback_type: 'type | None' = None,
+    **init_kwargs: 'Any'
+) -> 'Reference[Any]'
+
    +

    Field within the reference, if possible.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fieldNonethe field to search for.None
    fallback_typeNonethe type to use if we do not know a more specific one.None
    **init_kwargsNonearguments to use for constructing a new reference (via __make_reference__).None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA field-specific version of this reference.
    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    Lazy

    +
    class Lazy(
+    *args,
+    **kwargs
+)
+
    +

    Requirements for a lazy-evaluatable function.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    Descendants

    +
      +
    • dewret.workflow.LazyEvaluation
      • +
    +

    LazyEvaluation

    +
    class LazyEvaluation(
+    fn: 'Callable[..., RetType]'
+)
+
    +

    Tracks a single evaluation of a lazy function.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.workflow.Lazy
      • +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    +

    NestedStep

    +
    class NestedStep(
+    workflow: 'Workflow',
+    name: 'str',
+    subworkflow: 'Workflow',
+    arguments: 'Mapping[str, Basic | Reference[Any] | Raw]',
+    raw_as_parameter: 'bool' = False
+)
+
    +

    Calling out to a subworkflow.

    +

    Type of BaseStep to call a subworkflow, which holds a reference to it.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.workflow.BaseStep
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Class variables

    +
    positional_args
+
    +

    Instance variables

    +
    id
+
    +

    Consistent ID based on the value.

    +
    name
+
    +

    Name for this step.

    +

    May be remapped by the workflow to something nicer +than the ID.

    +
    return_type
+
    +

    Take the type of the wrapped function from the target.

    +

    Unwraps and inspects the signature, meaning that the original +wrapped function must have a typehint for the return value.

    +
    subworkflow
+
    +

    Subworkflow that is wrapped.

    +

    Methods

    +

    make_reference

    +
    def make_reference(
+    self,
+    **kwargs: 'Any'
+) -> "'StepReference[T]'"
+
    +

    Create a reference to this step.

    +

    Builds a reference to the (result of) this step, which will be iterable if appropriate.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    **kwargsNonearguments for reference constructor, which will be supplemented appropriately.None
    +

    set_workflow

    +
    def set_workflow(
+    self,
+    workflow: 'Workflow',
+    with_arguments: 'bool' = True
+) -> 'None'
+
    +

    Move the step reference to another workflow.

    +

    This method is primarily intended to be called by a step, allowing it to +switch to a new workflow. It also updates the workflow reference for any +arguments that are steps themselves, if specified.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneThe new target workflow to which the step should be moved.None
    with_argumentsNoneIf True, also update the workflow reference for the step's arguments.None
    +

    Parameter

    +
    class Parameter(
+    name: 'str',
+    default: 'T | UnsetType[T]',
+    tethered: 'Literal[False] | None | Step | Workflow' = None,
+    autoname: 'bool' = False,
+    typ: 'type[T] | Unset' = <dewret.utils.Unset object at 0x7f1e70dddb90>
+)
+
    +

    Global parameter.

    +

    Independent parameter that will be used when a task is spotted +reaching outside its scope. This wraps the variable it uses.

    +

    To allow for potential arithmetic operations, etc. it is a Sympy +symbol.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    nameNonename of the parameter.None
    defaultNonecaptured default value from the original value.None
    +

    Ancestors (in MRO)

    +
      +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    +

    Class variables

    +
    autoname
+
    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    is_loopable

    +
    def is_loopable(
+    typ: 'type'
+) -> 'bool'
+
    +

    Checks if this type can be looped over.

    +

    In particular, checks if this is an iterable that is NOT a str or bytes, possibly disguised +behind an Annotated.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    typNonetype to check.None
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    default
+
    +

    Retrieve default value for this parameter, or an unset token.

    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Extended name, suitable for rendering.

    +

    This attempts to create a unique name by tying the parameter to a step +if the user has not explicitly provided a name, ideally the one where +we discovered it.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    make_reference

    +
    def make_reference(
+    self,
+    **kwargs: 'Any'
+) -> "'ParameterReference[T]'"
+
    +

    Creates a new reference for the parameter.

    +

    The kwargs will be passed to the constructor, but the

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    typNonetype of the new reference's target.None
    **kwargsNonearguments to pass to the constructor.None
    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    register_caller

    +
    def register_caller(
+    self,
+    caller: 'BaseStep'
+) -> 'None'
+
    +

    Capture a step that uses this parameter.

    +

    Gathers together the steps using this parameter. The first found will +be recorded as the tethered step, and used for forming the name.

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    ParameterReference

    +
    class ParameterReference(
+    parameter: 'Parameter[U]',
+    *args: 'Any',
+    typ: 'type[U] | None' = None,
+    **kwargs: 'Any'
+)
+
    +

    Reference to an individual Parameter.

    +

    Allows us to refer to the outputs of a Parameter in subsequent Parameter +arguments.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    parameterNoneParameter referred to.None
    workflowNoneRelated workflow. In this case, as Parameters are generic
    but ParameterReferences are specific, this carries the actual workflow reference.    		None
    +

    Ancestors (in MRO)

    +
      +
    • dewret.workflow.FieldableMixin
      • +
    • dewret.core.Reference
      • +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Descendants

    +
      +
    • dewret.workflow.IterableParameterReference
      • +
    +

    Class variables

    +
    ParameterReferenceMetadata
+
    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Printable name of the reference.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    find_field

    +
    def find_field(
+    self: 'FieldableProtocol',
+    field: 'str | int',
+    fallback_type: 'type | None' = None,
+    **init_kwargs: 'Any'
+) -> 'Reference[Any]'
+
    +

    Field within the reference, if possible.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fieldNonethe field to search for.None
    fallback_typeNonethe type to use if we do not know a more specific one.None
    **init_kwargsNonearguments to use for constructing a new reference (via __make_reference__).None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA field-specific version of this reference.
    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    Step

    +
    class Step(
+    workflow: 'Workflow',
+    task: 'Task | Workflow',
+    arguments: 'Mapping[str, Reference[Any] | Raw]',
+    raw_as_parameter: 'bool' = False
+)
+
    +

    Regular step.

    +

    Ancestors (in MRO)

    +
      +
    • dewret.workflow.BaseStep
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Descendants

    +
      +
    • dewret.workflow.FactoryCall
      • +
    +

    Class variables

    +
    positional_args
+
    +

    Instance variables

    +
    id
+
    +

    Consistent ID based on the value.

    +
    name
+
    +

    Name for this step.

    +

    May be remapped by the workflow to something nicer +than the ID.

    +
    return_type
+
    +

    Take the type of the wrapped function from the target.

    +

    Unwraps and inspects the signature, meaning that the original +wrapped function must have a typehint for the return value.

    +

    Methods

    +

    make_reference

    +
    def make_reference(
+    self,
+    **kwargs: 'Any'
+) -> "'StepReference[T]'"
+
    +

    Create a reference to this step.

    +

    Builds a reference to the (result of) this step, which will be iterable if appropriate.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    **kwargsNonearguments for reference constructor, which will be supplemented appropriately.None
    +

    set_workflow

    +
    def set_workflow(
+    self,
+    workflow: 'Workflow',
+    with_arguments: 'bool' = True
+) -> 'None'
+
    +

    Move the step reference to another workflow.

    +

    This method is primarily intended to be called by a step, allowing it to +switch to a new workflow. It also updates the workflow reference for any +arguments that are steps themselves, if specified.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflowNoneThe new target workflow to which the step should be moved.None
    with_argumentsNoneIf True, also update the workflow reference for the step's arguments.None
    +

    StepReference

    +
    class StepReference(
+    step: 'BaseStep',
+    *args: 'Any',
+    typ: 'type[U] | None' = None,
+    **kwargs: 'Any'
+)
+
    +

    Reference to an individual Step.

    +

    Allows us to refer to the outputs of a Step in subsequent Step +arguments.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    _Nonemetadata wrapping the Step referred to.None
    +

    Ancestors (in MRO)

    +
      +
    • dewret.workflow.FieldableMixin
      • +
    • dewret.core.Reference
      • +
    • typing.Generic
      • +
    • sympy.core.symbol.Symbol
      • +
    • sympy.core.expr.AtomicExpr
      • +
    • sympy.core.basic.Atom
      • +
    • sympy.core.expr.Expr
      • +
    • sympy.logic.boolalg.Boolean
      • +
    • sympy.core.basic.Basic
      • +
    • sympy.printing.defaults.Printable
      • +
    • sympy.core.evalf.EvalfMixin
      • +
    • dewret.core.WorkflowComponent
      • +
    +

    Descendants

    +
      +
    • dewret.workflow.IterableStepReference
      • +
    +

    Class variables

    +
    StepReferenceMetadata
+
    +
    default_assumptions
+
    +
    is_Add
+
    +
    is_AlgebraicNumber
+
    +
    is_Atom
+
    +
    is_Boolean
+
    +
    is_Derivative
+
    +
    is_Dummy
+
    +
    is_Equality
+
    +
    is_Float
+
    +
    is_Function
+
    +
    is_Indexed
+
    +
    is_Integer
+
    +
    is_MatAdd
+
    +
    is_MatMul
+
    +
    is_Matrix
+
    +
    is_Mul
+
    +
    is_Not
+
    +
    is_Number
+
    +
    is_NumberSymbol
+
    +
    is_Order
+
    +
    is_Piecewise
+
    +
    is_Point
+
    +
    is_Poly
+
    +
    is_Pow
+
    +
    is_Rational
+
    +
    is_Relational
+
    +
    is_Symbol
+
    +
    is_Vector
+
    +
    is_Wild
+
    +
    is_comparable
+
    +
    is_number
+
    +
    is_scalar
+
    +
    is_symbol
+
    +

    Static methods

    +

    class_key

    +
    def class_key(
+    
+)
+
    +

    Nice order of classes.

    +

    fromiter

    +
    def fromiter(
+    args,
+    **assumptions
+)
+
    +

    Create a new object from an iterable.

    +

    This is a convenience function that allows one to create objects from +any iterable, without having to convert to a list or tuple first.

    +

    Examples

    +
    +
    +
    +

    from sympy import Tuple +Tuple.fromiter(i for i in range(5)) +(0, 1, 2, 3, 4)

    +
    +
    +
    +

    Instance variables

    +
    args
+
    +

    Returns a tuple of arguments of 'self'.

    +

    Examples

    +
    +
    +
    +

    from sympy import cot +from sympy.abc import x, y

    +

    cot(x).args +(x,)

    +

    cot(x).args[0] +x

    +

    (x*y).args +(x, y)

    +

    (x*y).args[1] +y

    +
    +
    +
    +

    Notes

    +

    Never use self._args, always use self.args. +Only use _args in new when creating a new function. +Do not override .args() from Basic (so that it is easy to +change the interface in the future if needed).

    +
    assumptions0
+
    +
    binary_symbols
+
    +
    canonical_variables
+
    +

    Return a dictionary mapping any variable defined in

    +

    self.bound_symbols to Symbols that do not clash +with any free symbols in the expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x +Lambda(x, 2*x).canonical_variables +{x: _0}

    +
    +
    +
    +
    expr_free_symbols
+
    +
    free_symbols
+
    +
    func
+
    +

    The top-level function in an expression.

    +

    The following should hold for all objects::

    +
    >> x == x.func(*x.args)
+
    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +a = 2x +a.func + +a.args +(2, x) +a.func(a.args) +2x +a == a.func(a.args) +True

    +
    +
    +
    +
    is_algebraic
+
    +
    is_antihermitian
+
    +
    is_commutative
+
    +
    is_complex
+
    +
    is_composite
+
    +
    is_even
+
    +
    is_extended_negative
+
    +
    is_extended_nonnegative
+
    +
    is_extended_nonpositive
+
    +
    is_extended_nonzero
+
    +
    is_extended_positive
+
    +
    is_extended_real
+
    +
    is_finite
+
    +
    is_hermitian
+
    +
    is_imaginary
+
    +
    is_infinite
+
    +
    is_integer
+
    +
    is_irrational
+
    +
    is_negative
+
    +
    is_noninteger
+
    +
    is_nonnegative
+
    +
    is_nonpositive
+
    +
    is_nonzero
+
    +
    is_odd
+
    +
    is_polar
+
    +
    is_positive
+
    +
    is_prime
+
    +
    is_rational
+
    +
    is_real
+
    +
    is_transcendental
+
    +
    is_zero
+
    +
    kind
+
    +
    name
+
    +

    Printable name of the reference.

    +

    Methods

    +

    adjoint

    +
    def adjoint(
+    self
+)
+
    +

    apart

    +
    def apart(
+    self,
+    x=None,
+    **args
+)
+
    +

    See the apart function in sympy.polys

    +

    args_cnc

    +
    def args_cnc(
+    self,
+    cset=False,
+    warn=True,
+    split_1=True
+)
+
    +

    Return [commutative factors, non-commutative factors] of self.

    +

    Explanation

    +

    self is treated as a Mul and the ordering of the factors is maintained. +If cset is True the commutative factors will be returned in a set. +If there were repeated factors (as may happen with an unevaluated Mul) +then an error will be raised unless it is explicitly suppressed by +setting warn to False.

    +

    Note: -1 is always separated from a Number unless split_1 is False.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, oo +A, B = symbols('A B', commutative=0) +x, y = symbols('x y') +(-2xy).args_cnc() +[[-1, 2, x, y], []] +(-2.5x).args_cnc() +[[-1, 2.5, x], []] +(-2xABy).args_cnc() +[[-1, 2, x, y], [A, B]] +(-2xABy).args_cnc(split_1=False) +[[-2, x, y], [A, B]] +(-2x*y).args_cnc(cset=True) +[{-1, 2, x, y}, []]

    +
    +
    +
    +

    The arg is always treated as a Mul:

    +
    +
    +
    +

    (-2 + x + A).args_cnc() +[[], [x - 2 + A]] +(-oo).args_cnc() # -oo is a singleton +[[-1, oo], []]

    +
    +
    +
    +

    as_base_exp

    +
    def as_base_exp(
+    self
+) -> 'tuple[Expr, Expr]'
+
    +

    as_coeff_Add

    +
    def as_coeff_Add(
+    self,
+    rational=False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a summation.

    +

    as_coeff_Mul

    +
    def as_coeff_Mul(
+    self,
+    rational: 'bool' = False
+) -> "tuple['Number', Expr]"
+
    +

    Efficiently extract the coefficient of a product.

    +

    as_coeff_add

    +
    def as_coeff_add(
+    self,
+    *deps
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as an Add, a.

    +

    c should be a Rational added to any terms of the Add that are +independent of deps.

    +

    args should be a tuple of all other terms of a; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is an Add or not but +you want to treat self as an Add or if you want to process the +individual arguments of the tail of self as an Add.

    +
      +
    • if you know self is an Add and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail.
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_add() +(3, ()) +(3 + x).as_coeff_add() +(3, (x,)) +(3 + x + y).as_coeff_add(x) +(y + 3, (x,)) +(3 + y).as_coeff_add(x) +(y + 3, ())

    +
    +
    +
    +

    as_coeff_exponent

    +
    def as_coeff_exponent(
+    self,
+    x
+) -> 'tuple[Expr, Expr]'
+
    +

    c*x**e -> c,e where x can be any symbolic expression.

    +

    as_coeff_mul

    +
    def as_coeff_mul(
+    self,
+    *deps,
+    **kwargs
+) -> 'tuple[Expr, tuple[Expr, ...]]'
+
    +

    Return the tuple (c, args) where self is written as a Mul, m.

    +

    c should be a Rational multiplied by any factors of the Mul that are +independent of deps.

    +

    args should be a tuple of all other factors of m; args is empty +if self is a Number or if self is independent of deps (when given).

    +

    This should be used when you do not know if self is a Mul or not but +you want to treat self as a Mul or if you want to process the +individual arguments of the tail of self as a Mul.

    +
      +
    • if you know self is a Mul and want only the head, use self.args[0];
      • +
    • if you do not want to process the arguments of the tail but need the + tail then use self.as_two_terms() which gives the head and tail;
      • +
    • if you want to split self into an independent and dependent parts + use self.as_independent(*deps)
      • +
    +
    +
    +
    +

    from sympy import S +from sympy.abc import x, y +(S(3)).as_coeff_mul() +(3, ()) +(3xy).as_coeff_mul() +(3, (x, y)) +(3xy).as_coeff_mul(x) +(3y, (x,)) +(3y).as_coeff_mul(x) +(3*y, ())

    +
    +
    +
    +

    as_coefficient

    +
    def as_coefficient(
+    self,
+    expr
+)
+
    +

    Extracts symbolic coefficient at the given expression. In

    +

    other words, this functions separates 'self' into the product +of 'expr' and 'expr'-free coefficient. If such separation +is not possible it will return None.

    +

    Examples

    +
    +
    +
    +

    from sympy import E, pi, sin, I, Poly +from sympy.abc import x

    +

    E.as_coefficient(E) +1 +(2E).as_coefficient(E) +2 +(2sin(E)*E).as_coefficient(E)

    +
    +
    +
    +

    Two terms have E in them so a sum is returned. (If one were +desiring the coefficient of the term exactly matching E then +the constant from the returned expression could be selected. +Or, for greater precision, a method of Poly can be used to +indicate the desired term from which the coefficient is +desired.)

    +
    +
    +
    +

    (2E + xE).as_coefficient(E) +x + 2 +_.args[0] # just want the exact match +2 +p = Poly(2E + xE); p +Poly(xE + 2E, x, E, domain='ZZ') +p.coeff_monomial(E) +2 +p.nth(0, 1) +2

    +
    +
    +
    +

    Since the following cannot be written as a product containing +E as a factor, None is returned. (If the coefficient 2*x is +desired then the coeff method should be used.)

    +
    +
    +
    +

    (2Ex + x).as_coefficient(E) +(2Ex + x).coeff(E) +2*x

    +

    (E*(x + 1) + x).as_coefficient(E)

    +

    (2piI).as_coefficient(piI) +2 +(2I).as_coefficient(pi*I)

    +
    +
    +
    +

    See Also

    +

    coeff: return sum of terms have a given factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    as_coefficients_dict

    +
    def as_coefficients_dict(
+    self,
+    *syms
+)
+
    +

    Return a dictionary mapping terms to their Rational coefficient.

    +

    Since the dictionary is a defaultdict, inquiries about terms which +were not present will return a coefficient of 0.

    +

    If symbols syms are provided, any multiplicative terms +independent of them will be considered a coefficient and a +regular dictionary of syms-dependent generators as keys and +their corresponding coefficients as values will be returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import a, x, y +(3x + ax + 4).as_coefficients_dict() +{1: 4, x: 3, ax: 1} +_[a] +0 +(3ax).as_coefficients_dict() +{ax: 3} +(3ax).as_coefficients_dict(x) +{x: 3a} +(3ax).as_coefficients_dict(y) +{1: 3a*x}

    +
    +
    +
    +

    as_content_primitive

    +
    def as_content_primitive(
+    self,
+    radical=False,
+    clear=True
+)
+
    +

    This method should recursively remove a Rational from all arguments

    +

    and return that (content) and the new self (primitive). The content +should always be positive and Mul(*foo.as_content_primitive()) == foo. +The primitive need not be in canonical form and should try to preserve +the underlying structure if possible (i.e. expand_mul should not be +applied to self).

    +

    Examples

    +
    +
    +
    +

    from sympy import sqrt +from sympy.abc import x, y, z

    +

    eq = 2 + 2x + 2y(3 + 3y)

    +
    +
    +
    +

    The as_content_primitive function is recursive and retains structure:

    +
    +
    +
    +

    eq.as_content_primitive() +(2, x + 3y(y + 1) + 1)

    +
    +
    +
    +

    Integer powers will have Rationals extracted from the base:

    +
    +
    +
    +

    ((2 + 6x)2).as_content_primitive() +(4, (3x + 1)2) +((2 + 6*x)(2y)).as_content_primitive() +(1, (2(3x + 1))(2y))

    +
    +
    +
    +

    Terms may end up joining once their as_content_primitives are added:

    +
    +
    +
    +

    ((5(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(11, x(y + 1)) +((3(x(1 + y)) + 2x(3 + 3y))).as_content_primitive() +(9, x(y + 1)) +((3(z(1 + y)) + 2.0x(3 + 3y))).as_content_primitive() +(1, 6.0x(y + 1) + 3z(y + 1)) +((5(x(1 + y)) + 2x(3 + 3y))2).as_content_primitive() +(121, x2(y + 1)2) +((x(1 + y) + 0.4x(3 + 3y))2).as_content_primitive() +(1, 4.84x2*(y + 1)2)

    +
    +
    +
    +

    Radical content can also be factored out of the primitive:

    +
    +
    +
    +

    (2sqrt(2) + 4sqrt(10)).as_content_primitive(radical=True) +(2, sqrt(2)(1 + 2sqrt(5)))

    +
    +
    +
    +

    If clear=False (default is True) then content will not be removed +from an Add if it can be distributed to leave one or more +terms with integer coefficients.

    +
    +
    +
    +

    (x/2 + y).as_content_primitive() +(1/2, x + 2*y) +(x/2 + y).as_content_primitive(clear=False) +(1, x/2 + y)

    +
    +
    +
    +

    as_dummy

    +
    def as_dummy(
+    self
+)
+
    +

    Return the expression with any objects having structurally

    +

    bound symbols replaced with unique, canonical symbols within +the object in which they appear and having only the default +assumption for commutativity being True. When applied to a +symbol a new symbol having only the same commutativity will be +returned.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Symbol +from sympy.abc import x +r = Symbol('r', real=True) +Integral(r, (r, x)).as_dummy() +Integral(0, (_0, x)) +.variables[0].is_real is None +True +r.as_dummy() +_r

    +
    +
    +
    +

    Notes

    +

    Any object that has structurally bound variables should have +a property, bound_symbols that returns those symbols +appearing in the object.

    +

    as_expr

    +
    def as_expr(
+    self,
+    *gens
+)
+
    +

    Convert a polynomial to a SymPy expression.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    f = (x2 + x*y).as_poly(x, y) +f.as_expr() +x2 + x*y

    +

    sin(x).as_expr() +sin(x)

    +
    +
    +
    +

    as_independent

    +
    def as_independent(
+    self,
+    *deps,
+    **hint
+) -> 'tuple[Expr, Expr]'
+
    +

    A mostly naive separation of a Mul or Add into arguments that are not

    +

    are dependent on deps. To obtain as complete a separation of variables +as possible, use a separation method first, e.g.:

    +
      +
    • separatevars() to change Mul, Add and Pow (including exp) into Mul
      • +
    • .expand(mul=True) to change Add or Mul into Add
      • +
    • .expand(log=True) to change log expr into an Add
      • +
    +

    The only non-naive thing that is done here is to respect noncommutative +ordering of variables and to always return (0, 0) for self of zero +regardless of hints.

    +

    For nonzero self, the returned tuple (i, d) has the +following interpretation:

    +
      +
    • i will has no variable that appears in deps
      • +
    • d will either have terms that contain variables that are in deps, or + be equal to 0 (when self is an Add) or 1 (when self is a Mul)
      • +
    • if self is an Add then self = i + d
      • +
    • if self is a Mul then self = i*d
      • +
    • otherwise (self, S.One) or (S.One, self) is returned.
      • +
    +

    To force the expression to be treated as an Add, use the hint as_Add=True

    +

    Examples

    +

    -- self is an Add

    +
    +
    +
    +

    from sympy import sin, cos, exp +from sympy.abc import x, y, z

    +

    (x + xy).as_independent(x) +(0, xy + x) +(x + xy).as_independent(y) +(x, xy) +(2xsin(x) + y + x + z).as_independent(x) +(y + z, 2xsin(x) + x) +(2xsin(x) + y + x + z).as_independent(x, y) +(z, 2xsin(x) + x + y)

    +
    +
    +
    +

    -- self is a Mul

    +
    +
    +
    +

    (xsin(x)cos(y)).as_independent(x) +(cos(y), x*sin(x))

    +
    +
    +
    +

    non-commutative terms cannot always be separated out when self is a Mul

    +
    +
    +
    +

    from sympy import symbols +n1, n2, n3 = symbols('n1 n2 n3', commutative=False) +(n1 + n1n2).as_independent(n2) +(n1, n1n2) +(n2n1 + n1n2).as_independent(n2) +(0, n1n2 + n2n1) +(n1n2n3).as_independent(n1) +(1, n1n2n3) +(n1n2n3).as_independent(n2) +(n1, n2n3) +((x-n1)(x-y)).as_independent(x) +(1, (x - y)*(x - n1))

    +
    +
    +
    +

    -- self is anything else:

    +
    +
    +
    +

    (sin(x)).as_independent(x) +(1, sin(x)) +(sin(x)).as_independent(y) +(sin(x), 1) +exp(x+y).as_independent(x) +(1, exp(x + y))

    +
    +
    +
    +

    -- force self to be treated as an Add:

    +
    +
    +
    +

    (3x).as_independent(x, as_Add=True) +(0, 3x)

    +
    +
    +
    +

    -- force self to be treated as a Mul:

    +
    +
    +
    +

    (3+x).as_independent(x, as_Add=False) +(1, x + 3) +(-3+x).as_independent(x, as_Add=False) +(1, x - 3)

    +
    +
    +
    +

    Note how the below differs from the above in making the +constant on the dep term positive.

    +
    +
    +
    +

    (y*(-3+x)).as_independent(x) +(y, x - 3)

    +
    +
    +
    +

    -- use .as_independent() for true independence testing instead + of .has(). The former considers only symbols in the free + symbols while the latter considers all symbols

    +
    +
    +
    +

    from sympy import Integral +I = Integral(x, (x, 1, 2)) +I.has(x) +True +x in I.free_symbols +False +I.as_independent(x) == (I, 1) +True +(I + x).as_independent(x) == (I, x) +True

    +
    +
    +
    +

    Note: when trying to get independent terms, a separation method +might need to be used first. In this case, it is important to keep +track of what you send to this routine so you know how to interpret +the returned values

    +
    +
    +
    +

    from sympy import separatevars, log +separatevars(exp(x+y)).as_independent(x) +(exp(y), exp(x)) +(x + xy).as_independent(y) +(x, xy) +separatevars(x + xy).as_independent(y) +(x, y + 1) +(x(1 + y)).as_independent(y) +(x, y + 1) +(x(1 + y)).expand(mul=True).as_independent(y) +(x, xy) +a, b=symbols('a b', positive=True) +(log(a*b).expand(log=True)).as_independent(b) +(log(a), log(b))

    +
    +
    +
    +

    See Also

    +

    separatevars +expand_log +sympy.core.add.Add.as_two_terms +sympy.core.mul.Mul.as_two_terms +as_coeff_mul

    +

    as_leading_term

    +
    def as_leading_term(
+    self,
+    *symbols,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading (nonzero) term of the series expansion of self.

    +

    The _eval_as_leading_term routines are used to do this, and they must +always return a non-zero value.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1 + x + x2).as_leading_term(x) +1 +(1/x2 + x + x2).as_leading_term(x) +x(-2)

    +
    +
    +
    +

    as_numer_denom

    +
    def as_numer_denom(
+    self
+)
+
    +

    Return the numerator and the denominator of an expression.

    +

    expression -> a/b -> a, b

    +

    This is just a stub that should be defined by +an object's class methods to get anything else.

    +

    See Also

    +

    normal: return a/b instead of (a, b)

    +

    as_ordered_factors

    +
    def as_ordered_factors(
+    self,
+    order=None
+)
+
    +

    Return list of ordered factors (if Mul) else [self].

    +

    as_ordered_terms

    +
    def as_ordered_terms(
+    self,
+    order=None,
+    data=False
+)
+
    +

    Transform an expression to an ordered list of terms.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, cos +from sympy.abc import x

    +

    (sin(x)2*cos(x) + sin(x)2 + 1).as_ordered_terms() +[sin(x)2*cos(x), sin(x)2, 1]

    +
    +
    +
    +

    as_poly

    +
    def as_poly(
+    self,
+    *gens,
+    **args
+)
+
    +

    Converts self to a polynomial or returns None.

    +

    Explanation

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y

    +

    print((x2 + x*y).as_poly()) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x2 + x*y).as_poly(x, y)) +Poly(x2 + x*y, x, y, domain='ZZ')

    +

    print((x**2 + sin(y)).as_poly(x, y)) +None

    +
    +
    +
    +

    as_powers_dict

    +
    def as_powers_dict(
+    self
+)
+
    +

    Return self as a dictionary of factors with each factor being

    +

    treated as a power. The keys are the bases of the factors and the +values, the corresponding exponents. The resulting dictionary should +be used with caution if the expression is a Mul and contains non- +commutative factors since the order that they appeared will be lost in +the dictionary.

    +

    See Also

    +

    as_ordered_factors: An alternative for noncommutative applications, + returning an ordered list of factors. +args_cnc: Similar to as_ordered_factors, but guarantees separation + of commutative and noncommutative factors.

    +

    as_real_imag

    +
    def as_real_imag(
+    self,
+    deep=True,
+    **hints
+)
+
    +

    Performs complex expansion on 'self' and returns a tuple

    +

    containing collected both real and imaginary parts. This +method cannot be confused with re() and im() functions, +which does not perform complex expansion at evaluation.

    +

    However it is possible to expand both re() and im() +functions and get exactly the same results as with +a single call to this function.

    +
    +
    +
    +

    from sympy import symbols, I

    +

    x, y = symbols('x,y', real=True)

    +

    (x + y*I).as_real_imag() +(x, y)

    +

    from sympy.abc import z, w

    +

    (z + w*I).as_real_imag() +(re(z) - im(w), re(w) + im(z))

    +
    +
    +
    +

    as_set

    +
    def as_set(
+    self
+)
+
    +

    Rewrites Boolean expression in terms of real sets.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Eq, Or, And +x = Symbol('x', real=True) +Eq(x, 0).as_set() +{0} +(x > 0).as_set() +Interval.open(0, oo) +And(-2 < x, x < 2).as_set() +Interval.open(-2, 2) +Or(x < -2, 2 < x).as_set() +Union(Interval.open(-oo, -2), Interval.open(2, oo))

    +
    +
    +
    +

    as_terms

    +
    def as_terms(
+    self
+)
+
    +

    Transform an expression to a list of terms.

    +

    aseries

    +
    def aseries(
+    self,
+    x=None,
+    n=6,
+    bound=0,
+    hir=False
+)
+
    +

    Asymptotic Series expansion of self.

    +

    This is equivalent to self.series(x, oo, n).

    +

    Parameters

    +

    self : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    hir : Boolean + Set this parameter to be True to produce hierarchical series. + It stops the recursion at an early level and may provide nicer + and more useful results.

    +

    bound : Value, Integer + Use the bound parameter to give limit on rewriting + coefficients in its normalised form.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, exp +from sympy.abc import x

    +

    e = sin(1/x + exp(-x)) - sin(1/x)

    +

    e.aseries(x) +(1/(24x4) - 1/(2x2) + 1 + O(x(-6), (x, oo)))*exp(-x)

    +

    e.aseries(x, n=3, hir=True) +-exp(-2x)sin(1/x)/2 + exp(-x)cos(1/x) + O(exp(-3x), (x, oo))

    +

    e = exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x) +exp(exp(x)/(1 - 1/x))

    +

    e.aseries(x, bound=3) # doctest: +SKIP +exp(exp(x)/x2)exp(exp(x)/x)exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x2)*exp(exp(x))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).aseries(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr + Asymptotic series expansion of the expression.

    +

    Notes

    +

    This algorithm is directly induced from the limit computational algorithm provided by Gruntz. +It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first +to look for the most rapidly varying subexpression w of a given expression f and then expands f +in a series in w. Then same thing is recursively done on the leading coefficient +till we get constant coefficients.

    +

    If the most rapidly varying subexpression of a given expression f is f itself, +the algorithm tries to find a normalised representation of the mrv set and rewrites f +using this normalised representation.

    +

    If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) +where w belongs to the most rapidly varying expression of self.

    +

    References

    +

    .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. + In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. + pp. 239-244. +.. [2] Gruntz thesis - p90 +.. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion

    +

    See Also

    +

    Expr.aseries: See the docstring of this function for complete details of this wrapper.

    +

    atoms

    +
    def atoms(
+    self,
+    *types
+)
+
    +

    Returns the atoms that form the current object.

    +

    By default, only objects that are truly atomic and cannot +be divided into smaller pieces are returned: symbols, numbers, +and number symbols like I and pi. It is possible to request +atoms of any type, however, as demonstrated below.

    +

    Examples

    +
    +
    +
    +

    from sympy import I, pi, sin +from sympy.abc import x, y +(1 + x + 2sin(y + Ipi)).atoms() +{1, 2, I, pi, x, y}

    +
    +
    +
    +

    If one or more types are given, the results will contain only +those types of atoms.

    +
    +
    +
    +

    from sympy import Number, NumberSymbol, Symbol +(1 + x + 2sin(y + Ipi)).atoms(Symbol) +{x, y}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number) +{1, 2}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol) +{1, 2, pi}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Number, NumberSymbol, I) +{1, 2, I, pi}

    +
    +
    +
    +

    Note that I (imaginary unit) and zoo (complex infinity) are special +types of number symbols and are not part of the NumberSymbol class.

    +

    The type can be given implicitly, too:

    +
    +
    +
    +

    (1 + x + 2sin(y + Ipi)).atoms(x) # x is a Symbol +{x, y}

    +
    +
    +
    +

    Be careful to check your assumptions when using the implicit option +since S(1).is_Integer = True but type(S(1)) is One, a special type +of SymPy atom, while type(S(2)) is type Integer and will find all +integers in an expression:

    +
    +
    +
    +

    from sympy import S +(1 + x + 2sin(y + Ipi)).atoms(S(1)) +{1}

    +

    (1 + x + 2sin(y + Ipi)).atoms(S(2)) +{1, 2}

    +
    +
    +
    +

    Finally, arguments to atoms() can select more than atomic atoms: any +SymPy type (loaded in core/init.py) can be listed as an argument +and those types of "atoms" as found in scanning the arguments of the +expression recursively:

    +
    +
    +
    +

    from sympy import Function, Mul +from sympy.core.function import AppliedUndef +f = Function('f') +(1 + f(x) + 2sin(y + Ipi)).atoms(Function) +{f(x), sin(y + Ipi)} +(1 + f(x) + 2sin(y + I*pi)).atoms(AppliedUndef) +{f(x)}

    +

    (1 + x + 2sin(y + Ipi)).atoms(Mul) +{Ipi, 2sin(y + I*pi)}

    +
    +
    +
    +

    cancel

    +
    def cancel(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the cancel function in sympy.polys

    +

    coeff

    +
    def coeff(
+    self,
+    x,
+    n=1,
+    right=False,
+    _first=True
+)
+
    +

    Returns the coefficient from the term(s) containing x**n. If n

    +

    is zero then all terms independent of x will be returned.

    +

    Explanation

    +

    When x is noncommutative, the coefficient to the left (default) or +right of x can be returned. The keyword 'right' is ignored when +x is commutative.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols +from sympy.abc import x, y, z

    +
    +
    +
    +

    You can select terms that have an explicit negative in front of them:

    +
    +
    +
    +

    (-x + 2y).coeff(-1) +x +(x - 2y).coeff(-1) +2*y

    +
    +
    +
    +

    You can select terms with no Rational coefficient:

    +
    +
    +
    +

    (x + 2y).coeff(1) +x +(3 + 2x + 4x*2).coeff(1) +0

    +
    +
    +
    +

    You can select terms independent of x by making n=0; in this case +expr.as_independent(x)[0] is returned (and 0 will be returned instead +of None):

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x, 0) +3 +eq = ((x + 1)3).expand() + 1 +eq +x3 + 3*x2 + 3*x + 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 2] +eq -= 2 +[eq.coeff(x, i) for i in reversed(range(4))] +[1, 3, 3, 0]

    +
    +
    +
    +

    You can select terms that have a numerical term in front of them:

    +
    +
    +
    +

    (-x - 2y).coeff(2) +-y +from sympy import sqrt +(x + sqrt(2)x).coeff(sqrt(2)) +x

    +
    +
    +
    +

    The matching is exact:

    +
    +
    +
    +

    (3 + 2x + 4x2).coeff(x) +2 +(3 + 2x + 4x2).coeff(x2) +4 +(3 + 2x + 4x2).coeff(x3) +0 +(z*(x + y)2).coeff((x + y)2) +z +(z*(x + y)2).coeff(x + y) +0

    +
    +
    +
    +

    In addition, no factoring is done, so 1 + z*(1 + y) is not obtained +from the following:

    +
    +
    +
    +

    (x + z(x + xy)).coeff(x) +1

    +
    +
    +
    +

    If such factoring is desired, factor_terms can be used first:

    +
    +
    +
    +

    from sympy import factor_terms +factor_terms(x + z(x + xy)).coeff(x) +z*(y + 1) + 1

    +

    n, m, o = symbols('n m o', commutative=False) +n.coeff(n) +1 +(3n).coeff(n) +3 +(nm + mnm).coeff(n) # = (1 + m)nm +1 + m +(nm + mnm).coeff(n, right=True) # = (1 + m)n*m +m

    +
    +
    +
    +

    If there is more than one possible coefficient 0 is returned:

    +
    +
    +
    +

    (nm + mn).coeff(n) +0

    +
    +
    +
    +

    If there is only one possible coefficient, it is returned:

    +
    +
    +
    +

    (nm + xmn).coeff(mn) +x +(nm + xmn).coeff(mn, right=1) +1

    +
    +
    +
    +

    See Also

    +

    as_coefficient: separate the expression into a coefficient and factor +as_coeff_Add: separate the additive constant from an expression +as_coeff_Mul: separate the multiplicative constant from an expression +as_independent: separate x-dependent terms/factors from others +sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly +sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

    +

    collect

    +
    def collect(
+    self,
+    syms,
+    func=None,
+    evaluate=True,
+    exact=False,
+    distribute_order_term=True
+)
+
    +

    See the collect function in sympy.simplify

    +

    combsimp

    +
    def combsimp(
+    self
+)
+
    +

    See the combsimp function in sympy.simplify

    +

    compare

    +
    def compare(
+    self,
+    other
+)
+
    +

    Return -1, 0, 1 if the object is less than, equal,

    +

    or greater than other in a canonical sense. +Non-Basic are always greater than Basic. +If both names of the classes being compared appear +in the ordering_of_classes then the ordering will +depend on the appearance of the names there. +If either does not appear in that list, then the +comparison is based on the class name. +If the names are the same then a comparison is made +on the length of the hashable content. +Items of the equal-lengthed contents are then +successively compared using the same rules. If there +is never a difference then 0 is returned.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +x.compare(y) +-1 +x.compare(x) +0 +y.compare(x) +1

    +
    +
    +
    +

    compute_leading_term

    +
    def compute_leading_term(
+    self,
+    x,
+    logx=None
+)
+
    +

    Deprecated function to compute the leading term of a series.

    +

    as_leading_term is only allowed for results of .series() +This is a wrapper to compute a series first.

    +

    conjugate

    +
    def conjugate(
+    self
+)
+
    +

    Returns the complex conjugate of 'self'.

    +

    copy

    +
    def copy(
+    self
+)
+
    +

    could_extract_minus_sign

    +
    def could_extract_minus_sign(
+    self
+)
+
    +

    Return True if self has -1 as a leading factor or has

    +

    more literal negative signs than positive signs in a sum, +otherwise False.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = x - y +{i.could_extract_minus_sign() for i in (e, -e)} +{False, True}

    +
    +
    +
    +

    Though the y - x is considered like -(x - y), since it +is in a product without a leading factor of -1, the result is +false below:

    +
    +
    +
    +

    (x*(y - x)).could_extract_minus_sign() +False

    +
    +
    +
    +

    To put something in canonical form wrt to sign, use signsimp:

    +
    +
    +
    +

    from sympy import signsimp +signsimp(x(y - x)) +-x(x - y) +_.could_extract_minus_sign() +True

    +
    +
    +
    +

    count

    +
    def count(
+    self,
+    query
+)
+
    +

    Count the number of matching subexpressions.

    +

    count_ops

    +
    def count_ops(
+    self,
+    visual=None
+)
+
    +

    Wrapper for count_ops that returns the operation count.

    +

    diff

    +
    def diff(
+    self,
+    *symbols,
+    **assumptions
+)
+
    +

    dir

    +
    def dir(
+    self,
+    x,
+    cdir
+)
+
    +

    doit

    +
    def doit(
+    self,
+    **hints
+)
+
    +

    Evaluate objects that are not evaluated by default like limits,

    +

    integrals, sums and products. All objects of this kind will be +evaluated recursively, unless some species were excluded via 'hints' +or unless the 'deep' hint was set to 'False'.

    +
    +
    +
    +

    from sympy import Integral +from sympy.abc import x

    +

    2Integral(x, x) +2Integral(x, x)

    +

    (2Integral(x, x)).doit() +x*2

    +

    (2Integral(x, x)).doit(deep=False) +2Integral(x, x)

    +
    +
    +
    +

    dummy_eq

    +
    def dummy_eq(
+    self,
+    other,
+    symbol=None
+)
+
    +

    Compare two expressions and handle dummy symbols.

    +

    Examples

    +
    +
    +
    +

    from sympy import Dummy +from sympy.abc import x, y

    +

    u = Dummy('u')

    +

    (u2 + 1).dummy_eq(x2 + 1) +True +(u2 + 1) == (x2 + 1) +False

    +

    (u2 + y).dummy_eq(x2 + y, x) +True +(u2 + y).dummy_eq(x2 + y, y) +False

    +
    +
    +
    +

    equals

    +
    def equals(
+    self,
+    other,
+    failing_expression=False
+)
+
    +

    Return True if self == other, False if it does not, or None. If

    +

    failing_expression is True then the expression which did not simplify +to a 0 will be returned instead of None.

    +

    Explanation

    +

    If self is a Number (or complex number) that is not zero, then +the result is False.

    +

    If self is a number and has not evaluated to zero, evalf will be +used to test whether the expression evaluates to zero. If it does so +and the result has significance (i.e. the precision is either -1, for +a Rational result, or is greater than 1) then the evalf value will be +used to return True or False.

    +

    evalf

    +
    def evalf(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    expand

    +
    def expand(
+    self,
+    deep=True,
+    modulus=None,
+    power_base=True,
+    power_exp=True,
+    mul=True,
+    log=True,
+    multinomial=True,
+    basic=True,
+    **hints
+)
+
    +

    Expand an expression using hints.

    +

    See the docstring of the expand() function in sympy.core.function for +more information.

    +

    extract_additively

    +
    def extract_additively(
+    self,
+    c
+)
+
    +

    Return self - c if it's possible to subtract c from self and

    +

    make all matching coefficients move towards zero, else return None.

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x, y +e = 2x + 3 +e.extract_additively(x + 1) +x + 2 +e.extract_additively(3x) +e.extract_additively(4) +(y(x + 1)).extract_additively(x + 1) +((x + 1)(x + 2y + 1) + 3).extract_additively(x + 1) +(x + 1)(x + 2*y) + 3

    +
    +
    +
    +

    See Also

    +

    extract_multiplicatively +coeff +as_coefficient

    +

    extract_branch_factor

    +
    def extract_branch_factor(
+    self,
+    allow_half=False
+)
+
    +

    Try to write self as exp_polar(2*pi*I*n)*z in a nice way.

    +

    Return (z, n).

    +
    +
    +
    +

    from sympy import exp_polar, I, pi +from sympy.abc import x, y +exp_polar(Ipi).extract_branch_factor() +(exp_polar(Ipi), 0) +exp_polar(2Ipi).extract_branch_factor() +(1, 1) +exp_polar(-piI).extract_branch_factor() +(exp_polar(Ipi), -1) +exp_polar(3piI + x).extract_branch_factor() +(exp_polar(x + Ipi), 1) +(yexp_polar(-5piI)exp_polar(3piI + 2pix)).extract_branch_factor() +(yexp_polar(2pix), -1) +exp_polar(-Ipi/2).extract_branch_factor() +(exp_polar(-Ipi/2), 0)

    +
    +
    +
    +

    If allow_half is True, also extract exp_polar(I*pi):

    +
    +
    +
    +

    exp_polar(Ipi).extract_branch_factor(allow_half=True) +(1, 1/2) +exp_polar(2Ipi).extract_branch_factor(allow_half=True) +(1, 1) +exp_polar(3Ipi).extract_branch_factor(allow_half=True) +(1, 3/2) +exp_polar(-Ipi).extract_branch_factor(allow_half=True) +(1, -1/2)

    +
    +
    +
    +

    extract_multiplicatively

    +
    def extract_multiplicatively(
+    self,
+    c
+)
+
    +

    Return None if it's not possible to make self in the form

    +

    c * something in a nice way, i.e. preserving the properties +of arguments of self.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Rational

    +

    x, y = symbols('x,y', real=True)

    +

    ((xy)3).extract_multiplicatively(x2 * y) +xy**2

    +

    ((xy)3).extract_multiplicatively(x*4 * y)

    +

    (2*x).extract_multiplicatively(2) +x

    +

    (2*x).extract_multiplicatively(3)

    +

    (Rational(1, 2)*x).extract_multiplicatively(3) +x/6

    +
    +
    +
    +

    factor

    +
    def factor(
+    self,
+    *gens,
+    **args
+)
+
    +

    See the factor() function in sympy.polys.polytools

    +

    find

    +
    def find(
+    self,
+    query,
+    group=False
+)
+
    +

    Find all subexpressions matching a query.

    +

    find_field

    +
    def find_field(
+    self: 'FieldableProtocol',
+    field: 'str | int',
+    fallback_type: 'type | None' = None,
+    **init_kwargs: 'Any'
+) -> 'Reference[Any]'
+
    +

    Field within the reference, if possible.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fieldNonethe field to search for.None
    fallback_typeNonethe type to use if we do not know a more specific one.None
    **init_kwargsNonearguments to use for constructing a new reference (via __make_reference__).None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA field-specific version of this reference.
    +

    fourier_series

    +
    def fourier_series(
+    self,
+    limits=None
+)
+
    +

    Compute fourier sine/cosine series of self.

    +

    See the docstring of the :func:fourier_series in sympy.series.fourier +for more information.

    +

    fps

    +
    def fps(
+    self,
+    x=None,
+    x0=0,
+    dir=1,
+    hyper=True,
+    order=4,
+    rational=True,
+    full=False
+)
+
    +

    Compute formal power power series of self.

    +

    See the docstring of the :func:fps function in sympy.series.formal for +more information.

    +

    gammasimp

    +
    def gammasimp(
+    self
+)
+
    +

    See the gammasimp function in sympy.simplify

    +

    getO

    +
    def getO(
+    self
+)
+
    +

    Returns the additive O(..) symbol if there is one, else None.

    +

    getn

    +
    def getn(
+    self
+)
+
    +

    Returns the order of the expression.

    +

    Explanation

    +

    The order is determined either from the O(...) term. If there +is no O(...) term, it returns None.

    +

    Examples

    +
    +
    +
    +

    from sympy import O +from sympy.abc import x +(1 + x + O(x**2)).getn() +2 +(1 + x).getn()

    +
    +
    +
    +

    has

    +
    def has(
+    self,
+    *patterns
+)
+
    +

    Test whether any subexpression matches any of the patterns.

    +

    Examples

    +
    +
    +
    +

    from sympy import sin +from sympy.abc import x, y, z +(x2 + sin(x*y)).has(z) +False +(x2 + sin(x*y)).has(x, y, z) +True +x.has(x) +True

    +
    +
    +
    +

    Note has is a structural algorithm with no knowledge of +mathematics. Consider the following half-open interval:

    +
    +
    +
    +

    from sympy import Interval +i = Interval.Lopen(0, 5); i +Interval.Lopen(0, 5) +i.args +(0, 5, True, False) +i.has(4) # there is no "4" in the arguments +False +i.has(0) # there is a "0" in the arguments +True

    +
    +
    +
    +

    Instead, use contains to determine whether a number is in the +interval or not:

    +
    +
    +
    +

    i.contains(4) +True +i.contains(0) +False

    +
    +
    +
    +

    Note that expr.has(*patterns) is exactly equivalent to +any(expr.has(p) for p in patterns). In particular, False is +returned when the list of patterns is empty.

    +
    +
    +
    +

    x.has() +False

    +
    +
    +
    +

    has_free

    +
    def has_free(
+    self,
+    *patterns
+)
+
    +

    Return True if self has object(s) x as a free expression

    +

    else False.

    +

    Examples

    +
    +
    +
    +

    from sympy import Integral, Function +from sympy.abc import x, y +f = Function('f') +g = Function('g') +expr = Integral(f(x), (f(x), 1, g(y))) +expr.free_symbols +{y} +expr.has_free(g(y)) +True +expr.has_free(*(x, f(x))) +False

    +
    +
    +
    +

    This works for subexpressions and types, too:

    +
    +
    +
    +

    expr.has_free(g) +True +(x + y + 1).has_free(y + 1) +True

    +
    +
    +
    +

    has_xfree

    +
    def has_xfree(
+    self,
+    s: 'set[Basic]'
+)
+
    +

    Return True if self has any of the patterns in s as a

    +

    free argument, else False. This is like Basic.has_free +but this will only report exact argument matches.

    +

    Examples

    +
    +
    +
    +

    from sympy import Function +from sympy.abc import x, y +f = Function('f') +f(x).has_xfree({f}) +False +f(x).has_xfree({f(x)}) +True +f(x + 1).has_xfree({x}) +True +f(x + 1).has_xfree({x + 1}) +True +f(x + y + 1).has_xfree({x + 1}) +False

    +
    +
    +
    +

    integrate

    +
    def integrate(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the integrate function in sympy.integrals

    +

    invert

    +
    def invert(
+    self,
+    g,
+    *gens,
+    **args
+)
+
    +

    Return the multiplicative inverse of self mod g

    +

    where self (and g) may be symbolic expressions).

    +

    See Also

    +

    sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert

    +

    is_algebraic_expr

    +
    def is_algebraic_expr(
+    self,
+    *syms
+)
+
    +

    This tests whether a given expression is algebraic or not, in the

    +

    given symbols, syms. When syms is not given, all free symbols +will be used. The rational function does not have to be in expanded +or in any kind of canonical form.

    +

    This function returns False for expressions that are "algebraic +expressions" with symbolic exponents. This is a simple extension to the +is_rational_function, including rational exponentiation.

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sqrt +x = Symbol('x', real=True) +sqrt(1 + x).is_rational_function() +False +sqrt(1 + x).is_algebraic_expr() +True

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be an algebraic +expression to become one.

    +
    +
    +
    +

    from sympy import exp, factor +a = sqrt(exp(x)*2 + 2exp(x) + 1)/(exp(x) + 1) +a.is_algebraic_expr(x) +False +factor(a).is_algebraic_expr() +True

    +
    +
    +
    +

    See Also

    +

    is_rational_function

    +

    References

    +

    .. [1] https://en.wikipedia.org/wiki/Algebraic_expression

    +

    is_constant

    +
    def is_constant(
+    self,
+    *wrt,
+    **flags
+)
+
    +

    Return True if self is constant, False if not, or None if

    +

    the constancy could not be determined conclusively.

    +

    Explanation

    +

    If an expression has no free symbols then it is a constant. If +there are free symbols it is possible that the expression is a +constant, perhaps (but not necessarily) zero. To test such +expressions, a few strategies are tried:

    +

    1) numerical evaluation at two random points. If two such evaluations +give two different values and the values have a precision greater than +1 then self is not constant. If the evaluations agree or could not be +obtained with any precision, no decision is made. The numerical testing +is done only if wrt is different than the free symbols.

    +

    2) differentiation with respect to variables in 'wrt' (or all free +symbols if omitted) to see if the expression is constant or not. This +will not always lead to an expression that is zero even though an +expression is constant (see added test in test_expr.py). If +all derivatives are zero then self is constant with respect to the +given symbols.

    +

    3) finding out zeros of denominator expression with free_symbols. +It will not be constant if there are zeros. It gives more negative +answers for expression that are not constant.

    +

    If neither evaluation nor differentiation can prove the expression is +constant, None is returned unless two numerical values happened to be +the same and the flag failing_number is True -- in that case the +numerical value will be returned.

    +

    If flag simplify=False is passed, self will not be simplified; +the default is True since self should be simplified before testing.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, sin, Sum, S, pi +from sympy.abc import a, n, x, y +x.is_constant() +False +S(2).is_constant() +True +Sum(x, (x, 1, 10)).is_constant() +True +Sum(x, (x, 1, n)).is_constant() +False +Sum(x, (x, 1, n)).is_constant(y) +True +Sum(x, (x, 1, n)).is_constant(n) +False +Sum(x, (x, 1, n)).is_constant(x) +True +eq = acos(x)2 + asin(x)**2 - a +eq.is_constant() +True +eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 +True

    +

    (0x).is_constant() +False +x.is_constant() +False +(xx).is_constant() +False +one = cos(x)2 + sin(x)2 +one.is_constant() +True +((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 +True

    +
    +
    +
    +

    is_hypergeometric

    +
    def is_hypergeometric(
+    self,
+    k
+)
+
    +

    is_meromorphic

    +
    def is_meromorphic(
+    self,
+    x,
+    a
+)
+
    +

    This tests whether an expression is meromorphic as

    +

    a function of the given symbol x at the point a.

    +

    This method is intended as a quick test that will return +None if no decision can be made without simplification or +more detailed analysis.

    +

    Examples

    +
    +
    +
    +

    from sympy import zoo, log, sin, sqrt +from sympy.abc import x

    +

    f = 1/x2 + 1 - 2*x3 +f.is_meromorphic(x, 0) +True +f.is_meromorphic(x, 1) +True +f.is_meromorphic(x, zoo) +True

    +

    g = x**log(3) +g.is_meromorphic(x, 0) +False +g.is_meromorphic(x, 1) +True +g.is_meromorphic(x, zoo) +False

    +

    h = sin(1/x)x*2 +h.is_meromorphic(x, 0) +False +h.is_meromorphic(x, 1) +True +h.is_meromorphic(x, zoo) +True

    +
    +
    +
    +

    Multivalued functions are considered meromorphic when their +branches are meromorphic. Thus most functions are meromorphic +everywhere except at essential singularities and branch points. +In particular, they will be meromorphic also on branch cuts +except at their endpoints.

    +
    +
    +
    +

    log(x).is_meromorphic(x, -1) +True +log(x).is_meromorphic(x, 0) +False +sqrt(x).is_meromorphic(x, -1) +True +sqrt(x).is_meromorphic(x, 0) +False

    +
    +
    +
    +

    is_polynomial

    +
    def is_polynomial(
+    self,
+    *syms
+)
+
    +

    Return True if self is a polynomial in syms and False otherwise.

    +

    This checks if self is an exact polynomial in syms. This function +returns False for expressions that are "polynomials" with symbolic +exponents. Thus, you should be able to apply polynomial algorithms to +expressions for which this returns True, and Poly(expr, *syms) should +work if and only if expr.is_polynomial(*syms) returns True. The +polynomial does not have to be in expanded form. If no symbols are +given, all free symbols in the expression will be used.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', polynomial=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, Function +x = Symbol('x') +((x2 + 1)4).is_polynomial(x) +True +((x2 + 1)4).is_polynomial() +True +(2x + 1).is_polynomial(x) +False +(2x + 1).is_polynomial(2**x) +True +f = Function('f') +(f(x) + 1).is_polynomial(x) +False +(f(x) + 1).is_polynomial(f(x)) +True +(1/f(x) + 1).is_polynomial(f(x)) +False

    +

    n = Symbol('n', nonnegative=True, integer=True) +(x**n + 1).is_polynomial(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a polynomial to +become one.

    +
    +
    +
    +

    from sympy import sqrt, factor, cancel +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1) +a.is_polynomial(y) +False +factor(a) +y + 1 +factor(a).is_polynomial(y) +True

    +

    b = (y*2 + 2y + 1)/(y + 1) +b.is_polynomial(y) +False +cancel(b) +y + 1 +cancel(b).is_polynomial(y) +True

    +
    +
    +
    +

    See also .is_rational_function()

    +

    is_rational_function

    +
    def is_rational_function(
+    self,
+    *syms
+)
+
    +

    Test whether function is a ratio of two polynomials in the given

    +

    symbols, syms. When syms is not given, all free symbols will be used. +The rational function does not have to be in expanded or in any kind of +canonical form.

    +

    This function returns False for expressions that are "rational +functions" with symbolic exponents. Thus, you should be able to call +.as_numer_denom() and apply polynomial algorithms to the result for +expressions for which this returns True.

    +

    This is not part of the assumptions system. You cannot do +Symbol('z', rational_function=True).

    +

    Examples

    +
    +
    +
    +

    from sympy import Symbol, sin +from sympy.abc import x, y

    +

    (x/y).is_rational_function() +True

    +

    (x**2).is_rational_function() +True

    +

    (x/sin(y)).is_rational_function(y) +False

    +

    n = Symbol('n', integer=True) +(x**n + 1).is_rational_function(x) +False

    +
    +
    +
    +

    This function does not attempt any nontrivial simplifications that may +result in an expression that does not appear to be a rational function +to become one.

    +
    +
    +
    +

    from sympy import sqrt, factor +y = Symbol('y', positive=True) +a = sqrt(y*2 + 2y + 1)/y +a.is_rational_function(y) +False +factor(a) +(y + 1)/y +factor(a).is_rational_function(y) +True

    +
    +
    +
    +

    See also is_algebraic_expr().

    +

    is_same

    +
    def is_same(
+    a,
+    b,
+    approx=None
+)
+
    +

    Return True if a and b are structurally the same, else False.

    +

    If approx is supplied, it will be used to test whether two +numbers are the same or not. By default, only numbers of the +same type will compare equal, so S.Half != Float(0.5).

    +

    Examples

    +

    In SymPy (unlike Python) two numbers do not compare the same if they are +not of the same type:

    +
    +
    +
    +

    from sympy import S +2.0 == S(2) +False +0.5 == S.Half +False

    +
    +
    +
    +

    By supplying a function with which to compare two numbers, such +differences can be ignored. e.g. equal_valued will return True +for decimal numbers having a denominator that is a power of 2, +regardless of precision.

    +
    +
    +
    +

    from sympy import Float +from sympy.core.numbers import equal_valued +(S.Half/4).is_same(Float(0.125, 1), equal_valued) +True +Float(1, 2).is_same(Float(1, 10), equal_valued) +True

    +
    +
    +
    +

    But decimals without a power of 2 denominator will compare +as not being the same.

    +
    +
    +
    +

    Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) +False

    +
    +
    +
    +

    But arbitrary differences can be ignored by supplying a function +to test the equivalence of two numbers:

    +
    +
    +
    +

    import math +Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) +True

    +
    +
    +
    +

    Other objects might compare the same even though types are not the +same. This routine will only return True if two expressions are +identical in terms of class types.

    +
    +
    +
    +

    from sympy import eye, Basic +eye(1) == S(eye(1)) # mutable vs immutable +True +Basic.is_same(eye(1), S(eye(1))) +False

    +
    +
    +
    +

    leadterm

    +
    def leadterm(
+    self,
+    x,
+    logx=None,
+    cdir=0
+)
+
    +

    Returns the leading term ax*b as a tuple (a, b).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(1+x+x2).leadterm(x) +(1, 0) +(1/x2+x+x**2).leadterm(x) +(1, -2)

    +
    +
    +
    +

    limit

    +
    def limit(
+    self,
+    x,
+    xlim,
+    dir='+'
+)
+
    +

    Compute limit x->xlim.

    +

    lseries

    +
    def lseries(
+    self,
+    x=None,
+    x0=0,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper for series yielding an iterator of the terms of the series.

    +

    Note: an infinite series will yield an infinite iterator. The following, +for exaxmple, will never terminate. It will just keep printing terms +of the sin(x) series::

    +

    for term in sin(x).lseries(x): + print term

    +

    The advantage of lseries() over nseries() is that many times you are +just interested in the next term in the series (i.e. the first term for +example), but you do not know how many you should ask for in nseries() +using the "n" parameter.

    +

    See also nseries().

    +

    match

    +
    def match(
+    self,
+    pattern,
+    old=False
+)
+
    +

    Pattern matching.

    +

    Wild symbols match all.

    +

    Return None when expression (self) does not match +with pattern. Otherwise return a dictionary such that::

    +

    pattern.xreplace(self.match(pattern)) == self

    +

    Examples

    +
    +
    +
    +

    from sympy import Wild, Sum +from sympy.abc import x, y +p = Wild("p") +q = Wild("q") +r = Wild("r") +e = (x+y)(x+y) +e.match(pp) +{p_: x + y} +e.match(pq) +{p_: x + y, q_: x + y} +e = (2*x)2 +e.match(pqr) +{p_: 4, q_: x, r_: 2} +(pqr).xreplace(e.match(p*qr)) +4x*2

    +
    +
    +
    +

    Structurally bound symbols are ignored during matching:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) +{p_: 2}

    +
    +
    +
    +

    But they can be identified if desired:

    +
    +
    +
    +

    Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) +{p_: 2, q_: x}

    +
    +
    +
    +

    The old flag will give the old-style pattern matching where +expressions and patterns are essentially solved to give the +match. Both of the following give None unless old=True:

    +
    +
    +
    +

    (x - 2).match(p - x, old=True) +{p_: 2x - 2} +(2/x).match(px, old=True) +{p_: 2/x**2}

    +
    +
    +
    +

    matches

    +
    def matches(
+    self,
+    expr,
+    repl_dict=None,
+    old=False
+)
+
    +

    Helper method for match() that looks for a match between Wild symbols

    +

    in self and expressions in expr.

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, Wild, Basic +a, b, c = symbols('a b c') +x = Wild('x') +Basic(a + x, x).matches(Basic(a + b, c)) is None +True +Basic(a + x, x).matches(Basic(a + b + c, b + c)) +{x_: b + c}

    +
    +
    +
    +

    n

    +
    def n(
+    self,
+    n=15,
+    subs=None,
+    maxn=100,
+    chop=False,
+    strict=False,
+    quad=None,
+    verbose=False
+)
+
    +

    Evaluate the given formula to an accuracy of n digits.

    +

    Parameters

    +

    subs : dict, optional + Substitute numerical values for symbols, e.g. + subs={x:3, y:1+pi}. The substitutions must be given as a + dictionary.

    +

    maxn : int, optional + Allow a maximum temporary working precision of maxn digits.

    +

    chop : bool or number, optional + Specifies how to replace tiny real or imaginary parts in + subresults by exact zeros.

    +
    When ``True`` the chop value defaults to standard precision.
+
+Otherwise the chop value is used to determine the
+magnitude of "small" for purposes of chopping.
+
+>>> from sympy import N
+>>> x = 1e-4
+>>> N(x, chop=True)
+0.000100000000000000
+>>> N(x, chop=1e-5)
+0.000100000000000000
+>>> N(x, chop=1e-4)
+0
+
    +

    strict : bool, optional + Raise PrecisionExhausted if any subresult fails to + evaluate to full accuracy, given the available maxprec.

    +

    quad : str, optional + Choose algorithm for numerical quadrature. By default, + tanh-sinh quadrature is used. For oscillatory + integrals on an infinite interval, try quad='osc'.

    +

    verbose : bool, optional + Print debug information.

    +

    Notes

    +

    When Floats are naively substituted into an expression, +precision errors may adversely affect the result. For example, +adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is +then subtracted, the result will be 0. +That is exactly what happens in the following:

    +
    +
    +
    +

    from sympy.abc import x, y, z +values = {x: 1e16, y: 1, z: 1e16} +(x + y - z).subs(values) +0

    +
    +
    +
    +

    Using the subs argument for evalf is the accurate way to +evaluate such an expression:

    +
    +
    +
    +

    (x + y - z).evalf(subs=values) +1.00000000000000

    +
    +
    +
    +

    normal

    +
    def normal(
+    self
+)
+
    +

    Return the expression as a fraction.

    +

    expression -> a/b

    +

    See Also

    +

    as_numer_denom: return (a, b) instead of a/b

    +

    nseries

    +
    def nseries(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Wrapper to _eval_nseries if assumptions allow, else to series.

    +

    If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is +called. This calculates "n" terms in the innermost expressions and +then builds up the final series just by "cross-multiplying" everything +out.

    +

    The optional logx parameter can be used to replace any log(x) in the +returned series with a symbolic value to avoid evaluating log(x) at 0. A +symbol to use in place of log(x) should be provided.

    +

    Advantage -- it's fast, because we do not have to determine how many +terms we need to calculate in advance.

    +

    Disadvantage -- you may end up with less terms than you may have +expected, but the O(x**n) term appended will always be correct and +so the result, though perhaps shorter, will also be correct.

    +

    If any of those assumptions is not met, this is treated like a +wrapper to series which will try harder to return the correct +number of terms.

    +

    See also lseries().

    +

    Examples

    +
    +
    +
    +

    from sympy import sin, log, Symbol +from sympy.abc import x, y +sin(x).nseries(x, 0, 6) +x - x3/6 + x5/120 + O(x6) +log(x+1).nseries(x, 0, 5) +x - x2/2 + x3/3 - x4/4 + O(x**5)

    +
    +
    +
    +

    Handling of the logx parameter --- in the following example the +expansion fails since sin does not have an asymptotic expansion +at -oo (the limit of log(x) as x approaches 0):

    +
    +
    +
    +

    e = sin(log(x)) +e.nseries(x, 0, 6) +Traceback (most recent call last): +... +PoleError: ... +... +logx = Symbol('logx') +e.nseries(x, 0, 6, logx=logx) +sin(logx)

    +
    +
    +
    +

    In the following example, the expansion works but only returns self +unless the logx parameter is used:

    +
    +
    +
    +

    e = xy +e.nseries(x, 0, 2) +xy +e.nseries(x, 0, 2, logx=logx) +exp(logx*y)

    +
    +
    +
    +

    nsimplify

    +
    def nsimplify(
+    self,
+    constants=(),
+    tolerance=None,
+    full=False
+)
+
    +

    See the nsimplify function in sympy.simplify

    +

    powsimp

    +
    def powsimp(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the powsimp function in sympy.simplify

    +

    primitive

    +
    def primitive(
+    self
+)
+
    +

    Return the positive Rational that can be extracted non-recursively

    +

    from every term of self (i.e., self is treated like an Add). This is +like the as_coeff_Mul() method but primitive always extracts a positive +Rational (never a negative or a Float).

    +

    Examples

    +
    +
    +
    +

    from sympy.abc import x +(3(x + 1)2).primitive() +(3, (x + 1)2) +a = (6x + 2); a.primitive() +(2, 3x + 1) +b = (x/2 + 3); b.primitive() +(1/2, x + 6) +(ab).primitive() == (1, a*b) +True

    +
    +
    +
    +

    radsimp

    +
    def radsimp(
+    self,
+    **kwargs
+)
+
    +

    See the radsimp function in sympy.simplify

    +

    ratsimp

    +
    def ratsimp(
+    self
+)
+
    +

    See the ratsimp function in sympy.simplify

    +

    rcall

    +
    def rcall(
+    self,
+    *args
+)
+
    +

    Apply on the argument recursively through the expression tree.

    +

    This method is used to simulate a common abuse of notation for +operators. For instance, in SymPy the following will not work:

    +

    (x+Lambda(y, 2*y))(z) == x+2*z,

    +

    however, you can use:

    +
    +
    +
    +

    from sympy import Lambda +from sympy.abc import x, y, z +(x + Lambda(y, 2y)).rcall(z) +x + 2z

    +
    +
    +
    +

    refine

    +
    def refine(
+    self,
+    assumption=True
+)
+
    +

    See the refine function in sympy.assumptions

    +

    removeO

    +
    def removeO(
+    self
+)
+
    +

    Removes the additive O(..) symbol if there is one

    +

    replace

    +
    def replace(
+    self,
+    query,
+    value,
+    map=False,
+    simultaneous=True,
+    exact=None
+)
+
    +

    Replace matching subexpressions of self with value.

    +

    If map = True then also return the mapping {old: new} where old +was a sub-expression found with query and new is the replacement +value for it. If the expression itself does not match the query, then +the returned value will be self.xreplace(map) otherwise it should +be self.subs(ordered(map.items())).

    +

    Traverses an expression tree and performs replacement of matching +subexpressions from the bottom to the top of the tree. The default +approach is to do the replacement in a simultaneous fashion so +changes made are targeted only once. If this is not desired or causes +problems, simultaneous can be set to False.

    +

    In addition, if an expression containing more than one Wild symbol +is being used to match subexpressions and the exact flag is None +it will be set to True so the match will only succeed if all non-zero +values are received for each Wild that appears in the match pattern. +Setting this to False accepts a match of 0; while setting it True +accepts all matches that have a 0 in them. See example below for +cautions.

    +

    The list of possible combinations of queries and replacement values +is listed below:

    +

    Examples

    +

    Initial setup

    +
    +
    +
    +

    from sympy import log, sin, cos, tan, Wild, Mul, Add +from sympy.abc import x, y +f = log(sin(x)) + tan(sin(x**2))

    +
    +
    +
    +

    1.1. type -> type + obj.replace(type, newtype)

    +
    When object of type ``type`` is found, replace it with the
+result of passing its argument(s) to ``newtype``.
+
+>>> f.replace(sin, cos)
+log(cos(x)) + tan(cos(x**2))
+>>> sin(x).replace(sin, cos, map=True)
+(cos(x), {sin(x): cos(x)})
+>>> (x*y).replace(Mul, Add)
+x + y
+
    +

    1.2. type -> func + obj.replace(type, func)

    +
    When object of type ``type`` is found, apply ``func`` to its
+argument(s). ``func`` must be written to handle the number
+of arguments of ``type``.
+
+>>> f.replace(sin, lambda arg: sin(2*arg))
+log(sin(2*x)) + tan(sin(2*x**2))
+>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
+sin(2*x*y)
+
    +

    2.1. pattern -> expr + obj.replace(pattern(wild), expr(wild))

    +
    Replace subexpressions matching ``pattern`` with the expression
+written in terms of the Wild symbols in ``pattern``.
+
+>>> a, b = map(Wild, 'ab')
+>>> f.replace(sin(a), tan(a))
+log(tan(x)) + tan(tan(x**2))
+>>> f.replace(sin(a), tan(a/2))
+log(tan(x/2)) + tan(tan(x**2/2))
+>>> f.replace(sin(a), a)
+log(x) + tan(x**2)
+>>> (x*y).replace(a*x, a)
+y
+
+Matching is exact by default when more than one Wild symbol
+is used: matching fails unless the match gives non-zero
+values for all Wild symbols:
+
+>>> (2*x + y).replace(a*x + b, b - a)
+y - 2
+>>> (2*x).replace(a*x + b, b - a)
+2*x
+
+When set to False, the results may be non-intuitive:
+
+>>> (2*x).replace(a*x + b, b - a, exact=False)
+2/x
+
    +

    2.2. pattern -> func + obj.replace(pattern(wild), lambda wild: expr(wild))

    +
    All behavior is the same as in 2.1 but now a function in terms of
+pattern variables is used rather than an expression:
+
+>>> f.replace(sin(a), lambda a: sin(2*a))
+log(sin(2*x)) + tan(sin(2*x**2))
+
    +

    3.1. func -> func + obj.replace(filter, func)

    +
    Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
+is True.
+
+>>> g = 2*sin(x**3)
+>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
+4*sin(x**9)
+
    +

    The expression itself is also targeted by the query but is done in +such a fashion that changes are not made twice.

    +
    >>> e = x*(x*y + 1)
+>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
+2*x*(2*x*y + 1)
+
    +

    When matching a single symbol, exact will default to True, but +this may or may not be the behavior that is desired:

    +

    Here, we want exact=False:

    +
    +
    +
    +

    from sympy import Function +f = Function('f') +e = f(1) + f(0) +q = f(a), lambda a: f(a + 1) +e.replace(q, exact=False) +f(1) + f(2) +e.replace(q, exact=True) +f(0) + f(2)

    +
    +
    +
    +

    But here, the nature of matching makes selecting +the right setting tricky:

    +
    +
    +
    +

    e = x(1 + y) +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=False) +x +(x(1 + y)).replace(x(1 + a), lambda a: x-a, exact=True) +x(-x - y + 1) +(xy).replace(x(1 + a), lambda a: x-a, exact=False) +x +(xy).replace(x(1 + a), lambda a: x-a, exact=True) +x**(1 - y)

    +
    +
    +
    +

    It is probably better to use a different form of the query +that describes the target expression more precisely:

    +
    +
    +
    +

    (1 + x(1 + y)).replace( +... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, +... lambda x: x.base(1 - (x.exp - 1))) +... +x**(1 - y) + 1

    +
    +
    +
    +

    See Also

    +

    subs: substitution of subexpressions as defined by the objects + themselves. +xreplace: exact node replacement in expr tree; also capable of + using matching rules

    +

    rewrite

    +
    def rewrite(
+    self,
+    *args,
+    deep=True,
+    **hints
+)
+
    +

    Rewrite self using a defined rule.

    +

    Rewriting transforms an expression to another, which is mathematically +equivalent but structurally different. For example you can rewrite +trigonometric functions as complex exponentials or combinatorial +functions as gamma function.

    +

    This method takes a pattern and a rule as positional arguments. +pattern is optional parameter which defines the types of expressions +that will be transformed. If it is not passed, all possible expressions +will be rewritten. rule defines how the expression will be rewritten.

    +

    Parameters

    +

    args : Expr + A rule, or pattern and rule. + - pattern is a type or an iterable of types. + - rule can be any object.

    +

    deep : bool, optional + If True, subexpressions are recursively transformed. Default is + True.

    +

    Examples

    +

    If pattern is unspecified, all possible expressions are transformed.

    +
    +
    +
    +

    from sympy import cos, sin, exp, I +from sympy.abc import x +expr = cos(x) + Isin(x) +expr.rewrite(exp) +exp(Ix)

    +
    +
    +
    +

    Pattern can be a type or an iterable of types.

    +
    +
    +
    +

    expr.rewrite(sin, exp) +exp(Ix)/2 + cos(x) - exp(-Ix)/2 +expr.rewrite([cos,], exp) +exp(Ix)/2 + Isin(x) + exp(-Ix)/2 +expr.rewrite([cos, sin], exp) +exp(Ix)

    +
    +
    +
    +

    Rewriting behavior can be implemented by defining _eval_rewrite() +method.

    +
    +
    +
    +

    from sympy import Expr, sqrt, pi +class MySin(Expr): +... def _eval_rewrite(self, rule, args, hints): +... x, = args +... if rule == cos: +... return cos(pi/2 - x, evaluate=False) +... if rule == sqrt: +... return sqrt(1 - cos(x)2) +MySin(MySin(x)).rewrite(cos) +cos(-cos(-x + pi/2) + pi/2) +MySin(x).rewrite(sqrt) +sqrt(1 - cos(x)**2)

    +
    +
    +
    +

    Defining _eval_rewrite_as_[...]() method is supported for backwards +compatibility reason. This may be removed in the future and using it is +discouraged.

    +
    +
    +
    +

    class MySin(Expr): +... def _eval_rewrite_as_cos(self, args, *hints): +... x, = args +... return cos(pi/2 - x, evaluate=False) +MySin(x).rewrite(cos) +cos(-x + pi/2)

    +
    +
    +
    +

    round

    +
    def round(
+    self,
+    n=None
+)
+
    +

    Return x rounded to the given decimal place.

    +

    If a complex number would results, apply round to the real +and imaginary components of the number.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, E, I, S, Number +pi.round() +3 +pi.round(2) +3.14 +(2pi + EI).round() +6 + 3*I

    +
    +
    +
    +

    The round method has a chopping effect:

    +
    +
    +
    +

    (2pi + I/10).round() +6 +(pi/10 + 2I).round() +2I +(pi/10 + EI).round(2) +0.31 + 2.72*I

    +
    +
    +
    +

    Notes

    +

    The Python round function uses the SymPy round method so it +will always return a SymPy number (not a Python float or int):

    +
    +
    +
    +

    isinstance(round(S(123), -2), Number) +True

    +
    +
    +
    +

    separate

    +
    def separate(
+    self,
+    deep=False,
+    force=False
+)
+
    +

    See the separate function in sympy.simplify

    +

    series

    +
    def series(
+    self,
+    x=None,
+    x0=0,
+    n=6,
+    dir='+',
+    logx=None,
+    cdir=0
+)
+
    +

    Series expansion of "self" around x = x0 yielding either terms of

    +

    the series one by one (the lazy series given when n=None), else +all the terms at once when n != None.

    +

    Returns the series expansion of "self" around the point x = x0 +with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

    +

    If x=None and self is univariate, the univariate symbol will +be supplied, otherwise an error will be raised.

    +

    Parameters

    +

    expr : Expression + The expression whose series is to be expanded.

    +

    x : Symbol + It is the variable of the expression to be calculated.

    +

    x0 : Value + The value around which x is calculated. Can be any value + from -oo to oo.

    +

    n : Value + The value used to represent the order in terms of x**n, + up to which the series is to be expanded.

    +

    dir : String, optional + The series-expansion can be bi-directional. If dir="+", + then (x->x0+). If dir="-", then (x->x0-). For infinitex0(ooor-oo), thedirargument is determined + from the direction of the infinity (i.e.,dir="-"foroo``).

    +

    logx : optional + It is used to replace any log(x) in the returned series with a + symbolic value rather than evaluating the actual value.

    +

    cdir : optional + It stands for complex direction, and indicates the direction + from which the expansion needs to be evaluated.

    +

    Examples

    +
    +
    +
    +

    from sympy import cos, exp, tan +from sympy.abc import x, y +cos(x).series() +1 - x2/2 + x4/24 + O(x6) +cos(x).series(n=4) +1 - x2/2 + O(x4) +cos(x).series(x, x0=1, n=2) +cos(1) - (x - 1)*sin(1) + O((x - 1)2, (x, 1)) +e = cos(x + exp(y)) +e.series(y, n=2) +cos(x + 1) - ysin(x + 1) + O(y2) +e.series(x, n=2) +cos(exp(y)) - xsin(exp(y)) + O(x**2)

    +
    +
    +
    +

    If n=None then a generator of the series terms will be returned.

    +
    +
    +
    +

    term=cos(x).series(n=None) +[next(term) for i in range(2)] +[1, -x**2/2]

    +
    +
    +
    +

    For dir=+ (default) the series is calculated from the right and +for dir=- the series from the left. For smooth functions this +flag will not alter the results.

    +
    +
    +
    +

    abs(x).series(dir="+") +x +abs(x).series(dir="-") +-x +f = tan(x) +f.series(x, 2, 6, "+") +tan(2) + (1 + tan(2)2)*(x - 2) + (x - 2)2(tan(2)3 + tan(2)) + +(x - 2)3(1/3 + 4tan(2)2/3 + tan(2)4) + (x - 2)4(tan(2)5 + +5*tan(2)3/3 + 2tan(2)/3) + (x - 2)5(2/15 + 17tan(2)2/15 + +2tan(2)4 + tan(2)6) + O((x - 2)**6, (x, 2))

    +

    f.series(x, 2, 3, "-") +tan(2) + (2 - x)(-tan(2)2 - 1) + (2 - x)2(tan(2)3 + tan(2)) ++ O((x - 2)3, (x, 2))

    +
    +
    +
    +

    For rational expressions this method may return original expression without the Order term.

    +
    +
    +
    +

    (1/x).series(x, n=8) +1/x

    +
    +
    +
    +

    Returns

    +

    Expr : Expression + Series expansion of the expression about x0

    +

    Raises

    +

    TypeError + If "n" and "x0" are infinity objects

    +

    PoleError + If "x0" is an infinity object

    +

    simplify

    +
    def simplify(
+    self,
+    **kwargs
+)
+
    +

    See the simplify function in sympy.simplify

    +

    sort_key

    +
    def sort_key(
+    self,
+    order=None
+)
+
    +

    Return a sort key.

    +

    Examples

    +
    +
    +
    +

    from sympy import S, I

    +

    sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) +[1/2, -I, I]

    +

    S("[x, 1/x, 1/x2, x2, x(1/2), x(1/4), x(3/2)]") +[x, 1/x, x(-2), x2, sqrt(x), x(1/4), x(3/2)] +sorted(_, key=lambda x: x.sort_key()) +[x(-2), 1/x, x(1/4), sqrt(x), x, x(3/2), x**2]

    +
    +
    +
    +

    subs

    +
    def subs(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    Substitutes old for new in an expression after sympifying args.

    +

    args is either: + - two arguments, e.g. foo.subs(old, new) + - one iterable argument, e.g. foo.subs(iterable). The iterable may be + o an iterable container with (old, new) pairs. In this case the + replacements are processed in the order given with successive + patterns possibly affecting replacements already made. + o a dict or set whose key/value items correspond to old/new pairs. + In this case the old/new pairs will be sorted by op count and in + case of a tie, by number of args and the default_sort_key. The + resulting sorted list is then processed as an iterable container + (see previous).

    +

    If the keyword simultaneous is True, the subexpressions will not be +evaluated until all the substitutions have been made.

    +

    Examples

    +
    +
    +
    +

    from sympy import pi, exp, limit, oo +from sympy.abc import x, y +(1 + xy).subs(x, pi) +piy + 1 +(1 + xy).subs({x:pi, y:2}) +1 + 2pi +(1 + xy).subs([(x, pi), (y, 2)]) +1 + 2pi +reps = [(y, x2), (x, 2)] +(x + y).subs(reps) +6 +(x + y).subs(reversed(reps)) +x2 + 2

    +

    (x2 + x4).subs(x2, y) +y2 + y

    +
    +
    +
    +

    To replace only the x2 but not the x4, use xreplace:

    +
    +
    +
    +

    (x2 + x4).xreplace({x2: y}) +x4 + y

    +
    +
    +
    +

    To delay evaluation until all substitutions have been made, +set the keyword simultaneous to True:

    +
    +
    +
    +

    (x/y).subs([(x, 0), (y, 0)]) +0 +(x/y).subs([(x, 0), (y, 0)], simultaneous=True) +nan

    +
    +
    +
    +

    This has the added feature of not allowing subsequent substitutions +to affect those already made:

    +
    +
    +
    +

    ((x + y)/y).subs({x + y: y, y: x + y}) +1 +((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) +y/(x + y)

    +
    +
    +
    +

    In order to obtain a canonical result, unordered iterables are +sorted by count_op length, number of arguments and by the +default_sort_key to break any ties. All other iterables are left +unsorted.

    +
    +
    +
    +

    from sympy import sqrt, sin, cos +from sympy.abc import a, b, c, d, e

    +

    A = (sqrt(sin(2x)), a) +B = (sin(2x), b) +C = (cos(2*x), c) +D = (x, d) +E = (exp(x), e)

    +

    expr = sqrt(sin(2x))sin(exp(x)x)cos(2x) + sin(2x)

    +

    expr.subs(dict([A, B, C, D, E])) +acsin(d*e) + b

    +
    +
    +
    +

    The resulting expression represents a literal replacement of the +old arguments with the new arguments. This may not reflect the +limiting behavior of the expression:

    +
    +
    +
    +

    (x*3 - 3x).subs({x: oo}) +nan

    +

    limit(x*3 - 3x, x, oo) +oo

    +
    +
    +
    +

    If the substitution will be followed by numerical +evaluation, it is better to pass the substitution to +evalf as

    +
    +
    +
    +

    (1/x).evalf(subs={x: 3.0}, n=21) +0.333333333333333333333

    +
    +
    +
    +

    rather than

    +
    +
    +
    +

    (1/x).subs({x: 3.0}).evalf(21) +0.333333333333333314830

    +
    +
    +
    +

    as the former will ensure that the desired level of precision is +obtained.

    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +xreplace: exact node replacement in expr tree; also capable of + using matching rules +sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

    +

    taylor_term

    +
    def taylor_term(
+    self,
+    n,
+    x,
+    *previous_terms
+)
+
    +

    General method for the taylor term.

    +

    This method is slow, because it differentiates n-times. Subclasses can +redefine it to make it faster by using the "previous_terms".

    +

    to_nnf

    +
    def to_nnf(
+    self,
+    simplify=True
+)
+
    +

    together

    +
    def together(
+    self,
+    *args,
+    **kwargs
+)
+
    +

    See the together function in sympy.polys

    +

    transpose

    +
    def transpose(
+    self
+)
+
    +

    trigsimp

    +
    def trigsimp(
+    self,
+    **args
+)
+
    +

    See the trigsimp function in sympy.simplify

    +

    xreplace

    +
    def xreplace(
+    self,
+    rule,
+    hack2=False
+)
+
    +

    Replace occurrences of objects within the expression.

    +

    Parameters

    +

    rule : dict-like + Expresses a replacement rule

    +

    Returns

    +

    xreplace : the result of the replacement

    +

    Examples

    +
    +
    +
    +

    from sympy import symbols, pi, exp +x, y, z = symbols('x y z') +(1 + xy).xreplace({x: pi}) +piy + 1 +(1 + xy).xreplace({x: pi, y: 2}) +1 + 2pi

    +
    +
    +
    +

    Replacements occur only if an entire node in the expression tree is +matched:

    +
    +
    +
    +

    (xy + z).xreplace({xy: pi}) +z + pi +(xyz).xreplace({xy: pi}) +xyz +(2x).xreplace({2x: y, x: z}) +y +(22x).xreplace({2x: y, x: z}) +4*z +(x + y + 2).xreplace({x + y: 2}) +x + y + 2 +(x + 2 + exp(x + 2)).xreplace({x + 2: y}) +x + exp(y) + 2

    +
    +
    +
    +

    xreplace does not differentiate between free and bound symbols. In the +following, subs(x, y) would not change x since it is a bound symbol, +but xreplace does:

    +
    +
    +
    +

    from sympy import Integral +Integral(x, (x, 1, 2x)).xreplace({x: y}) +Integral(y, (y, 1, 2y))

    +
    +
    +
    +

    Trying to replace x with an expression raises an error:

    +
    +
    +
    +

    Integral(x, (x, 1, 2x)).xreplace({x: 2y}) # doctest: +SKIP +ValueError: Invalid limits given: ((2y, 1, 4y),)

    +
    +
    +
    +

    See Also

    +

    replace: replacement capable of doing wildcard-like matching, + parsing of match, and conditional replacements +subs: substitution of subexpressions as defined by the objects + themselves.

    +

    Task

    +
    class Task(
+    name: 'str',
+    target: 'Lazy'
+)
+
    +

    Named wrapper of a lazy-evaluatable function.

    +

    Wraps a lazy-evaluatable function (dewret.workflow.Lazy) with any +metadata needed to render it later. At present, this is the name.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    nameNoneName of the lazy function as it will appear in the output workflow text.None
    targetNoneCallable that is wrapped.None
    +

    UnsetType

    +
    class UnsetType(
+    raw_type: 'type[T]'
+)
+
    +

    Unset variable with a specific type.

    +

    Attributes

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    typeNonetype of the variable.None
    +

    Ancestors (in MRO)

    +
      +
    • dewret.utils.Unset
      • +
    • typing.Generic
      • +
    +

    Workflow

    +
    class Workflow(
+    name: 'str | None' = None
+)
+
    +

    Overarching workflow concept.

    +

    Represents a whole workflow, as a singleton maintaining all +state information needed ahead of rendering. It is built up +as the lazy-evaluations are finally evaluated.

    +

    Attributes

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    stepsNonethe sequence of calls to lazy-evaluable functions,
    built as they are evaluated.    		None
    tasksNonethe mapping of names used in the steps to the actual
    Task wrappers they represent.    		None
    resultNonetarget reference to evaluate, if yet present.None
    +

    Static methods

    +

    assimilate

    +
    def assimilate(
+    *workflow_args: 'Workflow'
+) -> "'Workflow'"
+
    +

    Combine two Workflows into one Workflow.

    +

    Takes two workflows and unifies them by combining steps +and tasks. If it sees mismatched identifiers for the same +component, it will error. +This could happen if the hashing function is flawed +or some Python magic to do with Targets being passed.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    workflow_argsNoneworkflows to use as baseNone
    +

    from_result

    +
    def from_result(
+    result: 'StepReference[Any] | list[StepReference[Any]] | tuple[StepReference[Any], ...]',
+    simplify_ids: 'bool' = False,
+    nested: 'bool' = True
+) -> 'Workflow'
+
    +

    Create from a desired result.

    +

    Starts from a result, and builds a workflow to output it.

    +

    Instance variables

    +
    has_result
+
    +

    Confirms whether this workflow has a non-empty result.

    +

    Either None or an empty list/tuple are considered empty for this purpose.

    +

    Returns: True if the workflow has a result, False otherwise.

    +
    id
+
    +

    Consistent ID based off the step IDs.

    +
    indexed_steps
+
    +

    Steps mapped by ID.

    +

    Forces generation of IDs. Note that this effectively +freezes the steps, so it should not be used until we +are confident the steps are all ready to be hashed.

    +
    name
+
    +

    Get the name of the workflow.

    +
    result_type
+
    +

    Overall return type of this workflow.

    +
    steps
+
    +

    Get deduplicated steps.

    +

    Returns: steps for looping over without duplicates.

    +

    Methods

    +

    add_nested_step

    +
    def add_nested_step(
+    self,
+    name: 'str',
+    subworkflow: 'Workflow',
+    return_type: 'type | None',
+    kwargs: 'dict[str, Any]',
+    positional_args: 'dict[str, bool] | None' = None
+) -> 'StepReference[Any]'
+
    +

    Append a nested step.

    +

    Calls a subworkflow.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    nameNonename of the subworkflow.None
    subworkflowNonethe subworkflow itself.None
    return_typeNonea forced type for the return, or None.None
    kwargsNoneany key-value arguments to pass in the call.None
    positional_argsNonea mapping of arguments to bools, True if the argument is positional or otherwise False.None
    +

    add_step

    +
    def add_step(
+    self,
+    fn: 'Lazy',
+    kwargs: 'dict[str, Raw | Reference[Any]]',
+    raw_as_parameter: 'bool' = False,
+    is_factory: 'bool' = False,
+    positional_args: 'dict[str, bool] | None' = None
+) -> 'StepReference[Any]'
+
    +

    Append a step.

    +

    Adds a step, for running a target with key-value arguments, +to the workflow.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fnNonethe target function to turn into a step.None
    kwargsNoneany key-value arguments to pass in the call.None
    raw_as_parameterNonewhether to turn any discovered raw arguments into workflow parameters.None
    is_factoryNonewhether this step should be a Factory.None
    positional_argsNonea mapping of arguments to bools, True if the argument is positional or otherwise False.None
    +

    find_factories

    +
    def find_factories(
+    self
+) -> 'dict[str, FactoryCall]'
+
    +

    Steps that are factory calls.

    +

    find_parameters

    +
    def find_parameters(
+    self,
+    include_factory_calls: 'bool' = True
+) -> 'set[Parameter[Any]]'
+
    +

    Crawl steps for parameter references.

    +

    As the workflow does not hold its own list of parameters, this +dynamically finds them.

    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneSet of all references to parameters across the steps.
    +

    register_task

    +
    def register_task(
+    self,
+    fn: 'Lazy'
+) -> 'Task'
+
    +

    Note the existence of a lazy-evaluatable function, and wrap it as a Task.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    fnNonethe wrapped function.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneA new Task that wraps the function, and is retained in the Workflow.tasks
    dict.
    +

    remap

    +
    def remap(
+    self,
+    step_id: 'str'
+) -> 'str'
+
    +

    Apply name simplification if requested.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    step_idNonestep to check.None
    +

    Returns:

    + + + + + + + + + + + + + +
    TypeDescription
    NoneSame ID or a remapped name.
    +

    set_result

    +
    def set_result(
+    self,
+    result: 'Basic | list[Basic] | tuple[Basic]'
+) -> 'None'
+
    +

    Choose the result step.

    +

    Sets a step as being the result for the entire workflow. +When we evaluate a dynamic workflow, the engine (e.g. dask) +creates a graph to realize the result of a single collection. +Similarly, in the static case, we need to have a result that +drives the calculation.

    +

    Parameters:

    + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    resultNonereference to the chosen step.None
    +

    simplify_ids

    +
    def simplify_ids(
+    self,
+    infix: 'list[str] | None' = None
+) -> 'None'
+
    +

    Work out mapping to simple ints from hashes.

    +

    Goes through and numbers each step by the order of use of its task.

    +

    WorkflowLinkedComponent

    +
    class WorkflowLinkedComponent(
+    *args,
+    **kwargs
+)
+
    +

    Protocol for objects dynamically tied to a Workflow.

    +

    Ancestors (in MRO)

    +
      +
    • typing.Protocol
      • +
    • typing.Generic
      • +
    + + + + + + +
    +
    + + +
    + +
    + + + + +
    +
    +
    +
    + + + + + + + + + \ No newline at end of file diff --git a/search/search_index.json b/search/search_index.json new file mode 100644 index 00000000..9e5bc053 --- /dev/null +++ b/search/search_index.json @@ -0,0 +1 @@ +{"config":{"indexing":"full","lang":["en"],"min_search_length":3,"prebuild_index":false,"separator":"[\\s\\-]+"},"docs":[{"location":"","text":"dewret DEclarative Workflow REndering Tool Pron : durr-it, like \"durable\" Introduction Dewret allows certain workflows written in a dynamic style to be rendered to a static representation. Advantages of doing so include: git-versionable workflows : while code can be versioned, the changes of a dynamic workflow do not necessarily clearly correspond to changes in the executed workflow. This maintains a precise trackable history. plan and play : the workflow can be rapidly iterated, analysed and optimized before it is sent for real execution on expensive or restricted HPC hardware. optimization : creating the workflow explicitly opens up possibilities for static analysis and refactoring before real execution. debugging : a number of classes of workflow planning bugs will not appear until late in a simulation run that might take days or weeks. This catches them before startup. continuous integration and testing : complex dynamic workflows can be rapidly sense-checked in CI without needing all the hardware and internal algorithms present to run them. Documentation For further information, see the documentation .","title":"Home"},{"location":"#dewret","text":"DEclarative Workflow REndering Tool Pron : durr-it, like \"durable\"","title":"dewret"},{"location":"#introduction","text":"Dewret allows certain workflows written in a dynamic style to be rendered to a static representation. Advantages of doing so include: git-versionable workflows : while code can be versioned, the changes of a dynamic workflow do not necessarily clearly correspond to changes in the executed workflow. This maintains a precise trackable history. plan and play : the workflow can be rapidly iterated, analysed and optimized before it is sent for real execution on expensive or restricted HPC hardware. optimization : creating the workflow explicitly opens up possibilities for static analysis and refactoring before real execution. debugging : a number of classes of workflow planning bugs will not appear until late in a simulation run that might take days or weeks. This catches them before startup. continuous integration and testing : complex dynamic workflows can be rapidly sense-checked in CI without needing all the hardware and internal algorithms present to run them.","title":"Introduction"},{"location":"#documentation","text":"For further information, see the documentation .","title":"Documentation"},{"location":"docs/glossary/","text":"Glossary Construct To construct a workflow in dewret is to pull the connected steps into a single structure. Sub Workflow A subworkflow is a nested or hierarchical workflow. It is a workflow defined within another workflow, allowing for the encapsulation and reuse of complex operations as a single, higher-level step in the parent workflow. Specific type of task designed to encapsulate multiple tasks. Nested tasks are the culmination (or result) of multiple tasks represented as a single task in a dewret workflow . Render To render a workflow is to generate an executable workflow in a specific workflow language such as CWL and Snakemake. Step A step in a dewret workflow represents a single unit of work. It contains a single task and the arguments for that task. Corresponds to a CWL Step or a Snakemake Rule Task A task is the function scheduled to be executed later. Corresponds to a CWL Process Workflow A workflow is designed to define, manage, and execute a series of tasks that make use of both local and global parameters.","title":"Glossary"},{"location":"docs/glossary/#glossary","text":"","title":"Glossary"},{"location":"docs/glossary/#construct","text":"To construct a workflow in dewret is to pull the connected steps into a single structure.","title":"Construct"},{"location":"docs/glossary/#sub-workflow","text":"A subworkflow is a nested or hierarchical workflow. It is a workflow defined within another workflow, allowing for the encapsulation and reuse of complex operations as a single, higher-level step in the parent workflow. Specific type of task designed to encapsulate multiple tasks. Nested tasks are the culmination (or result) of multiple tasks represented as a single task in a dewret workflow .","title":"Sub Workflow"},{"location":"docs/glossary/#render","text":"To render a workflow is to generate an executable workflow in a specific workflow language such as CWL and Snakemake.","title":"Render"},{"location":"docs/glossary/#step","text":"A step in a dewret workflow represents a single unit of work. It contains a single task and the arguments for that task. Corresponds to a CWL Step or a Snakemake Rule","title":"Step"},{"location":"docs/glossary/#task","text":"A task is the function scheduled to be executed later. Corresponds to a CWL Process","title":"Task"},{"location":"docs/glossary/#workflow","text":"A workflow is designed to define, manage, and execute a series of tasks that make use of both local and global parameters.","title":"Workflow"},{"location":"docs/quickstart/","text":"Quickstart Introduction Description Dewret is a tool designed for creating complex workflows, written in a dynamic style, to be rendered to a static representation. Dewret provides a programmatic python interface to multiple declarative workflow engines, where workflows are often written in a yaml-like syntax. It makes it easier for users to define tasks and organize them into workflows. Currently, Dewret supports two renderers: Snakemake and CWL, which generate yamls in the corresponding workflow languages. What are Workflows? Workflows are a collection of tasks or steps designed to automate complex processes. These processes are common in fields like data science, scientific computing and software development, where you can ensure automation. Traditionally, managing workflows can be challenging due to the diversity of backend systems and the complexity of configurations involved. What Makes Dewret Unique? Why should I use Dewret? Dewret stands out by providing a unified and simplified interface for workflow management, making it accessible to users with varying levels of experience. Here are some key features that make Dewret unique: - Consistency: offers a consistent interface for defining tasks and workflows. - Optimization: creating a declarative workflow opens up possibilities for static analysis and refactoring before execution. - Customization: dewret offers the ability to create custom renderers for workflows in desired languages. This includes default support for CWL and Snakemake workflow languages. The capability to render a single workflow into multiple declarative languages enables users to experiment with different workflow engines. - Git-versionable workflows: while code can be versioned, changes in a dynamic workflow may not clearly correspond to changes in the executed workflow. By defining a static workflow that is rendered from the dynamic or programmatic workflow, we maintain a precise and trackable history. - Default Renderers: Snakemake and CWL. - Debugging: a number of classes of workflow planning bugs will not appear until late in a simulation run that might take days or weeks. Having a declarative and static workflow definition document post-render provides enhanced possibilities for static analysis, helping to catch these issues before startup. - Continuous Integration and Testing: complex dynamic workflows can be rapidly sense-checked in CI without needing all the hardware and internal algorithms present to run them. Installation for pure users If you simply want to use Dewret to run workflows, you can install it from PyPI or Conda. From PyPI: pip install dewret From Conda: conda install conda-forge::dewret Installation for developers From a cloned repository: pip install -e . Usage You can render a simple Common Workflow Language CWL workflow from a graph composed of one or more tasks as follows: # workflow.py from dewret.tasks import task @task () def increment ( num : int ) -> int : return num + 1 $ python -m dewret --pretty workflow.py increment num:3 class : Workflow cwlVersion : 1.2 outputs : out : outputSource : increment-e138626779553199eb2bd678356b640f-num type : int steps : increment-e138626779553199eb2bd678356b640f-num in : num : default : 3 out : - out run : increment By default dewret uses a dask backend so that dewret.task wraps a dask.delayed , and renders a CWL workflow. Programmatic Usage Building and rendering may be done programmatically, which provides the opportunity to use custom renderers and backends, as well as bespoke serialization or formatting. >>> import sys >>> import yaml >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 >>> >>> result = increment ( num = 3 ) >>> workflow = construct ( result , simplify_ids = True ) >>> cwl = render ( workflow )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : increment - 1 - num : default : 3 label : num type : int outputs : out : label : out outputSource : increment - 1 / out type : int steps : increment - 1 : in : num : source : increment - 1 - num out : - out run : increment","title":"Quickstart"},{"location":"docs/quickstart/#quickstart","text":"","title":"Quickstart"},{"location":"docs/quickstart/#introduction","text":"","title":"Introduction"},{"location":"docs/quickstart/#description","text":"Dewret is a tool designed for creating complex workflows, written in a dynamic style, to be rendered to a static representation. Dewret provides a programmatic python interface to multiple declarative workflow engines, where workflows are often written in a yaml-like syntax. It makes it easier for users to define tasks and organize them into workflows. Currently, Dewret supports two renderers: Snakemake and CWL, which generate yamls in the corresponding workflow languages.","title":"Description"},{"location":"docs/quickstart/#what-are-workflows","text":"Workflows are a collection of tasks or steps designed to automate complex processes. These processes are common in fields like data science, scientific computing and software development, where you can ensure automation. Traditionally, managing workflows can be challenging due to the diversity of backend systems and the complexity of configurations involved.","title":"What are Workflows?"},{"location":"docs/quickstart/#what-makes-dewret-unique-why-should-i-use-dewret","text":"Dewret stands out by providing a unified and simplified interface for workflow management, making it accessible to users with varying levels of experience. Here are some key features that make Dewret unique: - Consistency: offers a consistent interface for defining tasks and workflows. - Optimization: creating a declarative workflow opens up possibilities for static analysis and refactoring before execution. - Customization: dewret offers the ability to create custom renderers for workflows in desired languages. This includes default support for CWL and Snakemake workflow languages. The capability to render a single workflow into multiple declarative languages enables users to experiment with different workflow engines. - Git-versionable workflows: while code can be versioned, changes in a dynamic workflow may not clearly correspond to changes in the executed workflow. By defining a static workflow that is rendered from the dynamic or programmatic workflow, we maintain a precise and trackable history. - Default Renderers: Snakemake and CWL. - Debugging: a number of classes of workflow planning bugs will not appear until late in a simulation run that might take days or weeks. Having a declarative and static workflow definition document post-render provides enhanced possibilities for static analysis, helping to catch these issues before startup. - Continuous Integration and Testing: complex dynamic workflows can be rapidly sense-checked in CI without needing all the hardware and internal algorithms present to run them.","title":"What Makes Dewret Unique? Why should I use Dewret?"},{"location":"docs/quickstart/#installation-for-pure-users","text":"If you simply want to use Dewret to run workflows, you can install it from PyPI or Conda.","title":"Installation for pure users"},{"location":"docs/quickstart/#from-pypi","text":"pip install dewret","title":"From PyPI:"},{"location":"docs/quickstart/#from-conda","text":"conda install conda-forge::dewret","title":"From Conda:"},{"location":"docs/quickstart/#installation-for-developers","text":"From a cloned repository: pip install -e .","title":"Installation for developers"},{"location":"docs/quickstart/#usage","text":"You can render a simple Common Workflow Language CWL workflow from a graph composed of one or more tasks as follows: # workflow.py from dewret.tasks import task @task () def increment ( num : int ) -> int : return num + 1 $ python -m dewret --pretty workflow.py increment num:3 class : Workflow cwlVersion : 1.2 outputs : out : outputSource : increment-e138626779553199eb2bd678356b640f-num type : int steps : increment-e138626779553199eb2bd678356b640f-num in : num : default : 3 out : - out run : increment By default dewret uses a dask backend so that dewret.task wraps a dask.delayed , and renders a CWL workflow.","title":"Usage"},{"location":"docs/quickstart/#programmatic-usage","text":"Building and rendering may be done programmatically, which provides the opportunity to use custom renderers and backends, as well as bespoke serialization or formatting. >>> import sys >>> import yaml >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 >>> >>> result = increment ( num = 3 ) >>> workflow = construct ( result , simplify_ids = True ) >>> cwl = render ( workflow )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : increment - 1 - num : default : 3 label : num type : int outputs : out : label : out outputSource : increment - 1 / out type : int steps : increment - 1 : in : num : source : increment - 1 - num out : - out run : increment","title":"Programmatic Usage"},{"location":"docs/renderer_tutorial/","text":"Step-by-Step Guide to Writing a Custom Renderer 1. Understand the Target Workflow Language Before writing any code, it is essential to fully understand the target workflow language. This includes syntax, structure, and specific requirements. By breaking down each key dewret.workflow task into smaller components, you can better map your workflow definitions to the target language. Example: In Snakemake, a workflow task is generally created by: 1. Defining the task. - rule process_data 2. Defining the input required for the rule to run(dependencies). - input: \"data/raw_data.txt\" 3. Defining the output required for the rule to be considered finished. - output: \"data/processed_data.txt\" 4. Defining the actual work that the task will do. - in this case: shell: ... rule process_data: # Example Snakemake rule/task input: \"data/raw_data.txt\" output: output_file = \"data/processed_data.txt\" run: with open ( output.output_file, \"w\" ) as f: f.write ( \"data\" ) return output_file 2. Create WorkflowDefinition. The WorkflowDefinition class is responsible for transforming each step from a constructed dewret workflow into an executable step in the target workflow language (e.g. a Snakemake rule). This class should encapsulate workflow-level information, such as the list of steps to be executed, and any workflow-scope input/ouput. It should also contain a class method that initializes the WorkflowDefinition from an dewret Workflow (such as from_workflow below), and a method that renders the workflow as a Python dict (as in the render method below). Example: @define class WorkflowDefinition : steps : list [ StepDefinition ] # Returns a WorkflowDefinition instanace. # Steps contains all of the tasks you want to convert to the target WL tasks. @classmethod def from_workflow ( cls , workflow : Workflow ) -> \"WorkflowDefinition\" : return cls ( steps = [ StepDefinition . from_step ( step ) for step in workflow . steps ]) # Returns each task as a Snakemake executable rule. def render ( self ) -> dict [ str , RawType ]: return { f \"rule { step . name . replace ( \"-\" , \"_\" ) } \" : step . render () for step in self . steps } 3. Create a StepDefinition. Create a StepsDefinition class create each of the code blocks needed for a rule(step) to be executable in Snakemake. When you have defined each block in your target workflow language task from step 1 , you can go ahead and create, for each of the code blocks required to run a Snakemake rule, a BlockDefinition to handle the rendering of each block. Example: In the Snakemake example, we have created: 1. InputDefinition - Handles the input block, which contains what is required for a rule to be executed. It also handles the params block since the code for extracting the input and params blocks is similar. 2. RunDefinition - Handles the run block which contains the instructions needed for this specific task. 3. OutputDefinition - Handles the output block which is required for the rule to be considered successfully finished. @define class StepDefinition : \"\"\"Represents a Snakemake-renderable step definition in a dewret workflow. Attributes: name (str): The name of the step. run (str): The run block definition for the step. params (List[str]): The parameter definitions for the step. output (list[str]: The output definition for the step. input (List[str]): The input definitions for the step. Methods: from_step(cls, step: Step) -> \"StepDefinition\": Constructs a StepDefinition object from a Step object, extracting step information and components from the step and converting them to Snakemake format. render(self) -> dict[str, MainTypes]: Renders the step definition as a dictionary suitable for use in Snakemake workflows. \"\"\" # You can consider each step as a separate rule. # Each field in this class represents a separate block in the rule definition name : str # name of the rule input : list [ str ] # Input block params : list [ str ] # Params block output : list [ str ] # Output block run : list [ str ] # Run block - where the instructions for the task are @classmethod def from_step ( cls , step : Step ) -> \"StepDefinition\" : \"\"\"Constructs a StepDefinition object from a Step. Args: step (Step): The Step object from which step information and components are extracted. Returns: StepDefinition: A StepDefinition object containing the converted step information and components. \"\"\" input_block = InputDefinition . from_step ( step ) . render () run_block = RunDefinition . from_task ( step . task ) . render () output_block = OutputDefinition . from_step ( step ) . render () return cls ( name = step . name , run = run_block , params = input_block [ \"params\" ], input = input_block [ \"inputs\" ], output = output_block , ) def render ( self ) -> dict [ str , MainTypes ]: \"\"\"Renders the step definition as a dictionary. Returns: dict[str, MainTypes]: A dictionary containing the components of the step definition, for use in Snakemake workflows. \"\"\" return { \"run\" : self . run , \"input\" : self . input , \"params\" : self . params , \"output\" : self . output , } 4. Create the Separate block definitions. In this step, you'll define classes to handle the rendering of each code block required for a rule (step) to be executable in the target workflow language. Each of these classes will encapsulate the logic for converting parts of a workflow step into the target language format. Example: For the Snakemake workflow language, we will define: InputDefinition: Handles the input block and parameter block. RunDefinition: Handles the run block. OutputDefinition: Handles the output block. InputDefinition: The InputDefinition class is responsible for rendering the inputs and parameters required for a Snakemake rule. @define class InputDefinition : \"\"\"Represents input and parameter definitions block for a Snakemake-renderable workflow step. Attributes: inputs (List[str]): A list of input definitions. params (List[str]): A list of parameter definitions. Methods: from_step(cls, step: Step) -> \"InputDefinition\": Constructs an InputDefinition object from a Step object, extracting inputs and parameters and converting them to Snakemake-compatible format. render(self) -> dict[str, str]: Renders the input and parameter definitions as a dictionary for use in Snakemake Input and Params blocks. \"\"\" # As we already mention input and params block have similar generation # So it made sence to encapsulate them into one Definition inputs : list [ str ] params : list [ str ] @classmethod def from_step ( cls , step : Step ) -> \"InputDefinition\" : \"\"\"Constructs an InputDefinition object from a Step. Args: step (Step): The Step object from which input and parameter block definitions are extracted. Returns: InputDefinition: An InputDefinition object. \"\"\" params = [] inputs = [] # The keys represent the names of the arguments of the @tasks in our snakemake_workflow.py. # The params represent the values. for key , param in step . arguments . items (): # We check if the param is a reference. # If it is then it's an input requirement for the rule to run, so we put it in the input block if isinstance ( param , Reference ): ref = ReferenceDefinition . from_reference ( param ) . render () . replace ( \"-\" , \"_\" ) . replace ( \"/out\" , \".output\" ) input = f \" { key } =rules. { ref } .output_file\" inputs . append ( input ) params . append ( input + \",\" ) # If it's not - we put it in the params block for use in the RunDefinition elif isinstance ( param , Raw ): customized = f \" { key } = { to_snakemake_type ( param ) } ,\" params . append ( customized ) # Since the params must be comma separated except the last one - we remove the last comma if params : params [ len ( params ) - 1 ] = params [ len ( params ) - 1 ] . replace ( \",\" , \"\" ) return cls ( inputs = inputs , params = params ) def render ( self ) -> dict [ str , list [ str ]]: \"\"\"Renders the input and parameter definitions as a dictionary. Returns: dict[str, list[MainTypes]]: A dictionary containing the input and parameter definitions, for use in Snakemake Input and Params blocks. \"\"\" return { \"inputs\" : self . inputs , \"params\" : self . params } RunDefinition: The RunDefinition class is responsible for rendering the run block, which contains the actual instructions for the task. @define class RunDefinition : # This is where we handle the execution of the task itself. \"\"\"Represents a Snakemake-renderable run block for a dewret workflow step. Attributes: method_name (str): The name of the method to be executed in the snakefile run block. rel_import (str): The relative import path of the method. args (List[str]): The arguments to be passed to the method. Methods: from_task(cls, task: Task) -> \"RunDefinition\": Constructs a RunDefinition object from a Task object, extracting method information and arguments from the task and converting them to Snakemake-compatible format. render(self) -> list[str]: A list containing the import statement and the method call statement, for use in Snakemake run block. \"\"\" method_name : str rel_import : str args : list [ str ] @classmethod def from_task ( cls , task : Task ) -> \"RunDefinition\" : \"\"\"Constructs a RunDefinition object from a Task. Args: task (Task): The Task object from which method information and arguments are extracted. Returns: RunDefinition: A RunDefinition object containing the converted method information and arguments. \"\"\" # Since we can import our snakemake_workflow.py @tasks we need the relative path relative_path = get_method_rel_path ( task . target ) # If we need to make any customization to the import rel_import = f \" { relative_path } \" args = get_method_args ( task . target ) signature = [ f \" { param_name } =params. { param_name } \" for param_name in args . parameters . keys () ] return cls ( method_name = task . name , rel_import = rel_import , args = signature ) def render ( self ) -> list [ str ]: \"\"\"Renders the run block as a list of strings. Returns: list[str]: A list containing the import statement and the method call statement, for use in Snakemake run block. \"\"\" signature = \", \" . join ( f \" { arg } \" for arg in self . args ) # The comma after the last element is mandatory for the structure of rule onces it's used in yaml.dump return [ f \"import { self . rel_import } \\n \" , f \" { self . rel_import } . { self . method_name } ( { signature } ) \\n \" , ] OutputDefinition: The OutputDefinition class is responsible for rendering the output block, which specifies the output files or results that indicate the rule has successfully completed. @define class OutputDefinition : \"\"\"Represents the output definition block for a Snakemake-renderable workflow step. Attributes: output_file (str): The output file definition. Methods: from_step(cls, step: Step) -> \"OutputDefinition\": Constructs an OutputDefinition object from a Step object, extracting and converting the output file definition to Snakemake-compatible format. render(self) -> list[str]: Renders the output definition as a list suitable for use in Snakemake Output block. \"\"\" output_file : str @classmethod def from_step ( cls , step : Step ) -> \"OutputDefinition\" : \"\"\"Constructs an OutputDefinition object from a Step. Args: step (Step): The Step object from which the output file definition is extracted. Returns: OutputDefinition: An OutputDefinition object, for use in Snakemake Output block. \"\"\" # Since snakemake commonly communicates using files. # Output file must always be called - `output_file` # Further code could be added to handled if it's a reference in case we want take care of multiple tasks writing to the same output file. output_file = step . arguments [ \"output_file\" ] if isinstance ( output_file , Raw ): args = to_snakemake_type ( output_file ) return cls ( output_file = args ) def render ( self ) -> list [ str ]: \"\"\"Renders the output definition as a list. Returns: list[str]: A list containing the output file definition, for use in a Snakemake Output block. \"\"\" # The comma after the last element is mandatory for the structure of rule onces it's used in yaml.dump # It adds the new line to the output block return [ f \"output_file= { self . output_file } \" , ] Integrate these block definitions into the StepDefinition class as demonstrated in Step 3 . Each StepDefinition will use these block definitions to render the complete step in the target workflow language. 5. Helper methods. In this step, you'll define helper methods that will assist you in converting workflow components into the target workflow language format. In our case these methods will handle type conversion, extracting method arguments, and computing relative paths. Example: We'll define the following helper methods for our Snakemake renderer: to_snakemake_type(param: Raw) -> str: Converts a raw type to a Snakemake-compatible Python type. get_method_args(func: Lazy) -> inspect.Signature: Retrieves the argument names and types of a lazy-evaluatable function. get_method_rel_path(func: Lazy) -> str: Computes the relative path of the module containing the given function. Type Conversion Helper: # Basic types returned from dewret will look like this \"str|valueOfParam\". # We'll need to convert them. def to_Snakemake_type ( param : Raw ) -> str : typ = str ( param ) if typ . __contains__ ( \"str\" ): return f '\" { typ . replace ( \"str|\" , \"\" ) } \"' elif typ . __contains__ ( \"bool\" ): return typ . replace ( \"bool|\" , \"\" ) elif typ . __contains__ ( \"dict\" ): return typ . replace ( \"dict|\" , \"\" ) elif typ . __contains__ ( \"list\" ): return typ . replace ( \"list|\" , \"\" ) elif typ . __contains__ ( \"float\" ): return typ . replace ( \"float|\" , \"\" ) elif typ . __contains__ ( \"int\" ): return typ . replace ( \"int|\" , \"\" ) else : raise TypeError ( f \"Cannot render complex type ( { typ } )\" ) Argument Extraction Helper: # We need to get the signature of the method. def get_method_args ( func : Lazy ) -> inspect . Signature : args = inspect . signature ( func ) return args Relative Path Computation Helper: # Computes the relative path def get_method_rel_path ( func : Lazy ) -> str : source_file = inspect . getsourcefile ( func ) if source_file : relative_path = os . path . relpath ( source_file , start = os . getcwd ()) module_name = os . path . splitext ( relative_path )[ 0 ] . replace ( os . path . sep , \".\" ) return module_name Imports and custom types required in the SMK example: import os import yaml import inspect import typing from attrs import define from dewret.utils import Raw , BasicType from dewret.workflow import Lazy from dewret.workflow import Reference , Workflow , Step , Task RawType = typing . Union [ BasicType , list [ str ], list [ \"RawType\" ], dict [ str , \"RawType\" ]] To run this example: Import the snakemake renderer into your @tasks file There's an example in snakemake_tasks.py Run it: python snakemake_tasks.py Q: Should I add a brief description of dewret in step 1? Should link dewret types/docs etc here? A: Get details on how that happens and probably yes.","title":"Renderer Tutorial"},{"location":"docs/renderer_tutorial/#step-by-step-guide-to-writing-a-custom-renderer","text":"","title":"Step-by-Step Guide to Writing a Custom Renderer"},{"location":"docs/renderer_tutorial/#1-understand-the-target-workflow-language","text":"Before writing any code, it is essential to fully understand the target workflow language. This includes syntax, structure, and specific requirements. By breaking down each key dewret.workflow task into smaller components, you can better map your workflow definitions to the target language.","title":"1. Understand the Target Workflow Language"},{"location":"docs/renderer_tutorial/#example","text":"In Snakemake, a workflow task is generally created by: 1. Defining the task. - rule process_data 2. Defining the input required for the rule to run(dependencies). - input: \"data/raw_data.txt\" 3. Defining the output required for the rule to be considered finished. - output: \"data/processed_data.txt\" 4. Defining the actual work that the task will do. - in this case: shell: ... rule process_data: # Example Snakemake rule/task input: \"data/raw_data.txt\" output: output_file = \"data/processed_data.txt\" run: with open ( output.output_file, \"w\" ) as f: f.write ( \"data\" ) return output_file","title":"Example:"},{"location":"docs/renderer_tutorial/#2-create-workflowdefinition","text":"The WorkflowDefinition class is responsible for transforming each step from a constructed dewret workflow into an executable step in the target workflow language (e.g. a Snakemake rule). This class should encapsulate workflow-level information, such as the list of steps to be executed, and any workflow-scope input/ouput. It should also contain a class method that initializes the WorkflowDefinition from an dewret Workflow (such as from_workflow below), and a method that renders the workflow as a Python dict (as in the render method below).","title":"2. Create WorkflowDefinition."},{"location":"docs/renderer_tutorial/#example_1","text":"@define class WorkflowDefinition : steps : list [ StepDefinition ] # Returns a WorkflowDefinition instanace. # Steps contains all of the tasks you want to convert to the target WL tasks. @classmethod def from_workflow ( cls , workflow : Workflow ) -> \"WorkflowDefinition\" : return cls ( steps = [ StepDefinition . from_step ( step ) for step in workflow . steps ]) # Returns each task as a Snakemake executable rule. def render ( self ) -> dict [ str , RawType ]: return { f \"rule { step . name . replace ( \"-\" , \"_\" ) } \" : step . render () for step in self . steps }","title":"Example:"},{"location":"docs/renderer_tutorial/#3-create-a-stepdefinition","text":"Create a StepsDefinition class create each of the code blocks needed for a rule(step) to be executable in Snakemake. When you have defined each block in your target workflow language task from step 1 , you can go ahead and create, for each of the code blocks required to run a Snakemake rule, a BlockDefinition to handle the rendering of each block.","title":"3. Create a StepDefinition."},{"location":"docs/renderer_tutorial/#example_2","text":"In the Snakemake example, we have created: 1. InputDefinition - Handles the input block, which contains what is required for a rule to be executed. It also handles the params block since the code for extracting the input and params blocks is similar. 2. RunDefinition - Handles the run block which contains the instructions needed for this specific task. 3. OutputDefinition - Handles the output block which is required for the rule to be considered successfully finished. @define class StepDefinition : \"\"\"Represents a Snakemake-renderable step definition in a dewret workflow. Attributes: name (str): The name of the step. run (str): The run block definition for the step. params (List[str]): The parameter definitions for the step. output (list[str]: The output definition for the step. input (List[str]): The input definitions for the step. Methods: from_step(cls, step: Step) -> \"StepDefinition\": Constructs a StepDefinition object from a Step object, extracting step information and components from the step and converting them to Snakemake format. render(self) -> dict[str, MainTypes]: Renders the step definition as a dictionary suitable for use in Snakemake workflows. \"\"\" # You can consider each step as a separate rule. # Each field in this class represents a separate block in the rule definition name : str # name of the rule input : list [ str ] # Input block params : list [ str ] # Params block output : list [ str ] # Output block run : list [ str ] # Run block - where the instructions for the task are @classmethod def from_step ( cls , step : Step ) -> \"StepDefinition\" : \"\"\"Constructs a StepDefinition object from a Step. Args: step (Step): The Step object from which step information and components are extracted. Returns: StepDefinition: A StepDefinition object containing the converted step information and components. \"\"\" input_block = InputDefinition . from_step ( step ) . render () run_block = RunDefinition . from_task ( step . task ) . render () output_block = OutputDefinition . from_step ( step ) . render () return cls ( name = step . name , run = run_block , params = input_block [ \"params\" ], input = input_block [ \"inputs\" ], output = output_block , ) def render ( self ) -> dict [ str , MainTypes ]: \"\"\"Renders the step definition as a dictionary. Returns: dict[str, MainTypes]: A dictionary containing the components of the step definition, for use in Snakemake workflows. \"\"\" return { \"run\" : self . run , \"input\" : self . input , \"params\" : self . params , \"output\" : self . output , }","title":"Example:"},{"location":"docs/renderer_tutorial/#4-create-the-separate-block-definitions","text":"In this step, you'll define classes to handle the rendering of each code block required for a rule (step) to be executable in the target workflow language. Each of these classes will encapsulate the logic for converting parts of a workflow step into the target language format.","title":"4. Create the Separate block definitions."},{"location":"docs/renderer_tutorial/#example_3","text":"For the Snakemake workflow language, we will define: InputDefinition: Handles the input block and parameter block. RunDefinition: Handles the run block. OutputDefinition: Handles the output block.","title":"Example:"},{"location":"docs/renderer_tutorial/#inputdefinition","text":"The InputDefinition class is responsible for rendering the inputs and parameters required for a Snakemake rule. @define class InputDefinition : \"\"\"Represents input and parameter definitions block for a Snakemake-renderable workflow step. Attributes: inputs (List[str]): A list of input definitions. params (List[str]): A list of parameter definitions. Methods: from_step(cls, step: Step) -> \"InputDefinition\": Constructs an InputDefinition object from a Step object, extracting inputs and parameters and converting them to Snakemake-compatible format. render(self) -> dict[str, str]: Renders the input and parameter definitions as a dictionary for use in Snakemake Input and Params blocks. \"\"\" # As we already mention input and params block have similar generation # So it made sence to encapsulate them into one Definition inputs : list [ str ] params : list [ str ] @classmethod def from_step ( cls , step : Step ) -> \"InputDefinition\" : \"\"\"Constructs an InputDefinition object from a Step. Args: step (Step): The Step object from which input and parameter block definitions are extracted. Returns: InputDefinition: An InputDefinition object. \"\"\" params = [] inputs = [] # The keys represent the names of the arguments of the @tasks in our snakemake_workflow.py. # The params represent the values. for key , param in step . arguments . items (): # We check if the param is a reference. # If it is then it's an input requirement for the rule to run, so we put it in the input block if isinstance ( param , Reference ): ref = ReferenceDefinition . from_reference ( param ) . render () . replace ( \"-\" , \"_\" ) . replace ( \"/out\" , \".output\" ) input = f \" { key } =rules. { ref } .output_file\" inputs . append ( input ) params . append ( input + \",\" ) # If it's not - we put it in the params block for use in the RunDefinition elif isinstance ( param , Raw ): customized = f \" { key } = { to_snakemake_type ( param ) } ,\" params . append ( customized ) # Since the params must be comma separated except the last one - we remove the last comma if params : params [ len ( params ) - 1 ] = params [ len ( params ) - 1 ] . replace ( \",\" , \"\" ) return cls ( inputs = inputs , params = params ) def render ( self ) -> dict [ str , list [ str ]]: \"\"\"Renders the input and parameter definitions as a dictionary. Returns: dict[str, list[MainTypes]]: A dictionary containing the input and parameter definitions, for use in Snakemake Input and Params blocks. \"\"\" return { \"inputs\" : self . inputs , \"params\" : self . params }","title":"InputDefinition:"},{"location":"docs/renderer_tutorial/#rundefinition","text":"The RunDefinition class is responsible for rendering the run block, which contains the actual instructions for the task. @define class RunDefinition : # This is where we handle the execution of the task itself. \"\"\"Represents a Snakemake-renderable run block for a dewret workflow step. Attributes: method_name (str): The name of the method to be executed in the snakefile run block. rel_import (str): The relative import path of the method. args (List[str]): The arguments to be passed to the method. Methods: from_task(cls, task: Task) -> \"RunDefinition\": Constructs a RunDefinition object from a Task object, extracting method information and arguments from the task and converting them to Snakemake-compatible format. render(self) -> list[str]: A list containing the import statement and the method call statement, for use in Snakemake run block. \"\"\" method_name : str rel_import : str args : list [ str ] @classmethod def from_task ( cls , task : Task ) -> \"RunDefinition\" : \"\"\"Constructs a RunDefinition object from a Task. Args: task (Task): The Task object from which method information and arguments are extracted. Returns: RunDefinition: A RunDefinition object containing the converted method information and arguments. \"\"\" # Since we can import our snakemake_workflow.py @tasks we need the relative path relative_path = get_method_rel_path ( task . target ) # If we need to make any customization to the import rel_import = f \" { relative_path } \" args = get_method_args ( task . target ) signature = [ f \" { param_name } =params. { param_name } \" for param_name in args . parameters . keys () ] return cls ( method_name = task . name , rel_import = rel_import , args = signature ) def render ( self ) -> list [ str ]: \"\"\"Renders the run block as a list of strings. Returns: list[str]: A list containing the import statement and the method call statement, for use in Snakemake run block. \"\"\" signature = \", \" . join ( f \" { arg } \" for arg in self . args ) # The comma after the last element is mandatory for the structure of rule onces it's used in yaml.dump return [ f \"import { self . rel_import } \\n \" , f \" { self . rel_import } . { self . method_name } ( { signature } ) \\n \" , ]","title":"RunDefinition:"},{"location":"docs/renderer_tutorial/#outputdefinition","text":"The OutputDefinition class is responsible for rendering the output block, which specifies the output files or results that indicate the rule has successfully completed. @define class OutputDefinition : \"\"\"Represents the output definition block for a Snakemake-renderable workflow step. Attributes: output_file (str): The output file definition. Methods: from_step(cls, step: Step) -> \"OutputDefinition\": Constructs an OutputDefinition object from a Step object, extracting and converting the output file definition to Snakemake-compatible format. render(self) -> list[str]: Renders the output definition as a list suitable for use in Snakemake Output block. \"\"\" output_file : str @classmethod def from_step ( cls , step : Step ) -> \"OutputDefinition\" : \"\"\"Constructs an OutputDefinition object from a Step. Args: step (Step): The Step object from which the output file definition is extracted. Returns: OutputDefinition: An OutputDefinition object, for use in Snakemake Output block. \"\"\" # Since snakemake commonly communicates using files. # Output file must always be called - `output_file` # Further code could be added to handled if it's a reference in case we want take care of multiple tasks writing to the same output file. output_file = step . arguments [ \"output_file\" ] if isinstance ( output_file , Raw ): args = to_snakemake_type ( output_file ) return cls ( output_file = args ) def render ( self ) -> list [ str ]: \"\"\"Renders the output definition as a list. Returns: list[str]: A list containing the output file definition, for use in a Snakemake Output block. \"\"\" # The comma after the last element is mandatory for the structure of rule onces it's used in yaml.dump # It adds the new line to the output block return [ f \"output_file= { self . output_file } \" , ] Integrate these block definitions into the StepDefinition class as demonstrated in Step 3 . Each StepDefinition will use these block definitions to render the complete step in the target workflow language.","title":"OutputDefinition:"},{"location":"docs/renderer_tutorial/#5-helper-methods","text":"In this step, you'll define helper methods that will assist you in converting workflow components into the target workflow language format. In our case these methods will handle type conversion, extracting method arguments, and computing relative paths.","title":"5. Helper methods."},{"location":"docs/renderer_tutorial/#example_4","text":"We'll define the following helper methods for our Snakemake renderer: to_snakemake_type(param: Raw) -> str: Converts a raw type to a Snakemake-compatible Python type. get_method_args(func: Lazy) -> inspect.Signature: Retrieves the argument names and types of a lazy-evaluatable function. get_method_rel_path(func: Lazy) -> str: Computes the relative path of the module containing the given function.","title":"Example:"},{"location":"docs/renderer_tutorial/#type-conversion-helper","text":"# Basic types returned from dewret will look like this \"str|valueOfParam\". # We'll need to convert them. def to_Snakemake_type ( param : Raw ) -> str : typ = str ( param ) if typ . __contains__ ( \"str\" ): return f '\" { typ . replace ( \"str|\" , \"\" ) } \"' elif typ . __contains__ ( \"bool\" ): return typ . replace ( \"bool|\" , \"\" ) elif typ . __contains__ ( \"dict\" ): return typ . replace ( \"dict|\" , \"\" ) elif typ . __contains__ ( \"list\" ): return typ . replace ( \"list|\" , \"\" ) elif typ . __contains__ ( \"float\" ): return typ . replace ( \"float|\" , \"\" ) elif typ . __contains__ ( \"int\" ): return typ . replace ( \"int|\" , \"\" ) else : raise TypeError ( f \"Cannot render complex type ( { typ } )\" )","title":"Type Conversion Helper:"},{"location":"docs/renderer_tutorial/#argument-extraction-helper","text":"# We need to get the signature of the method. def get_method_args ( func : Lazy ) -> inspect . Signature : args = inspect . signature ( func ) return args","title":"Argument Extraction Helper:"},{"location":"docs/renderer_tutorial/#relative-path-computation-helper","text":"# Computes the relative path def get_method_rel_path ( func : Lazy ) -> str : source_file = inspect . getsourcefile ( func ) if source_file : relative_path = os . path . relpath ( source_file , start = os . getcwd ()) module_name = os . path . splitext ( relative_path )[ 0 ] . replace ( os . path . sep , \".\" ) return module_name","title":"Relative Path Computation Helper:"},{"location":"docs/renderer_tutorial/#imports-and-custom-types-required-in-the-smk-example","text":"import os import yaml import inspect import typing from attrs import define from dewret.utils import Raw , BasicType from dewret.workflow import Lazy from dewret.workflow import Reference , Workflow , Step , Task RawType = typing . Union [ BasicType , list [ str ], list [ \"RawType\" ], dict [ str , \"RawType\" ]]","title":"Imports and custom types required in the SMK example:"},{"location":"docs/renderer_tutorial/#to-run-this-example","text":"Import the snakemake renderer into your @tasks file There's an example in snakemake_tasks.py Run it: python snakemake_tasks.py","title":"To run this example:"},{"location":"docs/renderer_tutorial/#q-should-i-add-a-brief-description-of-dewret-in-step-1-should-link-dewret-typesdocs-etc-here","text":"","title":"Q: Should I add a brief description of dewret in step 1? Should link dewret types/docs etc here?"},{"location":"docs/renderer_tutorial/#a-get-details-on-how-that-happens-and-probably-yes","text":"","title":"A: Get details on how that happens and probably yes."},{"location":"docs/renderers/","text":"Renderers Renderers are a function that takes a task , which can be assumed to have a __workflow__ member of type Workflow , and return a YAML-serializable nested dict structure. CWL The default renderer is for the Common Workflow Language. It implements a very small subset of functionality, and is not yet strictly standards compliant. It assumes that all run names can be interpreted in the context of the workflow module's global scope. Custom ...","title":"Renderers"},{"location":"docs/renderers/#renderers","text":"Renderers are a function that takes a task , which can be assumed to have a __workflow__ member of type Workflow , and return a YAML-serializable nested dict structure.","title":"Renderers"},{"location":"docs/renderers/#cwl","text":"The default renderer is for the Common Workflow Language. It implements a very small subset of functionality, and is not yet strictly standards compliant. It assumes that all run names can be interpreted in the context of the workflow module's global scope.","title":"CWL"},{"location":"docs/renderers/#custom","text":"...","title":"Custom"},{"location":"docs/workflows/","text":"Workflows Description A dewret workflow is composed of one or more steps that may make use of both local and global parameters. Each step is defined by a dewret task that is created by using the @task() decorator, and each task may be used by multiple steps. Setup We can pull in dewret tools to produce CWL with a small number of imports. >>> import sys >>> import yaml >>> from dewret.tasks import task , construct >>> from dewret.workflow import param >>> from dewret.renderers.cwl import render Dependencies Specifying step interdependencies is possible by combining lazy-evaluated function calls. The output series of steps is not guaranteed to be in order of execution. Dewret hashes the parameters to identify and unify steps. This lets you do, for example: graph TD A[increment] --> B[double] A[increment] --> C[mod10] B[double] --> D[sum] C[mod10] --> D[sum] In code, this would be: >>> import sys >>> import yaml >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @task () ... def increment ( num : int ) -> int : ... \"\"\"Increment an integer.\"\"\" ... return num + 1 >>> >>> @task () ... def double ( num : int ) -> int : ... \"\"\"Double an integer.\"\"\" ... return 2 * num >>> >>> @task () ... def mod10 ( num : int ) -> int : ... \"\"\"Take num mod 10.\"\"\" ... return num % 10 >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... \"\"\"Add two integers.\"\"\" ... return left + right >>> >>> result = sum ( ... left = double ( num = increment ( num = 23 )), ... right = mod10 ( num = increment ( num = 23 )) ... ) >>> wkflw = construct ( result , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : increment - 1 - num : default : 23 label : num type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : double - 1 : in : num : source : increment - 1 / out out : - out run : double increment - 1 : in : num : source : increment - 1 - num out : - out run : increment mod10 - 1 : in : num : source : increment - 1 / out out : - out run : mod10 sum - 1 : in : left : source : double - 1 / out right : source : mod10 - 1 / out out : - out run : sum Notice that the increment tasks appears twice in the CWL workflow definition, being referenced twice in the python code above. This duplication can be avoided by explicitly indicating that the parameters are the same, with the param function. >>> import sys >>> import yaml >>> from dewret.workflow import param >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @task () ... def increment ( num : int ) -> int : ... \"\"\"Increment an integer.\"\"\" ... return num + 1 >>> >>> @task () ... def double ( num : int ) -> int : ... \"\"\"Double an integer.\"\"\" ... return 2 * num >>> >>> @task () ... def mod10 ( num : int ) -> int : ... \"\"\"Take num mod 10.\"\"\" ... return num % 10 >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... \"\"\"Add two integers.\"\"\" ... return left + right >>> >>> num = param ( \"num\" , default = 3 ) >>> result = sum ( ... left = double ( num = increment ( num = num )), ... right = mod10 ( num = increment ( num = num )) ... ) >>> wkflw = construct ( result , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : num : default : 3 label : num type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : double - 1 : in : num : source : increment - 1 / out out : - out run : double increment - 1 : in : num : source : num out : - out run : increment mod10 - 1 : in : num : source : increment - 1 / out out : - out run : mod10 sum - 1 : in : left : source : double - 1 / out right : source : mod10 - 1 / out out : - out run : sum Parameters The tool will spot global variables that you have used when building your tasks, and treat them as parameters. It will try to get the type from the typehint, or the value that you have set it to. This only works for basic types (and dict/lists of those). While global variables are implicit input to the Python function note that : in CWL, they will be rendered as explicit global input to a step as input, they are read-only, and must not be updated For example: >>> import sys >>> import yaml >>> from dewret.workflow import param >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> INPUT_NUM = 3 >>> @task () ... def rotate ( num : int ) -> int : ... \"\"\"Rotate an integer.\"\"\" ... return ( num + INPUT_NUM ) % INPUT_NUM >>> >>> result = rotate ( num = 5 ) >>> wkflw = construct ( result , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : INPUT_NUM : default : 3 label : INPUT_NUM type : int rotate - 1 - num : default : 5 label : num type : int outputs : out : label : out outputSource : rotate - 1 / out type : int steps : rotate - 1 : in : INPUT_NUM : source : INPUT_NUM num : source : rotate - 1 - num out : - out run : rotate Nested tasks When you wish to combine tasks together programmatically, you can use nested tasks. These are run at render time, not execution time. In other words, they do not appear in the final graph, and so must only combine other tasks. For example: graph TD A[rotate] --> B[rotate] B[rotate] --> C[double_rotate] As code: >>> import sys >>> import yaml >>> from dewret.core import set_configuration >>> from dewret.tasks import task , construct , workflow >>> from dewret.renderers.cwl import render >>> INPUT_NUM = 3 >>> @task () ... def rotate ( num : int ) -> int : ... \"\"\"Rotate an integer.\"\"\" ... return ( num + INPUT_NUM ) % INPUT_NUM >>> >>> @workflow () ... def double_rotate ( num : int ) -> int : ... \"\"\"Rotate an integer twice.\"\"\" ... return rotate ( num = rotate ( num = num )) >>> >>> with set_configuration ( flatten_all_nested = True ): ... result = double_rotate ( num = 3 ) ... wkflw = construct ( result , simplify_ids = True ) ... cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : INPUT_NUM : default : 3 label : INPUT_NUM type : int num : default : 3 label : num type : int outputs : out : label : out outputSource : rotate - 1 / out type : int steps : rotate - 1 : in : INPUT_NUM : source : INPUT_NUM num : source : rotate - 2 / out out : - out run : rotate rotate - 2 : in : INPUT_NUM : source : INPUT_NUM num : source : num out : - out run : rotate Note that, as with all dewret calculations, only the steps necessary to achieve the ultimate output are included in the final graph. Therefore, nested tasks must return a step execution (task that is being called) that forces any other calculations you wish to happen. In other words, if a task in a nested task does not have an impact on the return value, it will disappear . For example, the following code renders the same workflow as in the previous example: @workflow () def double_rotate ( num : int ) -> int : \"\"\"Rotate an integer twice.\"\"\" unused_var = increment ( num = num ) return rotate ( num = rotate ( num = num )) Step Output Fields Each step, by default, is treated as having a single result. However, we allow a mechanism for specifying multiple fields, using attrs or dataclasses . Where needed, fields can be accessed outside of tasks by dot notation and dewret will map that access to a specific output field in CWL. Note that in the example below, shuffle is still only seen once in the graph: graph TD A[shuffle] --> B[hearts] A[shuffle] --> C[diamonds] B[hearts] --> D[sum] C[diamonds] --> D[sum] As code: >>> import sys >>> import yaml >>> from attrs import define >>> from numpy import random >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> red_total = sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> wkflw = construct ( red_total , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : shuffle - 1 - max_cards_per_suit : default : 13 label : max_cards_per_suit type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : shuffle - 1 : in : max_cards_per_suit : source : shuffle - 1 - max_cards_per_suit out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : shuffle sum - 1 : in : left : source : shuffle - 1 / hearts right : source : shuffle - 1 / diamonds out : - out run : sum Here, we show the same example with dataclasses . >>> import sys >>> import yaml >>> from dataclasses import dataclass >>> from numpy import random >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @dataclass ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> red_total = sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> wkflw = construct ( red_total , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : shuffle - 1 - max_cards_per_suit : default : 13 label : max_cards_per_suit type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : shuffle - 1 : in : max_cards_per_suit : source : shuffle - 1 - max_cards_per_suit out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : shuffle sum - 1 : in : left : source : shuffle - 1 / hearts right : source : shuffle - 1 / diamonds out : - out run : sum Subworkflow A special form of nested task is available to help divide up more complex workflows: the subworkflow . By wrapping logic in subflows, dewret will produce multiple output workflows that reference each other. >>> import sys >>> import yaml >>> from attrs import define >>> from numpy import random >>> from dewret.tasks import task , construct , workflow >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @workflow () ... def red_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> @workflow () ... def black_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . spades , ... right = shuffle ( max_cards_per_suit = 13 ) . clubs ... ) >>> total = sum ( left = red_total (), right = black_total ()) >>> wkflw = construct ( total , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : {} outputs : out : label : out outputSource : sum - 1 / out type : int steps : black_total - 1 : in : {} out : - out run : black_total red_total - 1 : in : {} out : - out run : red_total sum - 1 : in : left : source : red_total - 1 / out right : source : black_total - 1 / out out : - out run : sum As we have used subworkflow to wrap the colour totals, the outer workflow contains references to them only. The subworkflows are now returned by render as a second term. >>> import sys >>> import yaml >>> from attrs import define >>> from numpy import random >>> from dewret.tasks import task , construct , workflow >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @workflow () ... def red_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> @workflow () ... def black_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . spades , ... right = shuffle ( max_cards_per_suit = 13 ) . clubs ... ) >>> total = sum ( left = red_total (), right = black_total ()) >>> wkflw = construct ( total , simplify_ids = True ) >>> cwl = render ( wkflw ) >>> yaml . dump ( cwl [ \"red_total-1\" ], sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : {} outputs : out : label : out outputSource : sum - 1 - 1 / out type : int steps : shuffle - 1 - 1 : in : max_cards_per_suit : default : 13 out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : shuffle sum - 1 - 1 : in : left : source : shuffle - 1 - 1 / hearts right : source : shuffle - 1 - 1 / diamonds out : - out run : sum Input Factories Sometimes we want to take complex Python input, not just raw types. Not all serialization support this, but the factory function lets us wrap a simple call, usually a constructor, that takes only raw arguments. This can then rendered as either a step or a parameter depending on whether the chosen renderer has the capability. Below is the default output, treating Pack as a task. >>> import sys >>> import yaml >>> from dewret.tasks import workflow , factory , workflow , construct , task >>> from attrs import define >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> Pack = factory ( PackResult ) >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @workflow () ... def black_total ( pack : PackResult ) -> int : ... return sum ( ... left = pack . spades , ... right = pack . clubs ... ) >>> pack = Pack ( hearts = 13 , spades = 13 , diamonds = 13 , clubs = 13 ) >>> wkflw = construct ( black_total ( pack = pack ), simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : PackResult - 1 - clubs : default : 13 label : clubs type : int PackResult - 1 - diamonds : default : 13 label : diamonds type : int PackResult - 1 - hearts : default : 13 label : hearts type : int PackResult - 1 - spades : default : 13 label : spades type : int outputs : out : label : out outputSource : black_total - 1 / out type : int steps : PackResult - 1 : in : clubs : source : PackResult - 1 - clubs diamonds : source : PackResult - 1 - diamonds hearts : source : PackResult - 1 - hearts spades : source : PackResult - 1 - spades out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : PackResult black_total - 1 : in : pack : source : PackResult - 1 / out out : - out run : black_total The CWL renderer is also able to treat pack as a parameter, if complex types are allowed. >>> import sys >>> import yaml >>> from dewret.tasks import task , factory , workflow , construct >>> from attrs import define >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> Pack = factory ( PackResult ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @workflow () ... def black_total ( pack : PackResult ) -> int : ... return sum ( ... left = pack . spades , ... right = pack . clubs ... ) >>> pack = Pack ( hearts = 13 , spades = 13 , diamonds = 13 , clubs = 13 ) >>> wkflw = construct ( black_total ( pack = pack ), simplify_ids = True ) >>> cwl = render ( wkflw , allow_complex_types = True , factories_as_params = True )[ \"black_total-1\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : pack : label : pack type : record outputs : out : label : out outputSource : sum - 1 - 1 / out type : int steps : sum - 1 - 1 : in : left : source : pack / spades right : source : pack / clubs out : - out run : sum","title":"Workflows"},{"location":"docs/workflows/#workflows","text":"","title":"Workflows"},{"location":"docs/workflows/#description","text":"A dewret workflow is composed of one or more steps that may make use of both local and global parameters. Each step is defined by a dewret task that is created by using the @task() decorator, and each task may be used by multiple steps.","title":"Description"},{"location":"docs/workflows/#setup","text":"We can pull in dewret tools to produce CWL with a small number of imports. >>> import sys >>> import yaml >>> from dewret.tasks import task , construct >>> from dewret.workflow import param >>> from dewret.renderers.cwl import render","title":"Setup"},{"location":"docs/workflows/#dependencies","text":"Specifying step interdependencies is possible by combining lazy-evaluated function calls. The output series of steps is not guaranteed to be in order of execution. Dewret hashes the parameters to identify and unify steps. This lets you do, for example: graph TD A[increment] --> B[double] A[increment] --> C[mod10] B[double] --> D[sum] C[mod10] --> D[sum] In code, this would be: >>> import sys >>> import yaml >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @task () ... def increment ( num : int ) -> int : ... \"\"\"Increment an integer.\"\"\" ... return num + 1 >>> >>> @task () ... def double ( num : int ) -> int : ... \"\"\"Double an integer.\"\"\" ... return 2 * num >>> >>> @task () ... def mod10 ( num : int ) -> int : ... \"\"\"Take num mod 10.\"\"\" ... return num % 10 >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... \"\"\"Add two integers.\"\"\" ... return left + right >>> >>> result = sum ( ... left = double ( num = increment ( num = 23 )), ... right = mod10 ( num = increment ( num = 23 )) ... ) >>> wkflw = construct ( result , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : increment - 1 - num : default : 23 label : num type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : double - 1 : in : num : source : increment - 1 / out out : - out run : double increment - 1 : in : num : source : increment - 1 - num out : - out run : increment mod10 - 1 : in : num : source : increment - 1 / out out : - out run : mod10 sum - 1 : in : left : source : double - 1 / out right : source : mod10 - 1 / out out : - out run : sum Notice that the increment tasks appears twice in the CWL workflow definition, being referenced twice in the python code above. This duplication can be avoided by explicitly indicating that the parameters are the same, with the param function. >>> import sys >>> import yaml >>> from dewret.workflow import param >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @task () ... def increment ( num : int ) -> int : ... \"\"\"Increment an integer.\"\"\" ... return num + 1 >>> >>> @task () ... def double ( num : int ) -> int : ... \"\"\"Double an integer.\"\"\" ... return 2 * num >>> >>> @task () ... def mod10 ( num : int ) -> int : ... \"\"\"Take num mod 10.\"\"\" ... return num % 10 >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... \"\"\"Add two integers.\"\"\" ... return left + right >>> >>> num = param ( \"num\" , default = 3 ) >>> result = sum ( ... left = double ( num = increment ( num = num )), ... right = mod10 ( num = increment ( num = num )) ... ) >>> wkflw = construct ( result , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : num : default : 3 label : num type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : double - 1 : in : num : source : increment - 1 / out out : - out run : double increment - 1 : in : num : source : num out : - out run : increment mod10 - 1 : in : num : source : increment - 1 / out out : - out run : mod10 sum - 1 : in : left : source : double - 1 / out right : source : mod10 - 1 / out out : - out run : sum","title":"Dependencies"},{"location":"docs/workflows/#parameters","text":"The tool will spot global variables that you have used when building your tasks, and treat them as parameters. It will try to get the type from the typehint, or the value that you have set it to. This only works for basic types (and dict/lists of those). While global variables are implicit input to the Python function note that : in CWL, they will be rendered as explicit global input to a step as input, they are read-only, and must not be updated For example: >>> import sys >>> import yaml >>> from dewret.workflow import param >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> INPUT_NUM = 3 >>> @task () ... def rotate ( num : int ) -> int : ... \"\"\"Rotate an integer.\"\"\" ... return ( num + INPUT_NUM ) % INPUT_NUM >>> >>> result = rotate ( num = 5 ) >>> wkflw = construct ( result , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : INPUT_NUM : default : 3 label : INPUT_NUM type : int rotate - 1 - num : default : 5 label : num type : int outputs : out : label : out outputSource : rotate - 1 / out type : int steps : rotate - 1 : in : INPUT_NUM : source : INPUT_NUM num : source : rotate - 1 - num out : - out run : rotate","title":"Parameters"},{"location":"docs/workflows/#nested-tasks","text":"When you wish to combine tasks together programmatically, you can use nested tasks. These are run at render time, not execution time. In other words, they do not appear in the final graph, and so must only combine other tasks. For example: graph TD A[rotate] --> B[rotate] B[rotate] --> C[double_rotate] As code: >>> import sys >>> import yaml >>> from dewret.core import set_configuration >>> from dewret.tasks import task , construct , workflow >>> from dewret.renderers.cwl import render >>> INPUT_NUM = 3 >>> @task () ... def rotate ( num : int ) -> int : ... \"\"\"Rotate an integer.\"\"\" ... return ( num + INPUT_NUM ) % INPUT_NUM >>> >>> @workflow () ... def double_rotate ( num : int ) -> int : ... \"\"\"Rotate an integer twice.\"\"\" ... return rotate ( num = rotate ( num = num )) >>> >>> with set_configuration ( flatten_all_nested = True ): ... result = double_rotate ( num = 3 ) ... wkflw = construct ( result , simplify_ids = True ) ... cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : INPUT_NUM : default : 3 label : INPUT_NUM type : int num : default : 3 label : num type : int outputs : out : label : out outputSource : rotate - 1 / out type : int steps : rotate - 1 : in : INPUT_NUM : source : INPUT_NUM num : source : rotate - 2 / out out : - out run : rotate rotate - 2 : in : INPUT_NUM : source : INPUT_NUM num : source : num out : - out run : rotate Note that, as with all dewret calculations, only the steps necessary to achieve the ultimate output are included in the final graph. Therefore, nested tasks must return a step execution (task that is being called) that forces any other calculations you wish to happen. In other words, if a task in a nested task does not have an impact on the return value, it will disappear . For example, the following code renders the same workflow as in the previous example: @workflow () def double_rotate ( num : int ) -> int : \"\"\"Rotate an integer twice.\"\"\" unused_var = increment ( num = num ) return rotate ( num = rotate ( num = num ))","title":"Nested tasks"},{"location":"docs/workflows/#step-output-fields","text":"Each step, by default, is treated as having a single result. However, we allow a mechanism for specifying multiple fields, using attrs or dataclasses . Where needed, fields can be accessed outside of tasks by dot notation and dewret will map that access to a specific output field in CWL. Note that in the example below, shuffle is still only seen once in the graph: graph TD A[shuffle] --> B[hearts] A[shuffle] --> C[diamonds] B[hearts] --> D[sum] C[diamonds] --> D[sum] As code: >>> import sys >>> import yaml >>> from attrs import define >>> from numpy import random >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> red_total = sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> wkflw = construct ( red_total , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : shuffle - 1 - max_cards_per_suit : default : 13 label : max_cards_per_suit type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : shuffle - 1 : in : max_cards_per_suit : source : shuffle - 1 - max_cards_per_suit out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : shuffle sum - 1 : in : left : source : shuffle - 1 / hearts right : source : shuffle - 1 / diamonds out : - out run : sum Here, we show the same example with dataclasses . >>> import sys >>> import yaml >>> from dataclasses import dataclass >>> from numpy import random >>> from dewret.tasks import task , construct >>> from dewret.renderers.cwl import render >>> @dataclass ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> red_total = sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> wkflw = construct ( red_total , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : shuffle - 1 - max_cards_per_suit : default : 13 label : max_cards_per_suit type : int outputs : out : label : out outputSource : sum - 1 / out type : int steps : shuffle - 1 : in : max_cards_per_suit : source : shuffle - 1 - max_cards_per_suit out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : shuffle sum - 1 : in : left : source : shuffle - 1 / hearts right : source : shuffle - 1 / diamonds out : - out run : sum","title":"Step Output Fields"},{"location":"docs/workflows/#subworkflow","text":"A special form of nested task is available to help divide up more complex workflows: the subworkflow . By wrapping logic in subflows, dewret will produce multiple output workflows that reference each other. >>> import sys >>> import yaml >>> from attrs import define >>> from numpy import random >>> from dewret.tasks import task , construct , workflow >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @workflow () ... def red_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> @workflow () ... def black_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . spades , ... right = shuffle ( max_cards_per_suit = 13 ) . clubs ... ) >>> total = sum ( left = red_total (), right = black_total ()) >>> wkflw = construct ( total , simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : {} outputs : out : label : out outputSource : sum - 1 / out type : int steps : black_total - 1 : in : {} out : - out run : black_total red_total - 1 : in : {} out : - out run : red_total sum - 1 : in : left : source : red_total - 1 / out right : source : black_total - 1 / out out : - out run : sum As we have used subworkflow to wrap the colour totals, the outer workflow contains references to them only. The subworkflows are now returned by render as a second term. >>> import sys >>> import yaml >>> from attrs import define >>> from numpy import random >>> from dewret.tasks import task , construct , workflow >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> @task () ... def shuffle ( max_cards_per_suit : int ) -> PackResult : ... \"\"\"Fill a random pile from a card deck, suit by suit.\"\"\" ... return PackResult ( ... hearts = random . randint ( max_cards_per_suit ), ... clubs = random . randint ( max_cards_per_suit ), ... spades = random . randint ( max_cards_per_suit ), ... diamonds = random . randint ( max_cards_per_suit ) ... ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @workflow () ... def red_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . hearts , ... right = shuffle ( max_cards_per_suit = 13 ) . diamonds ... ) >>> @workflow () ... def black_total () -> int : ... return sum ( ... left = shuffle ( max_cards_per_suit = 13 ) . spades , ... right = shuffle ( max_cards_per_suit = 13 ) . clubs ... ) >>> total = sum ( left = red_total (), right = black_total ()) >>> wkflw = construct ( total , simplify_ids = True ) >>> cwl = render ( wkflw ) >>> yaml . dump ( cwl [ \"red_total-1\" ], sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : {} outputs : out : label : out outputSource : sum - 1 - 1 / out type : int steps : shuffle - 1 - 1 : in : max_cards_per_suit : default : 13 out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : shuffle sum - 1 - 1 : in : left : source : shuffle - 1 - 1 / hearts right : source : shuffle - 1 - 1 / diamonds out : - out run : sum","title":"Subworkflow"},{"location":"docs/workflows/#input-factories","text":"Sometimes we want to take complex Python input, not just raw types. Not all serialization support this, but the factory function lets us wrap a simple call, usually a constructor, that takes only raw arguments. This can then rendered as either a step or a parameter depending on whether the chosen renderer has the capability. Below is the default output, treating Pack as a task. >>> import sys >>> import yaml >>> from dewret.tasks import workflow , factory , workflow , construct , task >>> from attrs import define >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> Pack = factory ( PackResult ) >>> >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @workflow () ... def black_total ( pack : PackResult ) -> int : ... return sum ( ... left = pack . spades , ... right = pack . clubs ... ) >>> pack = Pack ( hearts = 13 , spades = 13 , diamonds = 13 , clubs = 13 ) >>> wkflw = construct ( black_total ( pack = pack ), simplify_ids = True ) >>> cwl = render ( wkflw )[ \"__root__\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : PackResult - 1 - clubs : default : 13 label : clubs type : int PackResult - 1 - diamonds : default : 13 label : diamonds type : int PackResult - 1 - hearts : default : 13 label : hearts type : int PackResult - 1 - spades : default : 13 label : spades type : int outputs : out : label : out outputSource : black_total - 1 / out type : int steps : PackResult - 1 : in : clubs : source : PackResult - 1 - clubs diamonds : source : PackResult - 1 - diamonds hearts : source : PackResult - 1 - hearts spades : source : PackResult - 1 - spades out : clubs : label : clubs type : int diamonds : label : diamonds type : int hearts : label : hearts type : int spades : label : spades type : int run : PackResult black_total - 1 : in : pack : source : PackResult - 1 / out out : - out run : black_total The CWL renderer is also able to treat pack as a parameter, if complex types are allowed. >>> import sys >>> import yaml >>> from dewret.tasks import task , factory , workflow , construct >>> from attrs import define >>> from dewret.renderers.cwl import render >>> @define ... class PackResult : ... hearts : int ... clubs : int ... spades : int ... diamonds : int >>> >>> Pack = factory ( PackResult ) >>> @task () ... def sum ( left : int , right : int ) -> int : ... return left + right >>> >>> @workflow () ... def black_total ( pack : PackResult ) -> int : ... return sum ( ... left = pack . spades , ... right = pack . clubs ... ) >>> pack = Pack ( hearts = 13 , spades = 13 , diamonds = 13 , clubs = 13 ) >>> wkflw = construct ( black_total ( pack = pack ), simplify_ids = True ) >>> cwl = render ( wkflw , allow_complex_types = True , factories_as_params = True )[ \"black_total-1\" ] >>> yaml . dump ( cwl , sys . stdout , indent = 2 ) class : Workflow cwlVersion : 1.2 inputs : pack : label : pack type : record outputs : out : label : out outputSource : sum - 1 - 1 / out type : int steps : sum - 1 - 1 : in : left : source : pack / spades right : source : pack / clubs out : - out run : sum","title":"Input Factories"},{"location":"reference/dewret/","text":"Module dewret dewret: DEclarative Workflow REndering Tool. Dewret allows certain workflows written in a dynamic style to be rendered to a static representation. Sub-modules dewret.annotations dewret.backends dewret.core dewret.render dewret.renderers dewret.tasks dewret.utils dewret.workflow","title":"Index"},{"location":"reference/dewret/#module-dewret","text":"dewret: DEclarative Workflow REndering Tool. Dewret allows certain workflows written in a dynamic style to be rendered to a static representation.","title":"Module dewret"},{"location":"reference/dewret/#sub-modules","text":"dewret.annotations dewret.backends dewret.core dewret.render dewret.renderers dewret.tasks dewret.utils dewret.workflow","title":"Sub-modules"},{"location":"reference/dewret/annotations/","text":"Module dewret.annotations Tooling for managing annotations. Provides FunctionAnalyser , a toolkit that takes a Callable and can interrogate it for annotations, with some intelligent searching beyond the obvious location. Variables AtRender Fixed T Classes FunctionAnalyser class FunctionAnalyser ( fn : Callable [ ... , Any ] ) Convenience class for analysing a function with reduced duplication of effort. Attributes Name Type Description Default _fn None the wrapped callable None _annotations None stored annotations for the function. None Instance variables free_vars Get the free variables for this Callable. globals Get the globals for this Callable. return_type Return type of the callable. Returns: expected type of the return value. Methods argument_has def argument_has ( self , arg : str , annotation : type , exhaustive : bool = False ) -> bool Check if the named argument has the given annotation. Parameters: Name Type Description Default arg None argument to retrieve. None annotation None Annotated to search for. None exhaustive None whether to check the globals and other modules. None get_all_imported_names def get_all_imported_names ( self ) -> dict [ str , tuple [ module , str ]] Find all of the annotations that were imported into this module. get_all_module_names def get_all_module_names ( self ) -> dict [ str , typing . Any ] Find all of the annotations within this module. get_argument_annotation def get_argument_annotation ( self , arg : str , exhaustive : bool = False ) -> Any Retrieve the annotations for this argument. Parameters: Name Type Description Default arg None name of the argument. None exhaustive None True if we should search outside the function itself, into the module globals, and into imported modules. None is_at_construct_arg def is_at_construct_arg ( self , arg : str , exhaustive : bool = False ) -> bool Convience function to check for AtConstruct , wrapping FunctionAnalyser.argument_has . with_new_globals def with_new_globals ( self , new_globals : dict [ str , typing . Any ] ) -> Callable [ ... , Any ] Create a Callable that will run the current Callable with new globals.","title":"Annotations"},{"location":"reference/dewret/annotations/#module-dewretannotations","text":"Tooling for managing annotations. Provides FunctionAnalyser , a toolkit that takes a Callable and can interrogate it for annotations, with some intelligent searching beyond the obvious location.","title":"Module dewret.annotations"},{"location":"reference/dewret/annotations/#variables","text":"AtRender Fixed T","title":"Variables"},{"location":"reference/dewret/annotations/#classes","text":"","title":"Classes"},{"location":"reference/dewret/annotations/#functionanalyser","text":"class FunctionAnalyser ( fn : Callable [ ... , Any ] ) Convenience class for analysing a function with reduced duplication of effort.","title":"FunctionAnalyser"},{"location":"reference/dewret/annotations/#attributes","text":"Name Type Description Default _fn None the wrapped callable None _annotations None stored annotations for the function. None","title":"Attributes"},{"location":"reference/dewret/annotations/#instance-variables","text":"free_vars Get the free variables for this Callable. globals Get the globals for this Callable. return_type Return type of the callable. Returns: expected type of the return value.","title":"Instance variables"},{"location":"reference/dewret/annotations/#methods","text":"","title":"Methods"},{"location":"reference/dewret/annotations/#argument_has","text":"def argument_has ( self , arg : str , annotation : type , exhaustive : bool = False ) -> bool Check if the named argument has the given annotation. Parameters: Name Type Description Default arg None argument to retrieve. None annotation None Annotated to search for. None exhaustive None whether to check the globals and other modules. None","title":"argument_has"},{"location":"reference/dewret/annotations/#get_all_imported_names","text":"def get_all_imported_names ( self ) -> dict [ str , tuple [ module , str ]] Find all of the annotations that were imported into this module.","title":"get_all_imported_names"},{"location":"reference/dewret/annotations/#get_all_module_names","text":"def get_all_module_names ( self ) -> dict [ str , typing . Any ] Find all of the annotations within this module.","title":"get_all_module_names"},{"location":"reference/dewret/annotations/#get_argument_annotation","text":"def get_argument_annotation ( self , arg : str , exhaustive : bool = False ) -> Any Retrieve the annotations for this argument. Parameters: Name Type Description Default arg None name of the argument. None exhaustive None True if we should search outside the function itself, into the module globals, and into imported modules. None","title":"get_argument_annotation"},{"location":"reference/dewret/annotations/#is_at_construct_arg","text":"def is_at_construct_arg ( self , arg : str , exhaustive : bool = False ) -> bool Convience function to check for AtConstruct , wrapping FunctionAnalyser.argument_has .","title":"is_at_construct_arg"},{"location":"reference/dewret/annotations/#with_new_globals","text":"def with_new_globals ( self , new_globals : dict [ str , typing . Any ] ) -> Callable [ ... , Any ] Create a Callable that will run the current Callable with new globals.","title":"with_new_globals"},{"location":"reference/dewret/core/","text":"Module dewret.core Base classes that need to be available everywhere. Mainly tooling around configuration, protocols and superclasses for References and WorkflowComponents, that are concretized elsewhere. Variables BasicType CONFIGURATION ExprType FirmType RawType RenderConfiguration T U Functions default_construct_config def default_construct_config ( ) -> dewret . core . ConstructConfiguration Gets the default construct-time configuration. This is the primary mechanism for configuring dewret internals, so these defaults should be carefully chosen and, if they change, that likely has an impact on backwards compatibility from a SemVer perspective. Returns: configuration dictionary with default construct values. default_renderer_config def default_renderer_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Gets the default renderer configuration. This may be called frequently, but is cached so note that any changes to the wrapped config function will not be reflected during the process. It is a light wrapper for default_config in the supplier renderer module. Returns: the default configuration dict for the chosen renderer. get_configuration def get_configuration ( key : str ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Retrieve the configuration or (silently) return the default. Helps avoid a proliferation of set_configuration calls by not erroring if it has not been called. However, the cost is that the user may accidentally put configuration-affected logic outside a set_configuration call and be surprised that the behaviour is inexplicibly not as expected. Parameters: Name Type Description Default key None configuration key to retrieve. None get_render_configuration def get_render_configuration ( key : str ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Retrieve configuration for the active renderer. Finds the current user-set configuration, defaulting back to the chosen renderer module's declared defaults. Parameters: Name Type Description Default key None configuration key to retrieve. None set_configuration def set_configuration ( ** kwargs : *< class ' dewret . core . ConstructConfigurationTypedDict '> ) -> Iterator [ _contextvars . ContextVar [ dewret . core . GlobalConfiguration ]] Sets the construct-time configuration. This is a context manager, so that a setting can be temporarily overridden and automatically restored. set_render_configuration def set_render_configuration ( kwargs : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) -> Iterator [ _contextvars . ContextVar [ dewret . core . GlobalConfiguration ]] Sets the render-time configuration. This is a context manager, so that a setting can be temporarily overridden and automatically restored. Returns: the yielded global configuration ContextVar. strip_annotations def strip_annotations ( parent_type : type ) -> tuple [ type , tuple [ str ]] Discovers and removes annotations from a parent type. Parameters: Name Type Description Default parent_type None a type, possibly Annotated. None Classes BaseRenderModule class BaseRenderModule ( * args , ** kwargs ) Common routines for all renderer modules. Ancestors (in MRO) typing.Protocol typing.Generic Descendants dewret.core.RawRenderModule dewret.core.StructuredRenderModule Static methods default_config def default_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Retrieve default settings. These will not change during execution, but can be overridden by dewret.core.set_render_configuration . Returns: a static, serializable dict. ConstructConfiguration class ConstructConfiguration ( flatten_all_nested : bool = False , allow_positional_args : bool = False , allow_plain_dict_fields : bool = False , field_separator : str = '/' , field_index_types : str = 'int' , simplify_ids : bool = False ) Basic configuration of the construction process. Holds configuration that may be relevant to construst(...) calls or, realistically, anything prior to rendering. It should hold generic configuration that is renderer-independent. Instance variables allow_plain_dict_fields allow_positional_args field_index_types field_separator flatten_all_nested simplify_ids ConstructConfigurationTypedDict class ConstructConfigurationTypedDict ( / , * args , ** kwargs ) Basic configuration of the construction process. Holds configuration that may be relevant to construst(...) calls or, realistically, anything prior to rendering. It should hold generic configuration that is renderer-independent. THIS MUST BE KEPT IDENTICAL TO ConstructConfiguration. Ancestors (in MRO) builtins.dict Methods clear def clear ( ... ) D.clear() -> None. Remove all items from D. copy def copy ( ... ) D.copy() -> a shallow copy of D fromkeys def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value. get def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default. items def items ( ... ) D.items() -> a set-like object providing a view on D's items keys def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys pop def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError. popitem def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty. setdefault def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default. update def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k] values def values ( ... ) D.values() -> an object providing a view on D's values GlobalConfiguration class GlobalConfiguration ( construct : dewret . core . ConstructConfiguration , render : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) Overall configuration structure. Having a single configuration dict allows us to manage only one ContextVar. Instance variables construct render IterableMixin class IterableMixin ( typ : type [ ~ U ] | None = None , ** kwargs : Any ) Functionality for iterating over references to give new references. Ancestors (in MRO) dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent Descendants dewret.workflow.IterableParameterReference dewret.workflow.IterableStepReference Class variables default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. IteratedGenerator class IteratedGenerator ( to_wrap : dewret . core . Reference [ ~ U ] ) Sentinel value for capturing that an iteration has occured without performing it. Allows us to lazily evaluate a loop, for instance, in the renderer. This may be relevant if the renderer wishes to postpone iteration to runtime, and simply record it is required, rather than evaluating the iterator. Ancestors (in MRO) typing.Generic Raw class Raw ( value : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] ) Value object for any raw types. This is able to hash raw types consistently and provides a single type for validating type-consistency. Attributes Name Type Description Default value None the real value, e.g. a str , int , ... None RawRenderModule class RawRenderModule ( * args , ** kwargs ) Render module that returns raw text. Ancestors (in MRO) dewret.core.BaseRenderModule typing.Protocol typing.Generic Static methods default_config def default_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Retrieve default settings. These will not change during execution, but can be overridden by dewret.core.set_render_configuration . Returns: a static, serializable dict. Methods render_raw def render_raw ( self , workflow : dewret . core . WorkflowProtocol , ** kwargs : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) -> dict [ str , str ] Turn a workflow into flat strings. Returns: one or more subworkflows with a __root__ key representing the outermost workflow, at least. Reference class Reference ( * args : Any , typ : type [ ~ U ] | None = None , ** kwargs : Any ) Superclass for all symbolic references to values. Ancestors (in MRO) typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent Descendants dewret.core.IterableMixin dewret.workflow.ParameterReference dewret.workflow.StepReference Class variables default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. RenderCall class RenderCall ( * args , ** kwargs ) Callable that will render out workflow(s). Ancestors (in MRO) typing.Protocol typing.Generic StructuredRenderModule class StructuredRenderModule ( * args , ** kwargs ) Render module that returns JSON/YAML-serializable structures. Ancestors (in MRO) dewret.core.BaseRenderModule typing.Protocol typing.Generic Static methods default_config def default_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Retrieve default settings. These will not change during execution, but can be overridden by dewret.core.set_render_configuration . Returns: a static, serializable dict. Methods render def render ( self , workflow : dewret . core . WorkflowProtocol , ** kwargs : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) -> dict [ str , dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]]] Turn a workflow into a serializable structure. Returns: one or more subworkflows with a __root__ key representing the outermost workflow, at least. UnevaluatableError class UnevaluatableError ( / , * args , ** kwargs ) Signposts that a user has tried to treat a reference as the real (runtime) value. For example, by comparing to a concrete integer or value, etc. Ancestors (in MRO) builtins.Exception builtins.BaseException Class variables args Methods add_note def add_note ( ... ) Exception.add_note(note) -- add a note to the exception with_traceback def with_traceback ( ... ) Exception.with_traceback(tb) -- set self. traceback to tb and return self. WorkflowComponent class WorkflowComponent ( * args : Any , workflow : dewret . core . WorkflowProtocol , ** kwargs : Any ) Base class for anything directly tied to an individual Workflow . Attributes Name Type Description Default workflow None the Workflow that this is tied to. None Descendants dewret.core.Reference dewret.workflow.BaseStep dewret.workflow.ParameterReference WorkflowProtocol class WorkflowProtocol ( * args , ** kwargs ) Expected structure for a workflow. We do not expect various workflow implementations, but this allows us to define the interface expected by the core classes. Ancestors (in MRO) typing.Protocol typing.Generic Methods remap def remap ( self , name : str ) -> str Perform any name-changing for steps, etc. in the workflow. This enables, for example, simplifying all the IDs to an integer sequence. Returns: remapped name. set_result def set_result ( self , result : sympy . core . basic . Basic | list [ sympy . core . basic . Basic ] | tuple [ sympy . core . basic . Basic ] ) -> None Set the step that should produce a result for the overall workflow. simplify_ids def simplify_ids ( self , infix : list [ str ] | None = None ) -> None Drop the non-human-readable IDs if possible, in favour of integer sequences. Parameters: Name Type Description Default infix None any inherited intermediary identifiers, to allow nesting, or None. None","title":"Core"},{"location":"reference/dewret/core/#module-dewretcore","text":"Base classes that need to be available everywhere. Mainly tooling around configuration, protocols and superclasses for References and WorkflowComponents, that are concretized elsewhere.","title":"Module dewret.core"},{"location":"reference/dewret/core/#variables","text":"BasicType CONFIGURATION ExprType FirmType RawType RenderConfiguration T U","title":"Variables"},{"location":"reference/dewret/core/#functions","text":"","title":"Functions"},{"location":"reference/dewret/core/#default_construct_config","text":"def default_construct_config ( ) -> dewret . core . ConstructConfiguration Gets the default construct-time configuration. This is the primary mechanism for configuring dewret internals, so these defaults should be carefully chosen and, if they change, that likely has an impact on backwards compatibility from a SemVer perspective. Returns: configuration dictionary with default construct values.","title":"default_construct_config"},{"location":"reference/dewret/core/#default_renderer_config","text":"def default_renderer_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Gets the default renderer configuration. This may be called frequently, but is cached so note that any changes to the wrapped config function will not be reflected during the process. It is a light wrapper for default_config in the supplier renderer module. Returns: the default configuration dict for the chosen renderer.","title":"default_renderer_config"},{"location":"reference/dewret/core/#get_configuration","text":"def get_configuration ( key : str ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Retrieve the configuration or (silently) return the default. Helps avoid a proliferation of set_configuration calls by not erroring if it has not been called. However, the cost is that the user may accidentally put configuration-affected logic outside a set_configuration call and be surprised that the behaviour is inexplicibly not as expected. Parameters: Name Type Description Default key None configuration key to retrieve. None","title":"get_configuration"},{"location":"reference/dewret/core/#get_render_configuration","text":"def get_render_configuration ( key : str ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Retrieve configuration for the active renderer. Finds the current user-set configuration, defaulting back to the chosen renderer module's declared defaults. Parameters: Name Type Description Default key None configuration key to retrieve. None","title":"get_render_configuration"},{"location":"reference/dewret/core/#set_configuration","text":"def set_configuration ( ** kwargs : *< class ' dewret . core . ConstructConfigurationTypedDict '> ) -> Iterator [ _contextvars . ContextVar [ dewret . core . GlobalConfiguration ]] Sets the construct-time configuration. This is a context manager, so that a setting can be temporarily overridden and automatically restored.","title":"set_configuration"},{"location":"reference/dewret/core/#set_render_configuration","text":"def set_render_configuration ( kwargs : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) -> Iterator [ _contextvars . ContextVar [ dewret . core . GlobalConfiguration ]] Sets the render-time configuration. This is a context manager, so that a setting can be temporarily overridden and automatically restored. Returns: the yielded global configuration ContextVar.","title":"set_render_configuration"},{"location":"reference/dewret/core/#strip_annotations","text":"def strip_annotations ( parent_type : type ) -> tuple [ type , tuple [ str ]] Discovers and removes annotations from a parent type. Parameters: Name Type Description Default parent_type None a type, possibly Annotated. None","title":"strip_annotations"},{"location":"reference/dewret/core/#classes","text":"","title":"Classes"},{"location":"reference/dewret/core/#baserendermodule","text":"class BaseRenderModule ( * args , ** kwargs ) Common routines for all renderer modules.","title":"BaseRenderModule"},{"location":"reference/dewret/core/#ancestors-in-mro","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#descendants","text":"dewret.core.RawRenderModule dewret.core.StructuredRenderModule","title":"Descendants"},{"location":"reference/dewret/core/#static-methods","text":"","title":"Static methods"},{"location":"reference/dewret/core/#default_config","text":"def default_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Retrieve default settings. These will not change during execution, but can be overridden by dewret.core.set_render_configuration . Returns: a static, serializable dict.","title":"default_config"},{"location":"reference/dewret/core/#constructconfiguration","text":"class ConstructConfiguration ( flatten_all_nested : bool = False , allow_positional_args : bool = False , allow_plain_dict_fields : bool = False , field_separator : str = '/' , field_index_types : str = 'int' , simplify_ids : bool = False ) Basic configuration of the construction process. Holds configuration that may be relevant to construst(...) calls or, realistically, anything prior to rendering. It should hold generic configuration that is renderer-independent.","title":"ConstructConfiguration"},{"location":"reference/dewret/core/#instance-variables","text":"allow_plain_dict_fields allow_positional_args field_index_types field_separator flatten_all_nested simplify_ids","title":"Instance variables"},{"location":"reference/dewret/core/#constructconfigurationtypeddict","text":"class ConstructConfigurationTypedDict ( / , * args , ** kwargs ) Basic configuration of the construction process. Holds configuration that may be relevant to construst(...) calls or, realistically, anything prior to rendering. It should hold generic configuration that is renderer-independent. THIS MUST BE KEPT IDENTICAL TO ConstructConfiguration.","title":"ConstructConfigurationTypedDict"},{"location":"reference/dewret/core/#ancestors-in-mro_1","text":"builtins.dict","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#methods","text":"","title":"Methods"},{"location":"reference/dewret/core/#clear","text":"def clear ( ... ) D.clear() -> None. Remove all items from D.","title":"clear"},{"location":"reference/dewret/core/#copy","text":"def copy ( ... ) D.copy() -> a shallow copy of D","title":"copy"},{"location":"reference/dewret/core/#fromkeys","text":"def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value.","title":"fromkeys"},{"location":"reference/dewret/core/#get","text":"def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default.","title":"get"},{"location":"reference/dewret/core/#items","text":"def items ( ... ) D.items() -> a set-like object providing a view on D's items","title":"items"},{"location":"reference/dewret/core/#keys","text":"def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys","title":"keys"},{"location":"reference/dewret/core/#pop","text":"def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError.","title":"pop"},{"location":"reference/dewret/core/#popitem","text":"def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty.","title":"popitem"},{"location":"reference/dewret/core/#setdefault","text":"def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default.","title":"setdefault"},{"location":"reference/dewret/core/#update","text":"def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k]","title":"update"},{"location":"reference/dewret/core/#values","text":"def values ( ... ) D.values() -> an object providing a view on D's values","title":"values"},{"location":"reference/dewret/core/#globalconfiguration","text":"class GlobalConfiguration ( construct : dewret . core . ConstructConfiguration , render : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) Overall configuration structure. Having a single configuration dict allows us to manage only one ContextVar.","title":"GlobalConfiguration"},{"location":"reference/dewret/core/#instance-variables_1","text":"construct render","title":"Instance variables"},{"location":"reference/dewret/core/#iterablemixin","text":"class IterableMixin ( typ : type [ ~ U ] | None = None , ** kwargs : Any ) Functionality for iterating over references to give new references.","title":"IterableMixin"},{"location":"reference/dewret/core/#ancestors-in-mro_2","text":"dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#descendants_1","text":"dewret.workflow.IterableParameterReference dewret.workflow.IterableStepReference","title":"Descendants"},{"location":"reference/dewret/core/#class-variables","text":"default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/core/#static-methods_1","text":"","title":"Static methods"},{"location":"reference/dewret/core/#class_key","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/core/#fromiter","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/core/#examples","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/core/#instance-variables_2","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/core/#examples_1","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/core/#notes","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/core/#examples_2","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/core/#examples_3","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference.","title":"Examples"},{"location":"reference/dewret/core/#methods_1","text":"","title":"Methods"},{"location":"reference/dewret/core/#adjoint","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/core/#apart","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/core/#args_cnc","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/core/#explanation","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/core/#examples_4","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/core/#as_base_exp","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/core/#as_coeff_add","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/core/#as_coeff_mul","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/core/#as_coeff_add_1","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/core/#as_coeff_exponent","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/core/#as_coeff_mul_1","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/core/#as_coefficient","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/core/#examples_5","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/core/#see-also","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/core/#as_coefficients_dict","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/core/#examples_6","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/core/#as_content_primitive","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/core/#examples_7","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/core/#as_dummy","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/core/#examples_8","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/core/#notes_1","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/core/#as_expr","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/core/#examples_9","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/core/#as_independent","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/core/#examples_10","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/core/#see-also_1","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/core/#as_leading_term","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/core/#examples_11","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/core/#as_numer_denom","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/core/#see-also_2","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/core/#as_ordered_factors","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/core/#as_ordered_terms","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/core/#examples_12","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/core/#as_poly","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/core/#explanation_1","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/core/#as_powers_dict","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/core/#see-also_3","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/core/#as_real_imag","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/core/#as_set","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/core/#examples_13","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/core/#as_terms","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/core/#aseries","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/core/#parameters","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/core/#examples_14","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/core/#returns","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/core/#notes_2","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/core/#references","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/core/#see-also_4","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/core/#atoms","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/core/#examples_15","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/core/#cancel","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/core/#coeff","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/core/#explanation_2","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/core/#examples_16","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/core/#see-also_5","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/core/#collect","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/core/#combsimp","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/core/#compare","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/core/#examples_17","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/core/#compute_leading_term","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/core/#conjugate","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/core/#copy_1","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/core/#could_extract_minus_sign","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/core/#examples_18","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/core/#count","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/core/#count_ops","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/core/#diff","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/core/#dir","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/core/#doit","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/core/#dummy_eq","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/core/#examples_19","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/core/#equals","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/core/#explanation_3","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/core/#evalf","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/core/#parameters_1","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/core/#notes_3","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/core/#expand","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/core/#extract_additively","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/core/#examples_20","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/core/#see-also_6","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/core/#extract_branch_factor","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/core/#extract_multiplicatively","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/core/#examples_21","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/core/#factor","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/core/#find","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/core/#fourier_series","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/core/#fps","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/core/#gammasimp","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/core/#geto","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/core/#getn","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/core/#explanation_4","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/core/#examples_22","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/core/#has","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/core/#examples_23","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/core/#has_free","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/core/#examples_24","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/core/#has_xfree","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/core/#examples_25","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/core/#integrate","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/core/#invert","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/core/#see-also_7","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/core/#is_algebraic_expr","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/core/#examples_26","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/core/#see-also_8","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/core/#references_1","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/core/#is_constant","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/core/#explanation_5","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/core/#examples_27","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/core/#is_hypergeometric","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/core/#is_meromorphic","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/core/#examples_28","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/core/#is_polynomial","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/core/#examples_29","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/core/#is_rational_function","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/core/#examples_30","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/core/#is_same","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/core/#examples_31","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/core/#leadterm","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/core/#examples_32","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/core/#limit","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/core/#lseries","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/core/#match","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/core/#examples_33","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/core/#matches","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/core/#examples_34","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/core/#n","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/core/#parameters_2","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/core/#notes_4","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/core/#normal","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/core/#see-also_9","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/core/#nseries","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/core/#examples_35","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/core/#nsimplify","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/core/#powsimp","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/core/#primitive","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/core/#examples_36","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/core/#radsimp","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/core/#ratsimp","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/core/#rcall","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/core/#refine","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/core/#removeo","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/core/#replace","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/core/#examples_37","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/core/#see-also_10","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/core/#rewrite","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/core/#parameters_3","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/core/#examples_38","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/core/#round","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/core/#examples_39","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/core/#notes_5","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/core/#separate","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/core/#series","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/core/#parameters_4","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/core/#examples_40","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/core/#returns_1","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/core/#raises","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/core/#simplify","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/core/#sort_key","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/core/#examples_41","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/core/#subs","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/core/#examples_42","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/core/#see-also_11","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/core/#taylor_term","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/core/#to_nnf","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/core/#together","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/core/#transpose","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/core/#trigsimp","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/core/#xreplace","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/core/#parameters_5","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/core/#returns_2","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/core/#examples_43","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/core/#see-also_12","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/core/#iteratedgenerator","text":"class IteratedGenerator ( to_wrap : dewret . core . Reference [ ~ U ] ) Sentinel value for capturing that an iteration has occured without performing it. Allows us to lazily evaluate a loop, for instance, in the renderer. This may be relevant if the renderer wishes to postpone iteration to runtime, and simply record it is required, rather than evaluating the iterator.","title":"IteratedGenerator"},{"location":"reference/dewret/core/#ancestors-in-mro_3","text":"typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#raw","text":"class Raw ( value : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] ) Value object for any raw types. This is able to hash raw types consistently and provides a single type for validating type-consistency.","title":"Raw"},{"location":"reference/dewret/core/#attributes","text":"Name Type Description Default value None the real value, e.g. a str , int , ... None","title":"Attributes"},{"location":"reference/dewret/core/#rawrendermodule","text":"class RawRenderModule ( * args , ** kwargs ) Render module that returns raw text.","title":"RawRenderModule"},{"location":"reference/dewret/core/#ancestors-in-mro_4","text":"dewret.core.BaseRenderModule typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#static-methods_2","text":"","title":"Static methods"},{"location":"reference/dewret/core/#default_config_1","text":"def default_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Retrieve default settings. These will not change during execution, but can be overridden by dewret.core.set_render_configuration . Returns: a static, serializable dict.","title":"default_config"},{"location":"reference/dewret/core/#methods_2","text":"","title":"Methods"},{"location":"reference/dewret/core/#render_raw","text":"def render_raw ( self , workflow : dewret . core . WorkflowProtocol , ** kwargs : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) -> dict [ str , str ] Turn a workflow into flat strings. Returns: one or more subworkflows with a __root__ key representing the outermost workflow, at least.","title":"render_raw"},{"location":"reference/dewret/core/#reference","text":"class Reference ( * args : Any , typ : type [ ~ U ] | None = None , ** kwargs : Any ) Superclass for all symbolic references to values.","title":"Reference"},{"location":"reference/dewret/core/#ancestors-in-mro_5","text":"typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#descendants_2","text":"dewret.core.IterableMixin dewret.workflow.ParameterReference dewret.workflow.StepReference","title":"Descendants"},{"location":"reference/dewret/core/#class-variables_1","text":"default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/core/#static-methods_3","text":"","title":"Static methods"},{"location":"reference/dewret/core/#class_key_1","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/core/#fromiter_1","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/core/#examples_44","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/core/#instance-variables_3","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/core/#examples_45","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/core/#notes_6","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/core/#examples_46","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/core/#examples_47","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference.","title":"Examples"},{"location":"reference/dewret/core/#methods_3","text":"","title":"Methods"},{"location":"reference/dewret/core/#adjoint_1","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/core/#apart_1","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/core/#args_cnc_1","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/core/#explanation_6","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/core/#examples_48","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/core/#as_base_exp_1","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/core/#as_coeff_add_2","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/core/#as_coeff_mul_2","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/core/#as_coeff_add_3","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/core/#as_coeff_exponent_1","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/core/#as_coeff_mul_3","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/core/#as_coefficient_1","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/core/#examples_49","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/core/#see-also_13","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/core/#as_coefficients_dict_1","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/core/#examples_50","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/core/#as_content_primitive_1","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/core/#examples_51","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/core/#as_dummy_1","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/core/#examples_52","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/core/#notes_7","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/core/#as_expr_1","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/core/#examples_53","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/core/#as_independent_1","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/core/#examples_54","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/core/#see-also_14","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/core/#as_leading_term_1","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/core/#examples_55","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/core/#as_numer_denom_1","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/core/#see-also_15","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/core/#as_ordered_factors_1","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/core/#as_ordered_terms_1","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/core/#examples_56","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/core/#as_poly_1","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/core/#explanation_7","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/core/#as_powers_dict_1","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/core/#see-also_16","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/core/#as_real_imag_1","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/core/#as_set_1","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/core/#examples_57","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/core/#as_terms_1","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/core/#aseries_1","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/core/#parameters_6","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/core/#examples_58","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/core/#returns_3","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/core/#notes_8","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/core/#references_2","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/core/#see-also_17","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/core/#atoms_1","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/core/#examples_59","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/core/#cancel_1","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/core/#coeff_1","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/core/#explanation_8","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/core/#examples_60","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/core/#see-also_18","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/core/#collect_1","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/core/#combsimp_1","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/core/#compare_1","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/core/#examples_61","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/core/#compute_leading_term_1","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/core/#conjugate_1","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/core/#copy_2","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/core/#could_extract_minus_sign_1","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/core/#examples_62","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/core/#count_1","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/core/#count_ops_1","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/core/#diff_1","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/core/#dir_1","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/core/#doit_1","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/core/#dummy_eq_1","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/core/#examples_63","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/core/#equals_1","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/core/#explanation_9","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/core/#evalf_1","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/core/#parameters_7","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/core/#notes_9","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/core/#expand_1","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/core/#extract_additively_1","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/core/#examples_64","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/core/#see-also_19","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/core/#extract_branch_factor_1","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/core/#extract_multiplicatively_1","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/core/#examples_65","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/core/#factor_1","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/core/#find_1","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/core/#fourier_series_1","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/core/#fps_1","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/core/#gammasimp_1","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/core/#geto_1","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/core/#getn_1","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/core/#explanation_10","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/core/#examples_66","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/core/#has_1","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/core/#examples_67","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/core/#has_free_1","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/core/#examples_68","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/core/#has_xfree_1","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/core/#examples_69","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/core/#integrate_1","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/core/#invert_1","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/core/#see-also_20","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/core/#is_algebraic_expr_1","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/core/#examples_70","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/core/#see-also_21","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/core/#references_3","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/core/#is_constant_1","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/core/#explanation_11","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/core/#examples_71","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/core/#is_hypergeometric_1","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/core/#is_meromorphic_1","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/core/#examples_72","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/core/#is_polynomial_1","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/core/#examples_73","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/core/#is_rational_function_1","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/core/#examples_74","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/core/#is_same_1","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/core/#examples_75","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/core/#leadterm_1","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/core/#examples_76","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/core/#limit_1","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/core/#lseries_1","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/core/#match_1","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/core/#examples_77","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/core/#matches_1","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/core/#examples_78","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/core/#n_1","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/core/#parameters_8","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/core/#notes_10","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/core/#normal_1","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/core/#see-also_22","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/core/#nseries_1","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/core/#examples_79","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/core/#nsimplify_1","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/core/#powsimp_1","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/core/#primitive_1","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/core/#examples_80","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/core/#radsimp_1","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/core/#ratsimp_1","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/core/#rcall_1","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/core/#refine_1","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/core/#removeo_1","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/core/#replace_1","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/core/#examples_81","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/core/#see-also_23","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/core/#rewrite_1","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/core/#parameters_9","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/core/#examples_82","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/core/#round_1","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/core/#examples_83","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/core/#notes_11","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/core/#separate_1","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/core/#series_1","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/core/#parameters_10","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/core/#examples_84","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/core/#returns_4","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/core/#raises_1","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/core/#simplify_1","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/core/#sort_key_1","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/core/#examples_85","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/core/#subs_1","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/core/#examples_86","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/core/#see-also_24","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/core/#taylor_term_1","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/core/#to_nnf_1","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/core/#together_1","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/core/#transpose_1","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/core/#trigsimp_1","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/core/#xreplace_1","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/core/#parameters_11","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/core/#returns_5","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/core/#examples_87","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/core/#see-also_25","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/core/#rendercall","text":"class RenderCall ( * args , ** kwargs ) Callable that will render out workflow(s).","title":"RenderCall"},{"location":"reference/dewret/core/#ancestors-in-mro_6","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#structuredrendermodule","text":"class StructuredRenderModule ( * args , ** kwargs ) Render module that returns JSON/YAML-serializable structures.","title":"StructuredRenderModule"},{"location":"reference/dewret/core/#ancestors-in-mro_7","text":"dewret.core.BaseRenderModule typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#static-methods_4","text":"","title":"Static methods"},{"location":"reference/dewret/core/#default_config_2","text":"def default_config ( ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Retrieve default settings. These will not change during execution, but can be overridden by dewret.core.set_render_configuration . Returns: a static, serializable dict.","title":"default_config"},{"location":"reference/dewret/core/#methods_4","text":"","title":"Methods"},{"location":"reference/dewret/core/#render","text":"def render ( self , workflow : dewret . core . WorkflowProtocol , ** kwargs : dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] ) -> dict [ str , dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]]] Turn a workflow into a serializable structure. Returns: one or more subworkflows with a __root__ key representing the outermost workflow, at least.","title":"render"},{"location":"reference/dewret/core/#unevaluatableerror","text":"class UnevaluatableError ( / , * args , ** kwargs ) Signposts that a user has tried to treat a reference as the real (runtime) value. For example, by comparing to a concrete integer or value, etc.","title":"UnevaluatableError"},{"location":"reference/dewret/core/#ancestors-in-mro_8","text":"builtins.Exception builtins.BaseException","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#class-variables_2","text":"args","title":"Class variables"},{"location":"reference/dewret/core/#methods_5","text":"","title":"Methods"},{"location":"reference/dewret/core/#add_note","text":"def add_note ( ... ) Exception.add_note(note) -- add a note to the exception","title":"add_note"},{"location":"reference/dewret/core/#with_traceback","text":"def with_traceback ( ... ) Exception.with_traceback(tb) -- set self. traceback to tb and return self.","title":"with_traceback"},{"location":"reference/dewret/core/#workflowcomponent","text":"class WorkflowComponent ( * args : Any , workflow : dewret . core . WorkflowProtocol , ** kwargs : Any ) Base class for anything directly tied to an individual Workflow .","title":"WorkflowComponent"},{"location":"reference/dewret/core/#attributes_1","text":"Name Type Description Default workflow None the Workflow that this is tied to. None","title":"Attributes"},{"location":"reference/dewret/core/#descendants_3","text":"dewret.core.Reference dewret.workflow.BaseStep dewret.workflow.ParameterReference","title":"Descendants"},{"location":"reference/dewret/core/#workflowprotocol","text":"class WorkflowProtocol ( * args , ** kwargs ) Expected structure for a workflow. We do not expect various workflow implementations, but this allows us to define the interface expected by the core classes.","title":"WorkflowProtocol"},{"location":"reference/dewret/core/#ancestors-in-mro_9","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/core/#methods_6","text":"","title":"Methods"},{"location":"reference/dewret/core/#remap","text":"def remap ( self , name : str ) -> str Perform any name-changing for steps, etc. in the workflow. This enables, for example, simplifying all the IDs to an integer sequence. Returns: remapped name.","title":"remap"},{"location":"reference/dewret/core/#set_result","text":"def set_result ( self , result : sympy . core . basic . Basic | list [ sympy . core . basic . Basic ] | tuple [ sympy . core . basic . Basic ] ) -> None Set the step that should produce a result for the overall workflow.","title":"set_result"},{"location":"reference/dewret/core/#simplify_ids","text":"def simplify_ids ( self , infix : list [ str ] | None = None ) -> None Drop the non-human-readable IDs if possible, in favour of integer sequences. Parameters: Name Type Description Default infix None any inherited intermediary identifiers, to allow nesting, or None. None","title":"simplify_ids"},{"location":"reference/dewret/render/","text":"Module dewret.render Utilities for building renderers. Provides the routines for calling varied renderers in a standard way, and for renderers to reuse to build up their own functionality. Variables T Functions base_render def base_render ( workflow : dewret . workflow . Workflow , build_cb : Callable [[ dewret . workflow . Workflow ], ~ T ] ) -> dict [ str , ~ T ] Render to a dict-like structure. Parameters: Name Type Description Default workflow None workflow to evaluate result. None build_cb None a callback to call for each workflow found. None Returns: Type Description None Reduced form as a native Python dict structure for serialization. get_render_method def get_render_method ( renderer : pathlib . Path | dewret . core . RawRenderModule | dewret . core . StructuredRenderModule , pretty : bool = False ) -> dewret . core . RenderCall Create a ready-made callable to render the workflow that is appropriate for the renderer module. Parameters: Name Type Description Default renderer None a module or path to a module. None pretty None whether the renderer should attempt to YAML-format the output (if relevant). None structured_to_raw def structured_to_raw ( rendered : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]], pretty : bool = False ) -> str Serialize a serializable structure to a string. Parameters: Name Type Description Default rendered None a possibly-nested, static basic Python structure. None pretty None whether to attempt YAML dumping with an indent of 2. None","title":"Render"},{"location":"reference/dewret/render/#module-dewretrender","text":"Utilities for building renderers. Provides the routines for calling varied renderers in a standard way, and for renderers to reuse to build up their own functionality.","title":"Module dewret.render"},{"location":"reference/dewret/render/#variables","text":"T","title":"Variables"},{"location":"reference/dewret/render/#functions","text":"","title":"Functions"},{"location":"reference/dewret/render/#base_render","text":"def base_render ( workflow : dewret . workflow . Workflow , build_cb : Callable [[ dewret . workflow . Workflow ], ~ T ] ) -> dict [ str , ~ T ] Render to a dict-like structure. Parameters: Name Type Description Default workflow None workflow to evaluate result. None build_cb None a callback to call for each workflow found. None Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"base_render"},{"location":"reference/dewret/render/#get_render_method","text":"def get_render_method ( renderer : pathlib . Path | dewret . core . RawRenderModule | dewret . core . StructuredRenderModule , pretty : bool = False ) -> dewret . core . RenderCall Create a ready-made callable to render the workflow that is appropriate for the renderer module. Parameters: Name Type Description Default renderer None a module or path to a module. None pretty None whether the renderer should attempt to YAML-format the output (if relevant). None","title":"get_render_method"},{"location":"reference/dewret/render/#structured_to_raw","text":"def structured_to_raw ( rendered : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]], pretty : bool = False ) -> str Serialize a serializable structure to a string. Parameters: Name Type Description Default rendered None a possibly-nested, static basic Python structure. None pretty None whether to attempt YAML dumping with an indent of 2. None","title":"structured_to_raw"},{"location":"reference/dewret/tasks/","text":"Module dewret.tasks Abstraction layer for task operations. Access dask, or other, backends consistently using this module. It provides decorators and execution calls that manage tasks. Note that the task decorator should be called with no arguments, and will return the appropriate decorator for the current backend. Typical usage example: >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 Variables DEFAULT_BACKEND Param RetType construct An alias pointing to an instance of the TaskManager class. Used for constructing a set of tasks into a dewret workflow instance. Functions ensure_lazy def ensure_lazy ( task : Any ) -> dewret . workflow . Lazy | None Evaluate a single task for a known workflow. As we mask our lazy-evaluable functions to appear as their original types to the type system (see dewret.tasks.task ), we must cast them back, to allow the type-checker to comb the remainder of the code. Parameters: Name Type Description Default task None the suspected task to check. None Returns: Type Description None Original task, cast to a Lazy, or None. evaluate def evaluate ( task : dewret . workflow . Lazy | list [ dewret . workflow . Lazy ] | tuple [ dewret . workflow . Lazy ], __workflow__ : dewret . workflow . Workflow , thread_pool : concurrent . futures . thread . ThreadPoolExecutor | None = None , ** kwargs : Any ) -> Any Evaluate a single task for a known workflow. Parameters: Name Type Description Default task None the task to evaluate. None workflow None workflow within which this exists. None thread_pool None existing pool of threads to run this in, or None. None **kwargs None any arguments to pass to the task. None factory def factory ( fn : collections . abc . Callable [ ... , ~ RetType ] ) -> collections . abc . Callable [ ... , ~ RetType ] Create a factory, that can be treated as complex input to a workflow. Parameters: Name Type Description Default fn None a callable to create the entity. None in_nested_task def in_nested_task ( ) -> Generator [ NoneType , NoneType , NoneType ] Informs the builder that we are within a nested task. This is only really relevant in the subworkflow context. TODO: check impact of ContextVar being thread-sensitive on build. is_in_nested_task def is_in_nested_task ( ) -> bool Check if we are within a nested task. Used, for example, to see if discovered parameters should be treated as \"local\" (i.e. should take a default to the step) or global (i.e. should be turned into a workflow parameter) if we are inside or outside a subworkflow, respectively. lazy def lazy ( ) -> collections . abc . Callable [[ collections . abc . Callable [ ... , typing . Any ]], dewret . workflow . Lazy ] Get the lazy decorator for this backend. Returns: Type Description None Real decorator for this backend. set_backend def set_backend ( backend : dewret . tasks . Backend ) -> None Choose a backend. Will raise an error if a backend is already chosen. Parameters: Name Type Description Default backend None chosen backend to use from here-on in. None task def task ( nested : bool = False , flatten_nested : bool = True , is_factory : bool = False ) -> collections . abc . Callable [[ collections . abc . Callable [ ~ Param , ~ RetType ]], collections . abc . Callable [ ~ Param , ~ RetType ]] Decorator factory abstracting backend's own task decorator. For example: >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 If the backend is dask (the default), it is will evaluate this as a dask.delayed . Note that, with any backend, dewret will hijack the decorator to record the attempted evalution rather than actually evaluating the lazy function. Nonetheless, this hijacking will still be executed with the backend's lazy executor, so dask.delayed will still be called, for example, in the dask case. Parameters: Name Type Description Default nested None whether this should be executed to find other tasks. None flatten_nested None (only relevant to nested tasks) should this nested task be considered a distinct subworkflow, or is it just organizational for the outer workflow. None is_factory None whether this task should be marked as a 'factory', rather than a normal step. None Returns: Type Description None Decorator for the current backend to mark lazy-executable tasks. Raises: Type Description TypeError if arguments are missing or incorrect, in line with usual Python behaviour. unwrap def unwrap ( task : dewret . workflow . Lazy ) -> collections . abc . Callable [ ... , typing . Any ] Unwraps a lazy-evaluated function to get the function. Ideally, we could use the __wrapped__ property but not all workflow engines support this, and most importantly, dask has only done so as of 2024.03. Parameters: Name Type Description Default task None task to be unwrapped. None Returns: Type Description None Original target. Raises: Type Description RuntimeError if the task is not a wrapped function. workflow def workflow ( ) -> collections . abc . Callable [[ collections . abc . Callable [ ~ Param , ~ RetType ]], collections . abc . Callable [ ~ Param , ~ RetType ]] Shortcut for marking a task as nested. A nested task is one which calls other tasks and does not do anything else important. It will not actually get called at runtime, but should map entirely into the graph. As such, arithmetic operations on results, etc. will cause errors at render-time. Combining tasks is acceptable, and intended. The effect of the nested task will be considered equivalent to whatever reaching whatever step reference is returned at the end. >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 >>> @workflow () ... def double_increment ( num : int ) -> int : ... return increment ( increment ( num = num )) Returns: Type Description None Task that runs at render, not execution, time. Classes Backend class Backend ( * args , ** kwds ) Stringy enum representing available backends. Ancestors (in MRO) enum.Enum Class variables DASK name value TaskException class TaskException ( task : dewret . workflow . Task | collections . abc . Callable [ ... , typing . Any ], dec_tb : traceback | None , tb : traceback | None , message : str , * args : Any , ** kwargs : Any ) Exception tied to a specific task. Primarily aimed at parsing issues, but this will ensure that a message is shown with useful debug information for the workflow writer. Ancestors (in MRO) builtins.Exception builtins.BaseException Class variables args Methods add_note def add_note ( ... ) Exception.add_note(note) -- add a note to the exception with_traceback def with_traceback ( ... ) Exception.with_traceback(tb) -- set self. traceback to tb and return self. TaskManager class TaskManager ( / , * args , ** kwargs ) Overarching backend-agnostic task manager. Gatekeeps the specific backend implementation. This can be instantiated without choosing a backend, but the first call to any of its methods will concretize that choice - either as the default, or the backend set via TaskManager.set_backend . It cannot be changed after this point. Methods backend def backend ( ... ) Import backend module. Cached property to load the backend module, if it has not been already. Returns: Type Description None Backend module for the specific choice of backend. ensure_lazy def ensure_lazy ( self , task : Any ) -> dewret . workflow . Lazy | None Evaluate a single task for a known workflow. As we mask our lazy-evaluable functions to appear as their original types to the type system (see dewret.tasks.task ), we must cast them back, to allow the type-checker to comb the remainder of the code. Parameters: Name Type Description Default task None the suspected task to check. None Returns: Type Description None Original task, cast to a Lazy, or None. evaluate def evaluate ( self , task : dewret . workflow . Lazy | list [ dewret . workflow . Lazy ] | tuple [ dewret . workflow . Lazy ], __workflow__ : dewret . workflow . Workflow , thread_pool : concurrent . futures . thread . ThreadPoolExecutor | None = None , ** kwargs : Any ) -> Any Evaluate a single task for a known workflow. Parameters: Name Type Description Default task None the task to evaluate. None workflow None workflow within which this exists. None thread_pool None existing pool of threads to run this in, or None. None **kwargs None any arguments to pass to the task. None make_lazy def make_lazy ( self ) -> collections . abc . Callable [[ collections . abc . Callable [ ... , typing . Any ]], dewret . workflow . Lazy ] Get the lazy decorator for this backend. Returns: Type Description None Real decorator for this backend. set_backend def set_backend ( self , backend : dewret . tasks . Backend ) -> dewret . tasks . Backend Choose a backend. Sets the backend, provided it has not already been loaded. Parameters: Name Type Description Default backend None chosen backend, to override the default. None Returns: Type Description None Backend that was set. Raises: Type Description RuntimeError when a backend has already been loaded. unwrap def unwrap ( self , task : dewret . workflow . Lazy ) -> collections . abc . Callable [ ... , typing . Any ] Unwraps a lazy-evaluated function to get the function. Ideally, we could use the __wrapped__ property but not all workflow engines support this, and most importantly, dask has only done so as of 2024.03. Parameters: Name Type Description Default task None task to be unwrapped. None Returns: Type Description None Original target. Raises: Type Description RuntimeError if the task is not a wrapped function.","title":"Tasks"},{"location":"reference/dewret/tasks/#module-dewrettasks","text":"Abstraction layer for task operations. Access dask, or other, backends consistently using this module. It provides decorators and execution calls that manage tasks. Note that the task decorator should be called with no arguments, and will return the appropriate decorator for the current backend. Typical usage example: >>> @task () ... def increment ( num : int ) -> int : ... return num + 1","title":"Module dewret.tasks"},{"location":"reference/dewret/tasks/#variables","text":"DEFAULT_BACKEND Param RetType construct An alias pointing to an instance of the TaskManager class. Used for constructing a set of tasks into a dewret workflow instance.","title":"Variables"},{"location":"reference/dewret/tasks/#functions","text":"","title":"Functions"},{"location":"reference/dewret/tasks/#ensure_lazy","text":"def ensure_lazy ( task : Any ) -> dewret . workflow . Lazy | None Evaluate a single task for a known workflow. As we mask our lazy-evaluable functions to appear as their original types to the type system (see dewret.tasks.task ), we must cast them back, to allow the type-checker to comb the remainder of the code. Parameters: Name Type Description Default task None the suspected task to check. None Returns: Type Description None Original task, cast to a Lazy, or None.","title":"ensure_lazy"},{"location":"reference/dewret/tasks/#evaluate","text":"def evaluate ( task : dewret . workflow . Lazy | list [ dewret . workflow . Lazy ] | tuple [ dewret . workflow . Lazy ], __workflow__ : dewret . workflow . Workflow , thread_pool : concurrent . futures . thread . ThreadPoolExecutor | None = None , ** kwargs : Any ) -> Any Evaluate a single task for a known workflow. Parameters: Name Type Description Default task None the task to evaluate. None workflow None workflow within which this exists. None thread_pool None existing pool of threads to run this in, or None. None **kwargs None any arguments to pass to the task. None","title":"evaluate"},{"location":"reference/dewret/tasks/#factory","text":"def factory ( fn : collections . abc . Callable [ ... , ~ RetType ] ) -> collections . abc . Callable [ ... , ~ RetType ] Create a factory, that can be treated as complex input to a workflow. Parameters: Name Type Description Default fn None a callable to create the entity. None","title":"factory"},{"location":"reference/dewret/tasks/#in_nested_task","text":"def in_nested_task ( ) -> Generator [ NoneType , NoneType , NoneType ] Informs the builder that we are within a nested task. This is only really relevant in the subworkflow context. TODO: check impact of ContextVar being thread-sensitive on build.","title":"in_nested_task"},{"location":"reference/dewret/tasks/#is_in_nested_task","text":"def is_in_nested_task ( ) -> bool Check if we are within a nested task. Used, for example, to see if discovered parameters should be treated as \"local\" (i.e. should take a default to the step) or global (i.e. should be turned into a workflow parameter) if we are inside or outside a subworkflow, respectively.","title":"is_in_nested_task"},{"location":"reference/dewret/tasks/#lazy","text":"def lazy ( ) -> collections . abc . Callable [[ collections . abc . Callable [ ... , typing . Any ]], dewret . workflow . Lazy ] Get the lazy decorator for this backend. Returns: Type Description None Real decorator for this backend.","title":"lazy"},{"location":"reference/dewret/tasks/#set_backend","text":"def set_backend ( backend : dewret . tasks . Backend ) -> None Choose a backend. Will raise an error if a backend is already chosen. Parameters: Name Type Description Default backend None chosen backend to use from here-on in. None","title":"set_backend"},{"location":"reference/dewret/tasks/#task","text":"def task ( nested : bool = False , flatten_nested : bool = True , is_factory : bool = False ) -> collections . abc . Callable [[ collections . abc . Callable [ ~ Param , ~ RetType ]], collections . abc . Callable [ ~ Param , ~ RetType ]] Decorator factory abstracting backend's own task decorator. For example: >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 If the backend is dask (the default), it is will evaluate this as a dask.delayed . Note that, with any backend, dewret will hijack the decorator to record the attempted evalution rather than actually evaluating the lazy function. Nonetheless, this hijacking will still be executed with the backend's lazy executor, so dask.delayed will still be called, for example, in the dask case. Parameters: Name Type Description Default nested None whether this should be executed to find other tasks. None flatten_nested None (only relevant to nested tasks) should this nested task be considered a distinct subworkflow, or is it just organizational for the outer workflow. None is_factory None whether this task should be marked as a 'factory', rather than a normal step. None Returns: Type Description None Decorator for the current backend to mark lazy-executable tasks. Raises: Type Description TypeError if arguments are missing or incorrect, in line with usual Python behaviour.","title":"task"},{"location":"reference/dewret/tasks/#unwrap","text":"def unwrap ( task : dewret . workflow . Lazy ) -> collections . abc . Callable [ ... , typing . Any ] Unwraps a lazy-evaluated function to get the function. Ideally, we could use the __wrapped__ property but not all workflow engines support this, and most importantly, dask has only done so as of 2024.03. Parameters: Name Type Description Default task None task to be unwrapped. None Returns: Type Description None Original target. Raises: Type Description RuntimeError if the task is not a wrapped function.","title":"unwrap"},{"location":"reference/dewret/tasks/#workflow","text":"def workflow ( ) -> collections . abc . Callable [[ collections . abc . Callable [ ~ Param , ~ RetType ]], collections . abc . Callable [ ~ Param , ~ RetType ]] Shortcut for marking a task as nested. A nested task is one which calls other tasks and does not do anything else important. It will not actually get called at runtime, but should map entirely into the graph. As such, arithmetic operations on results, etc. will cause errors at render-time. Combining tasks is acceptable, and intended. The effect of the nested task will be considered equivalent to whatever reaching whatever step reference is returned at the end. >>> @task () ... def increment ( num : int ) -> int : ... return num + 1 >>> @workflow () ... def double_increment ( num : int ) -> int : ... return increment ( increment ( num = num )) Returns: Type Description None Task that runs at render, not execution, time.","title":"workflow"},{"location":"reference/dewret/tasks/#classes","text":"","title":"Classes"},{"location":"reference/dewret/tasks/#backend","text":"class Backend ( * args , ** kwds ) Stringy enum representing available backends.","title":"Backend"},{"location":"reference/dewret/tasks/#ancestors-in-mro","text":"enum.Enum","title":"Ancestors (in MRO)"},{"location":"reference/dewret/tasks/#class-variables","text":"DASK name value","title":"Class variables"},{"location":"reference/dewret/tasks/#taskexception","text":"class TaskException ( task : dewret . workflow . Task | collections . abc . Callable [ ... , typing . Any ], dec_tb : traceback | None , tb : traceback | None , message : str , * args : Any , ** kwargs : Any ) Exception tied to a specific task. Primarily aimed at parsing issues, but this will ensure that a message is shown with useful debug information for the workflow writer.","title":"TaskException"},{"location":"reference/dewret/tasks/#ancestors-in-mro_1","text":"builtins.Exception builtins.BaseException","title":"Ancestors (in MRO)"},{"location":"reference/dewret/tasks/#class-variables_1","text":"args","title":"Class variables"},{"location":"reference/dewret/tasks/#methods","text":"","title":"Methods"},{"location":"reference/dewret/tasks/#add_note","text":"def add_note ( ... ) Exception.add_note(note) -- add a note to the exception","title":"add_note"},{"location":"reference/dewret/tasks/#with_traceback","text":"def with_traceback ( ... ) Exception.with_traceback(tb) -- set self. traceback to tb and return self.","title":"with_traceback"},{"location":"reference/dewret/tasks/#taskmanager","text":"class TaskManager ( / , * args , ** kwargs ) Overarching backend-agnostic task manager. Gatekeeps the specific backend implementation. This can be instantiated without choosing a backend, but the first call to any of its methods will concretize that choice - either as the default, or the backend set via TaskManager.set_backend . It cannot be changed after this point.","title":"TaskManager"},{"location":"reference/dewret/tasks/#methods_1","text":"","title":"Methods"},{"location":"reference/dewret/tasks/#backend_1","text":"def backend ( ... ) Import backend module. Cached property to load the backend module, if it has not been already. Returns: Type Description None Backend module for the specific choice of backend.","title":"backend"},{"location":"reference/dewret/tasks/#ensure_lazy_1","text":"def ensure_lazy ( self , task : Any ) -> dewret . workflow . Lazy | None Evaluate a single task for a known workflow. As we mask our lazy-evaluable functions to appear as their original types to the type system (see dewret.tasks.task ), we must cast them back, to allow the type-checker to comb the remainder of the code. Parameters: Name Type Description Default task None the suspected task to check. None Returns: Type Description None Original task, cast to a Lazy, or None.","title":"ensure_lazy"},{"location":"reference/dewret/tasks/#evaluate_1","text":"def evaluate ( self , task : dewret . workflow . Lazy | list [ dewret . workflow . Lazy ] | tuple [ dewret . workflow . Lazy ], __workflow__ : dewret . workflow . Workflow , thread_pool : concurrent . futures . thread . ThreadPoolExecutor | None = None , ** kwargs : Any ) -> Any Evaluate a single task for a known workflow. Parameters: Name Type Description Default task None the task to evaluate. None workflow None workflow within which this exists. None thread_pool None existing pool of threads to run this in, or None. None **kwargs None any arguments to pass to the task. None","title":"evaluate"},{"location":"reference/dewret/tasks/#make_lazy","text":"def make_lazy ( self ) -> collections . abc . Callable [[ collections . abc . Callable [ ... , typing . Any ]], dewret . workflow . Lazy ] Get the lazy decorator for this backend. Returns: Type Description None Real decorator for this backend.","title":"make_lazy"},{"location":"reference/dewret/tasks/#set_backend_1","text":"def set_backend ( self , backend : dewret . tasks . Backend ) -> dewret . tasks . Backend Choose a backend. Sets the backend, provided it has not already been loaded. Parameters: Name Type Description Default backend None chosen backend, to override the default. None Returns: Type Description None Backend that was set. Raises: Type Description RuntimeError when a backend has already been loaded.","title":"set_backend"},{"location":"reference/dewret/tasks/#unwrap_1","text":"def unwrap ( self , task : dewret . workflow . Lazy ) -> collections . abc . Callable [ ... , typing . Any ] Unwraps a lazy-evaluated function to get the function. Ideally, we could use the __wrapped__ property but not all workflow engines support this, and most importantly, dask has only done so as of 2024.03. Parameters: Name Type Description Default task None task to be unwrapped. None Returns: Type Description None Original target. Raises: Type Description RuntimeError if the task is not a wrapped function.","title":"unwrap"},{"location":"reference/dewret/utils/","text":"Module dewret.utils Utility module. General types and functions to centralize common logic. Functions crawl_raw def crawl_raw ( value : Any , action : Optional [ Callable [[ Any ], Any ]] = None ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Takes a Raw-like structure and makes it RawType. Particularly useful for squashing any TypedDicts. Parameters: Name Type Description Default value None value to squash None action None an callback to apply to each found entry, or None. None ensure_raw def ensure_raw ( value : Any , cast_tuple : bool = False ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Check if a variable counts as \"raw\". This works around a checking issue that isinstance of a union of types assigned to a variable, such as RawType , may throw errors even though Python 3.11+ does not. Instead, we explicitly make the full union in the statement below. firm_to_raw def firm_to_raw ( value : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], list [ 'FirmType' ], dict [ str , 'FirmType' ], tuple [ 'FirmType' , ... ]] ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Convenience wrapper for firm structures. Turns structures that would be raw, except for tuples, into raw structures by mapping any tuples to lists. Parameters: Name Type Description Default value None a firm structure (contains raw/tuple values). None flatten_if_set def flatten_if_set ( value : Any ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], dewret . utils . Unset ] Takes a Raw-like structure and makes it RawType or Unset. Flattens if the value is set, but otherwise returns the unset sentinel value as-is. Parameters: Name Type Description Default value None value to squash None hasher def hasher ( construct : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], list [ 'FirmType' ], dict [ str , 'FirmType' ], tuple [ 'FirmType' , ... ]] ) -> str Consistently hash a RawType or tuple structure. Turns a possibly nested structure of basic types, dicts, lists and tuples into a consistent hash. Parameters: Name Type Description Default construct None structure to hash. None Returns: Type Description None Hash string that should be unique to the construct. The limits of this uniqueness have not yet been explicitly calculated. is_expr def is_expr ( value : Any , permitted_references : type = < class ' dewret . core . Reference '> ) -> bool Confirms whether a structure has only raw or expression types. Parameters: Name Type Description Default value None a crawlable structure. None permitted_references None a class representing the allowed types of References. None is_firm def is_firm ( value : Any , check : Optional [ Callable [[ Any ], bool ]] = None ) -> bool Confirms whether a function is firm. That is, whether its contents are raw or tuples. Parameters: Name Type Description Default value None value to check. None check None any additional check to apply. None is_raw def is_raw ( value : Any , check : Optional [ Callable [[ Any ], bool ]] = None ) -> bool Check if a variable counts as \"raw\". This works around a checking issue that isinstance of a union of types assigned to a variable, such as RawType , may throw errors even though Python 3.11+ does not. Instead, we explicitly make the full union in the statement below. is_raw_type def is_raw_type ( typ : type ) -> bool Check if a type counts as \"raw\". load_module_or_package def load_module_or_package ( target_name : str , path : pathlib . Path ) -> module Convenience loader for modules. If an __init__.py is found in the same location as the target, it will try to load the renderer module as if it is contained in a package and, if it cannot, will fall back to loading the single file. Parameters: Name Type Description Default target_name None module name that should appear in sys.modules . None path None location of the module. None make_traceback def make_traceback ( skip : int = 2 ) -> traceback | None Creates a traceback for the current frame. Necessary to allow tracebacks to be prepped for potential errors in lazy-evaluated functions. Parameters: Name Type Description Default skip None number of frames to skip before starting traceback. None Classes DataclassProtocol class DataclassProtocol ( * args , ** kwargs ) Format of a dataclass. Since dataclasses do not expose a proper type, we use this to represent them. Ancestors (in MRO) typing.Protocol typing.Generic Unset class Unset ( / , * args , ** kwargs ) Unset variable, with no default value. Descendants dewret.workflow.UnsetType","title":"Utils"},{"location":"reference/dewret/utils/#module-dewretutils","text":"Utility module. General types and functions to centralize common logic.","title":"Module dewret.utils"},{"location":"reference/dewret/utils/#functions","text":"","title":"Functions"},{"location":"reference/dewret/utils/#crawl_raw","text":"def crawl_raw ( value : Any , action : Optional [ Callable [[ Any ], Any ]] = None ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Takes a Raw-like structure and makes it RawType. Particularly useful for squashing any TypedDicts. Parameters: Name Type Description Default value None value to squash None action None an callback to apply to each found entry, or None. None","title":"crawl_raw"},{"location":"reference/dewret/utils/#ensure_raw","text":"def ensure_raw ( value : Any , cast_tuple : bool = False ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Check if a variable counts as \"raw\". This works around a checking issue that isinstance of a union of types assigned to a variable, such as RawType , may throw errors even though Python 3.11+ does not. Instead, we explicitly make the full union in the statement below.","title":"ensure_raw"},{"location":"reference/dewret/utils/#firm_to_raw","text":"def firm_to_raw ( value : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], list [ 'FirmType' ], dict [ str , 'FirmType' ], tuple [ 'FirmType' , ... ]] ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]] Convenience wrapper for firm structures. Turns structures that would be raw, except for tuples, into raw structures by mapping any tuples to lists. Parameters: Name Type Description Default value None a firm structure (contains raw/tuple values). None","title":"firm_to_raw"},{"location":"reference/dewret/utils/#flatten_if_set","text":"def flatten_if_set ( value : Any ) -> Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], dewret . utils . Unset ] Takes a Raw-like structure and makes it RawType or Unset. Flattens if the value is set, but otherwise returns the unset sentinel value as-is. Parameters: Name Type Description Default value None value to squash None","title":"flatten_if_set"},{"location":"reference/dewret/utils/#hasher","text":"def hasher ( construct : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], list [ 'FirmType' ], dict [ str , 'FirmType' ], tuple [ 'FirmType' , ... ]] ) -> str Consistently hash a RawType or tuple structure. Turns a possibly nested structure of basic types, dicts, lists and tuples into a consistent hash. Parameters: Name Type Description Default construct None structure to hash. None Returns: Type Description None Hash string that should be unique to the construct. The limits of this uniqueness have not yet been explicitly calculated.","title":"hasher"},{"location":"reference/dewret/utils/#is_expr","text":"def is_expr ( value : Any , permitted_references : type = < class ' dewret . core . Reference '> ) -> bool Confirms whether a structure has only raw or expression types. Parameters: Name Type Description Default value None a crawlable structure. None permitted_references None a class representing the allowed types of References. None","title":"is_expr"},{"location":"reference/dewret/utils/#is_firm","text":"def is_firm ( value : Any , check : Optional [ Callable [[ Any ], bool ]] = None ) -> bool Confirms whether a function is firm. That is, whether its contents are raw or tuples. Parameters: Name Type Description Default value None value to check. None check None any additional check to apply. None","title":"is_firm"},{"location":"reference/dewret/utils/#is_raw","text":"def is_raw ( value : Any , check : Optional [ Callable [[ Any ], bool ]] = None ) -> bool Check if a variable counts as \"raw\". This works around a checking issue that isinstance of a union of types assigned to a variable, such as RawType , may throw errors even though Python 3.11+ does not. Instead, we explicitly make the full union in the statement below.","title":"is_raw"},{"location":"reference/dewret/utils/#is_raw_type","text":"def is_raw_type ( typ : type ) -> bool Check if a type counts as \"raw\".","title":"is_raw_type"},{"location":"reference/dewret/utils/#load_module_or_package","text":"def load_module_or_package ( target_name : str , path : pathlib . Path ) -> module Convenience loader for modules. If an __init__.py is found in the same location as the target, it will try to load the renderer module as if it is contained in a package and, if it cannot, will fall back to loading the single file. Parameters: Name Type Description Default target_name None module name that should appear in sys.modules . None path None location of the module. None","title":"load_module_or_package"},{"location":"reference/dewret/utils/#make_traceback","text":"def make_traceback ( skip : int = 2 ) -> traceback | None Creates a traceback for the current frame. Necessary to allow tracebacks to be prepped for potential errors in lazy-evaluated functions. Parameters: Name Type Description Default skip None number of frames to skip before starting traceback. None","title":"make_traceback"},{"location":"reference/dewret/utils/#classes","text":"","title":"Classes"},{"location":"reference/dewret/utils/#dataclassprotocol","text":"class DataclassProtocol ( * args , ** kwargs ) Format of a dataclass. Since dataclasses do not expose a proper type, we use this to represent them.","title":"DataclassProtocol"},{"location":"reference/dewret/utils/#ancestors-in-mro","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/utils/#unset","text":"class Unset ( / , * args , ** kwargs ) Unset variable, with no default value.","title":"Unset"},{"location":"reference/dewret/utils/#descendants","text":"dewret.workflow.UnsetType","title":"Descendants"},{"location":"reference/dewret/workflow/","text":"Module dewret.workflow Overarching workflow concepts. Basic constructs for describing a workflow. Variables AVAILABLE_TYPES CHECK_IDS LazyFactory RetType StepExecution T TYPE_CHECKING Target U UNSET logger Functions expr_to_references def expr_to_references ( expression : 'Any' , remap : 'Callable[[Any], Any] | None' = None ) -> 'tuple[ExprType, list[Reference[Any] | Parameter[Any]]]' Pull out any references, or other free symbols, from an expression. Parameters: Name Type Description Default expression None normally a reference that can be immediately returned, but may be a sympy expression or a dict/tuple/list/etc. of such. None remap None a callable to project certain values down before extracting symbols, or None. None is_task def is_task ( task : 'Lazy' ) -> 'bool' Decide whether this is a task. Checks whether the wrapped function has the magic attribute __step_expression__ set to True, which is done within task creation. Parameters: Name Type Description Default task None lazy-evaluated value, suspected to be a task. None Returns: Type Description None True if task is indeed a task. param def param ( name : 'str' , default : 'T | UnsetType[T] | Unset' = < dewret . utils . Unset object at 0x7f1e70dddb90 > , tethered : 'Literal[False] | None | Step | Workflow' = False , typ : 'type[T] | Unset' = < dewret . utils . Unset object at 0x7f1e70dddb90 > , autoname : 'bool' = False ) -> 'T' Create a parameter. Will cast so it looks like the original type. Returns: Type Description None Parameter class cast to the type of the supplied default. unify_workflows def unify_workflows ( expression : 'Any' , base_workflow : 'Workflow | None' , set_only : 'bool' = False ) -> 'tuple[Basic | None, Workflow | None]' Takes an expression and ensures all of its references exist in the same workflow. Parameters: Name Type Description Default expression None any valid argument to dewret.workflow.expr_to_references . None base_workflow None the desired workflow to align on, or None. None set_only None whether to bother assimilating all the workflows (False), or to assume that has been done (False). None Classes BaseStep class BaseStep ( workflow : 'Workflow' , task : 'Task | Workflow' , arguments : 'Mapping[str, Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Lazy-evaluated function call. Individual function call to a lazy-evaluatable function, tracked for building up the Workflow . Attributes Name Type Description Default task None the Task being called in this step. None arguments None key-value pairs of arguments to this step. None Ancestors (in MRO) dewret.core.WorkflowComponent Descendants dewret.workflow.NestedStep dewret.workflow.Step Class variables positional_args Instance variables id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value. Methods make_reference def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None set_workflow def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None FactoryCall class FactoryCall ( workflow : 'Workflow' , task : 'Task | Workflow' , arguments : 'Mapping[str, Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Call to a factory function. Ancestors (in MRO) dewret.workflow.Step dewret.workflow.BaseStep dewret.core.WorkflowComponent Class variables positional_args Instance variables id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value. Methods make_reference def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None set_workflow def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None FieldableMixin class FieldableMixin ( self : 'FieldableProtocol' , * args : 'Any' , field : 'str | int | tuple[str | int, ...] | None' = None , ** kwargs : 'Any' ) Tooling for enhancing a type with referenceable fields. Descendants dewret.workflow.ParameterReference dewret.workflow.StepReference Methods find_field def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference. FieldableProtocol class FieldableProtocol ( * args : 'Any' , field : 'str | None' = None , ** kwargs : 'Any' ) Expected interfaces for a type that can take fields. Attributes Name Type Description Default field None tuple representing the named fields, either strings or integers. None Ancestors (in MRO) typing.Protocol typing.Generic Instance variables name The name for the target, accounting for the field. IterableParameterReference class IterableParameterReference ( typ : type [ ~ U ] | None = None , ** kwargs : Any ) Iterable form of parameter references. Ancestors (in MRO) dewret.core.IterableMixin dewret.workflow.ParameterReference dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent Class variables ParameterReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. find_field def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. IterableStepReference class IterableStepReference ( typ : type [ ~ U ] | None = None , ** kwargs : Any ) Iterable form of a step reference. Ancestors (in MRO) dewret.core.IterableMixin dewret.workflow.StepReference dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent Class variables StepReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. find_field def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. Lazy class Lazy ( * args , ** kwargs ) Requirements for a lazy-evaluatable function. Ancestors (in MRO) typing.Protocol typing.Generic Descendants dewret.workflow.LazyEvaluation LazyEvaluation class LazyEvaluation ( fn : 'Callable[..., RetType]' ) Tracks a single evaluation of a lazy function. Ancestors (in MRO) dewret.workflow.Lazy typing.Protocol typing.Generic NestedStep class NestedStep ( workflow : 'Workflow' , name : 'str' , subworkflow : 'Workflow' , arguments : 'Mapping[str, Basic | Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Calling out to a subworkflow. Type of BaseStep to call a subworkflow, which holds a reference to it. Ancestors (in MRO) dewret.workflow.BaseStep dewret.core.WorkflowComponent Class variables positional_args Instance variables id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value. subworkflow Subworkflow that is wrapped. Methods make_reference def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None set_workflow def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None Parameter class Parameter ( name : 'str' , default : 'T | UnsetType[T]' , tethered : 'Literal[False] | None | Step | Workflow' = None , autoname : 'bool' = False , typ : 'type[T] | Unset' = < dewret . utils . Unset object at 0x7f1e70dddb90 > ) Global parameter. Independent parameter that will be used when a task is spotted reaching outside its scope. This wraps the variable it uses. To allow for potential arithmetic operations, etc. it is a Sympy symbol. Attributes Name Type Description Default name None name of the parameter. None default None captured default value from the original value. None Ancestors (in MRO) typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin Class variables autoname default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) is_loopable def is_loopable ( typ : 'type' ) -> 'bool' Checks if this type can be looped over. In particular, checks if this is an iterable that is NOT a str or bytes, possibly disguised behind an Annotated. Parameters: Name Type Description Default typ None type to check. None Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} default Retrieve default value for this parameter, or an unset token. expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Extended name, suitable for rendering. This attempts to create a unique name by tying the parameter to a step if the user has not explicitly provided a name, ideally the one where we discovered it. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). make_reference def make_reference ( self , ** kwargs : 'Any' ) -> \"'ParameterReference[T]'\" Creates a new reference for the parameter. The kwargs will be passed to the constructor, but the Parameters: Name Type Description Default typ None type of the new reference's target. None **kwargs None arguments to pass to the constructor. None match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions register_caller def register_caller ( self , caller : 'BaseStep' ) -> 'None' Capture a step that uses this parameter. Gathers together the steps using this parameter. The first found will be recorded as the tethered step, and used for forming the name. removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. ParameterReference class ParameterReference ( parameter : 'Parameter[U]' , * args : 'Any' , typ : 'type[U] | None' = None , ** kwargs : 'Any' ) Reference to an individual Parameter . Allows us to refer to the outputs of a Parameter in subsequent Parameter arguments. Attributes Name Type Description Default parameter None Parameter referred to. None workflow None Related workflow. In this case, as Parameters are generic but ParameterReferences are specific, this carries the actual workflow reference. None Ancestors (in MRO) dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent Descendants dewret.workflow.IterableParameterReference Class variables ParameterReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. find_field def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. Step class Step ( workflow : 'Workflow' , task : 'Task | Workflow' , arguments : 'Mapping[str, Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Regular step. Ancestors (in MRO) dewret.workflow.BaseStep dewret.core.WorkflowComponent Descendants dewret.workflow.FactoryCall Class variables positional_args Instance variables id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value. Methods make_reference def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None set_workflow def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None StepReference class StepReference ( step : 'BaseStep' , * args : 'Any' , typ : 'type[U] | None' = None , ** kwargs : 'Any' ) Reference to an individual Step . Allows us to refer to the outputs of a Step in subsequent Step arguments. Attributes Name Type Description Default _ None metadata wrapping the Step referred to. None Ancestors (in MRO) dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent Descendants dewret.workflow.IterableStepReference Class variables StepReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol Static methods class_key def class_key ( ) Nice order of classes. fromiter def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) Instance variables args Returns a tuple of arguments of 'self'. Examples from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y Notes Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression. Examples from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference. Methods adjoint def adjoint ( self ) apart def apart ( self , x = None , ** args ) See the apart function in sympy.polys args_cnc def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self. Explanation self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False. Examples from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []] as_base_exp def as_base_exp ( self ) -> 'tuple[Expr, Expr]' as_coeff_Add def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation. as_coeff_Mul def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product. as_coeff_add def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ()) as_coeff_exponent def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression. as_coeff_mul def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ()) as_coefficient def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. Examples from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I) See Also coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used as_coefficients_dict def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned. Examples from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x} as_content_primitive def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self). Examples from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y) as_dummy def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r Notes Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object. as_expr def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression. Examples from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x) as_independent def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples -- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See Also separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul as_leading_term def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value. Examples from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2) as_numer_denom def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else. See Also normal: return a/b instead of (a, b) as_ordered_factors def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self]. as_ordered_terms def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms. Examples from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1] as_poly def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None . Explanation from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None as_powers_dict def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary. See Also as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors. as_real_imag def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z)) as_set def as_set ( self ) Rewrites Boolean expression in terms of real sets. Examples from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo)) as_terms def as_terms ( self ) Transform an expression to a list of terms. aseries def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) . Parameters self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form. Examples from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x Returns Expr Asymptotic series expansion of the expression. Notes This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self . References .. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion See Also Expr.aseries: See the docstring of this function for complete details of this wrapper. atoms def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)} cancel def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys coeff def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned. Explanation When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1 See Also as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used collect def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify combsimp def combsimp ( self ) See the combsimp function in sympy.simplify compare def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1 compute_leading_term def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. conjugate def conjugate ( self ) Returns the complex conjugate of 'self'. copy def copy ( self ) could_extract_minus_sign def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False. Examples from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True count def count ( self , query ) Count the number of matching subexpressions. count_ops def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count. diff def diff ( self , * symbols , ** assumptions ) dir def dir ( self , x , cdir ) doit def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x) dummy_eq def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols. Examples from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False equals def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None. Explanation If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False. evalf def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 expand def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information. extract_additively def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None. Examples from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3 See Also extract_multiplicatively coeff as_coefficient extract_branch_factor def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2) extract_multiplicatively def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. Examples from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6 factor def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools find def find ( self , query , group = False ) Find all subexpressions matching a query. find_field def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference. fourier_series def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information. fps def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information. gammasimp def gammasimp ( self ) See the gammasimp function in sympy.simplify getO def getO ( self ) Returns the additive O(..) symbol if there is one, else None. getn def getn ( self ) Returns the order of the expression. Explanation The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Examples from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn() has def has ( self , * patterns ) Test whether any subexpression matches any of the patterns. Examples from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False has_free def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False. Examples from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True has_xfree def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches. Examples from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False integrate def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals invert def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions). See Also sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert is_algebraic_expr def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation. Examples from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True See Also is_rational_function References .. [1] https://en.wikipedia.org/wiki/Algebraic_expression is_constant def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively. Explanation If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing. Examples from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True is_hypergeometric def is_hypergeometric ( self , k ) is_meromorphic def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis. Examples from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False is_polynomial def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). Examples from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function() is_rational_function def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Examples from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr(). is_same def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False leadterm def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b). Examples from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2) limit def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim. lseries def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries(). match def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2} matches def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} n def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits. Parameters subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information. Notes When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000 normal def normal ( self ) Return the expression as a fraction. expression -> a/b See Also as_numer_denom: return (a, b) instead of a/b nseries def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). Examples from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y) nsimplify def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify powsimp def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify primitive def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True radsimp def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify ratsimp def ratsimp ( self ) See the ratsimp function in sympy.simplify rcall def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z refine def refine ( self , assumption = True ) See the refine function in sympy.assumptions removeO def removeO ( self ) Removes the additive O(..) symbol if there is one replace def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules rewrite def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten. Parameters args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True . Examples If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2) round def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number. Examples from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I Notes The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True separate def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify series def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised. Parameters expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated. Examples from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x Returns Expr : Expression Series expansion of the expression about x0 Raises TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object simplify def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify sort_key def sort_key ( self , order = None ) Return a sort key. Examples from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2] subs def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision taylor_term def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\". to_nnf def to_nnf ( self , simplify = True ) together def together ( self , * args , ** kwargs ) See the together function in sympy.polys transpose def transpose ( self ) trigsimp def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify xreplace def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression. Parameters rule : dict-like Expresses a replacement rule Returns xreplace : the result of the replacement Examples from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),) See Also replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. Task class Task ( name : 'str' , target : 'Lazy' ) Named wrapper of a lazy-evaluatable function. Wraps a lazy-evaluatable function ( dewret.workflow.Lazy ) with any metadata needed to render it later. At present, this is the name. Attributes Name Type Description Default name None Name of the lazy function as it will appear in the output workflow text. None target None Callable that is wrapped. None UnsetType class UnsetType ( raw_type : 'type[T]' ) Unset variable with a specific type. Attributes Name Type Description Default type None type of the variable. None Ancestors (in MRO) dewret.utils.Unset typing.Generic Workflow class Workflow ( name : 'str | None' = None ) Overarching workflow concept. Represents a whole workflow, as a singleton maintaining all state information needed ahead of rendering. It is built up as the lazy-evaluations are finally evaluated. Attributes Name Type Description Default steps None the sequence of calls to lazy-evaluable functions, built as they are evaluated. None tasks None the mapping of names used in the steps to the actual Task wrappers they represent. None result None target reference to evaluate, if yet present. None Static methods assimilate def assimilate ( * workflow_args : 'Workflow' ) -> \"'Workflow'\" Combine two Workflows into one Workflow. Takes two workflows and unifies them by combining steps and tasks. If it sees mismatched identifiers for the same component, it will error. This could happen if the hashing function is flawed or some Python magic to do with Targets being passed. Parameters: Name Type Description Default workflow_args None workflows to use as base None from_result def from_result ( result : 'StepReference[Any] | list[StepReference[Any]] | tuple[StepReference[Any], ...]' , simplify_ids : 'bool' = False , nested : 'bool' = True ) -> 'Workflow' Create from a desired result. Starts from a result, and builds a workflow to output it. Instance variables has_result Confirms whether this workflow has a non-empty result. Either None or an empty list/tuple are considered empty for this purpose. Returns: True if the workflow has a result, False otherwise. id Consistent ID based off the step IDs. indexed_steps Steps mapped by ID. Forces generation of IDs. Note that this effectively freezes the steps, so it should not be used until we are confident the steps are all ready to be hashed. name Get the name of the workflow. result_type Overall return type of this workflow. steps Get deduplicated steps. Returns: steps for looping over without duplicates. Methods add_nested_step def add_nested_step ( self , name : 'str' , subworkflow : 'Workflow' , return_type : 'type | None' , kwargs : 'dict[str, Any]' , positional_args : 'dict[str, bool] | None' = None ) -> 'StepReference[Any]' Append a nested step. Calls a subworkflow. Parameters: Name Type Description Default name None name of the subworkflow. None subworkflow None the subworkflow itself. None return_type None a forced type for the return, or None. None kwargs None any key-value arguments to pass in the call. None positional_args None a mapping of arguments to bools, True if the argument is positional or otherwise False. None add_step def add_step ( self , fn : 'Lazy' , kwargs : 'dict[str, Raw | Reference[Any]]' , raw_as_parameter : 'bool' = False , is_factory : 'bool' = False , positional_args : 'dict[str, bool] | None' = None ) -> 'StepReference[Any]' Append a step. Adds a step, for running a target with key-value arguments, to the workflow. Parameters: Name Type Description Default fn None the target function to turn into a step. None kwargs None any key-value arguments to pass in the call. None raw_as_parameter None whether to turn any discovered raw arguments into workflow parameters. None is_factory None whether this step should be a Factory. None positional_args None a mapping of arguments to bools, True if the argument is positional or otherwise False. None find_factories def find_factories ( self ) -> 'dict[str, FactoryCall]' Steps that are factory calls. find_parameters def find_parameters ( self , include_factory_calls : 'bool' = True ) -> 'set[Parameter[Any]]' Crawl steps for parameter references. As the workflow does not hold its own list of parameters, this dynamically finds them. Returns: Type Description None Set of all references to parameters across the steps. register_task def register_task ( self , fn : 'Lazy' ) -> 'Task' Note the existence of a lazy-evaluatable function, and wrap it as a Task . Parameters: Name Type Description Default fn None the wrapped function. None Returns: Type Description None A new Task that wraps the function, and is retained in the Workflow.tasks dict. remap def remap ( self , step_id : 'str' ) -> 'str' Apply name simplification if requested. Parameters: Name Type Description Default step_id None step to check. None Returns: Type Description None Same ID or a remapped name. set_result def set_result ( self , result : 'Basic | list[Basic] | tuple[Basic]' ) -> 'None' Choose the result step. Sets a step as being the result for the entire workflow. When we evaluate a dynamic workflow, the engine (e.g. dask) creates a graph to realize the result of a single collection. Similarly, in the static case, we need to have a result that drives the calculation. Parameters: Name Type Description Default result None reference to the chosen step. None simplify_ids def simplify_ids ( self , infix : 'list[str] | None' = None ) -> 'None' Work out mapping to simple ints from hashes. Goes through and numbers each step by the order of use of its task. WorkflowLinkedComponent class WorkflowLinkedComponent ( * args , ** kwargs ) Protocol for objects dynamically tied to a Workflow . 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None","title":"Attributes"},{"location":"reference/dewret/workflow/#ancestors-in-mro","text":"dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#descendants","text":"dewret.workflow.NestedStep dewret.workflow.Step","title":"Descendants"},{"location":"reference/dewret/workflow/#class-variables","text":"positional_args","title":"Class variables"},{"location":"reference/dewret/workflow/#instance-variables","text":"id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. 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None","title":"set_workflow"},{"location":"reference/dewret/workflow/#factorycall","text":"class FactoryCall ( workflow : 'Workflow' , task : 'Task | Workflow' , arguments : 'Mapping[str, Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Call to a factory function.","title":"FactoryCall"},{"location":"reference/dewret/workflow/#ancestors-in-mro_1","text":"dewret.workflow.Step dewret.workflow.BaseStep dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#class-variables_1","text":"positional_args","title":"Class variables"},{"location":"reference/dewret/workflow/#instance-variables_1","text":"id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value.","title":"Instance variables"},{"location":"reference/dewret/workflow/#methods_1","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#make_reference_1","text":"def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None","title":"make_reference"},{"location":"reference/dewret/workflow/#set_workflow_1","text":"def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None","title":"set_workflow"},{"location":"reference/dewret/workflow/#fieldablemixin","text":"class FieldableMixin ( self : 'FieldableProtocol' , * args : 'Any' , field : 'str | int | tuple[str | int, ...] | None' = None , ** kwargs : 'Any' ) Tooling for enhancing a type with referenceable fields.","title":"FieldableMixin"},{"location":"reference/dewret/workflow/#descendants_1","text":"dewret.workflow.ParameterReference dewret.workflow.StepReference","title":"Descendants"},{"location":"reference/dewret/workflow/#methods_2","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#find_field","text":"def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference.","title":"find_field"},{"location":"reference/dewret/workflow/#fieldableprotocol","text":"class FieldableProtocol ( * args : 'Any' , field : 'str | None' = None , ** kwargs : 'Any' ) Expected interfaces for a type that can take fields.","title":"FieldableProtocol"},{"location":"reference/dewret/workflow/#attributes_1","text":"Name Type Description Default field None tuple representing the named fields, either strings or integers. None","title":"Attributes"},{"location":"reference/dewret/workflow/#ancestors-in-mro_2","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#instance-variables_2","text":"name The name for the target, accounting for the field.","title":"Instance variables"},{"location":"reference/dewret/workflow/#iterableparameterreference","text":"class IterableParameterReference ( typ : type [ ~ U ] | None = None , ** kwargs : Any ) Iterable form of parameter references.","title":"IterableParameterReference"},{"location":"reference/dewret/workflow/#ancestors-in-mro_3","text":"dewret.core.IterableMixin dewret.workflow.ParameterReference dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#class-variables_2","text":"ParameterReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/workflow/#static-methods","text":"","title":"Static methods"},{"location":"reference/dewret/workflow/#class_key","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/workflow/#fromiter","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/workflow/#examples","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/workflow/#instance-variables_3","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/workflow/#examples_1","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/workflow/#notes","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/workflow/#examples_2","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/workflow/#examples_3","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference.","title":"Examples"},{"location":"reference/dewret/workflow/#methods_3","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#adjoint","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/workflow/#apart","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/workflow/#args_cnc","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/workflow/#explanation","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_4","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/workflow/#as_base_exp","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/workflow/#as_coeff_add","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/workflow/#as_coeff_mul","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/workflow/#as_coeff_add_1","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/workflow/#as_coeff_exponent","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/workflow/#as_coeff_mul_1","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/workflow/#as_coefficient","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/workflow/#examples_5","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#as_coefficients_dict","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/workflow/#examples_6","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/workflow/#as_content_primitive","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/workflow/#examples_7","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/workflow/#as_dummy","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/workflow/#examples_8","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/workflow/#notes_1","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/workflow/#as_expr","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/workflow/#examples_9","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/workflow/#as_independent","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/workflow/#examples_10","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_1","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/workflow/#as_leading_term","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/workflow/#examples_11","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/workflow/#as_numer_denom","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/workflow/#see-also_2","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/workflow/#as_ordered_factors","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/workflow/#as_ordered_terms","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/workflow/#examples_12","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/workflow/#as_poly","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/workflow/#explanation_1","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/workflow/#as_powers_dict","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/workflow/#see-also_3","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/workflow/#as_real_imag","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/workflow/#as_set","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/workflow/#examples_13","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/workflow/#as_terms","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/workflow/#aseries","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/workflow/#parameters","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_14","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/workflow/#notes_2","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/workflow/#references","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/workflow/#see-also_4","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/workflow/#atoms","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/workflow/#examples_15","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/workflow/#cancel","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/workflow/#coeff","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/workflow/#explanation_2","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_16","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_5","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#collect","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/workflow/#combsimp","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/workflow/#compare","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/workflow/#examples_17","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/workflow/#compute_leading_term","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/workflow/#conjugate","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/workflow/#copy","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/workflow/#could_extract_minus_sign","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/workflow/#examples_18","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/workflow/#count","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/workflow/#count_ops","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/workflow/#diff","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/workflow/#dir","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/workflow/#doit","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/workflow/#dummy_eq","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/workflow/#examples_19","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/workflow/#equals","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/workflow/#explanation_3","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/workflow/#evalf","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/workflow/#parameters_1","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_3","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#expand","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/workflow/#extract_additively","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/workflow/#examples_20","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_6","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/workflow/#extract_branch_factor","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/workflow/#extract_multiplicatively","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/workflow/#examples_21","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/workflow/#factor","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/workflow/#find","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/workflow/#find_field_1","text":"def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference.","title":"find_field"},{"location":"reference/dewret/workflow/#fourier_series","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/workflow/#fps","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/workflow/#gammasimp","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/workflow/#geto","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/workflow/#getn","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/workflow/#explanation_4","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_22","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/workflow/#has","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/workflow/#examples_23","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/workflow/#has_free","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/workflow/#examples_24","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/workflow/#has_xfree","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/workflow/#examples_25","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/workflow/#integrate","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/workflow/#invert","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/workflow/#see-also_7","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/workflow/#is_algebraic_expr","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/workflow/#examples_26","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_8","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/workflow/#references_1","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/workflow/#is_constant","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/workflow/#explanation_5","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_27","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/workflow/#is_hypergeometric","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/workflow/#is_meromorphic","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/workflow/#examples_28","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/workflow/#is_polynomial","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/workflow/#examples_29","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/workflow/#is_rational_function","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/workflow/#examples_30","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/workflow/#is_same","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/workflow/#examples_31","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/workflow/#leadterm","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/workflow/#examples_32","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/workflow/#limit","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/workflow/#lseries","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/workflow/#match","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/workflow/#examples_33","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/workflow/#matches","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/workflow/#examples_34","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/workflow/#n","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/workflow/#parameters_2","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_4","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#normal","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/workflow/#see-also_9","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/workflow/#nseries","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/workflow/#examples_35","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/workflow/#nsimplify","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/workflow/#powsimp","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/workflow/#primitive","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/workflow/#examples_36","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/workflow/#radsimp","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/workflow/#ratsimp","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/workflow/#rcall","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/workflow/#refine","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/workflow/#removeo","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/workflow/#replace","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/workflow/#examples_37","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_10","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/workflow/#rewrite","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/workflow/#parameters_3","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_38","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/workflow/#round","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/workflow/#examples_39","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/workflow/#notes_5","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/workflow/#separate","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/workflow/#series","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/workflow/#parameters_4","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_40","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_1","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/workflow/#raises","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/workflow/#simplify","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/workflow/#sort_key","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/workflow/#examples_41","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/workflow/#subs","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/workflow/#examples_42","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_11","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/workflow/#taylor_term","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/workflow/#to_nnf","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/workflow/#together","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/workflow/#transpose","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/workflow/#trigsimp","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/workflow/#xreplace","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/workflow/#parameters_5","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/workflow/#returns_2","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/workflow/#examples_43","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_12","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/workflow/#iterablestepreference","text":"class IterableStepReference ( typ : type [ ~ U ] | None = None , ** kwargs : Any ) Iterable form of a step reference.","title":"IterableStepReference"},{"location":"reference/dewret/workflow/#ancestors-in-mro_4","text":"dewret.core.IterableMixin dewret.workflow.StepReference dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#class-variables_3","text":"StepReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/workflow/#static-methods_1","text":"","title":"Static methods"},{"location":"reference/dewret/workflow/#class_key_1","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/workflow/#fromiter_1","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/workflow/#examples_44","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/workflow/#instance-variables_4","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/workflow/#examples_45","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/workflow/#notes_6","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/workflow/#examples_46","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/workflow/#examples_47","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference.","title":"Examples"},{"location":"reference/dewret/workflow/#methods_4","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#adjoint_1","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/workflow/#apart_1","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/workflow/#args_cnc_1","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/workflow/#explanation_6","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_48","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/workflow/#as_base_exp_1","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/workflow/#as_coeff_add_2","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/workflow/#as_coeff_mul_2","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/workflow/#as_coeff_add_3","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/workflow/#as_coeff_exponent_1","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/workflow/#as_coeff_mul_3","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/workflow/#as_coefficient_1","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/workflow/#examples_49","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_13","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#as_coefficients_dict_1","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/workflow/#examples_50","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/workflow/#as_content_primitive_1","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/workflow/#examples_51","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/workflow/#as_dummy_1","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/workflow/#examples_52","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/workflow/#notes_7","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/workflow/#as_expr_1","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/workflow/#examples_53","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/workflow/#as_independent_1","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/workflow/#examples_54","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_14","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/workflow/#as_leading_term_1","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/workflow/#examples_55","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/workflow/#as_numer_denom_1","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/workflow/#see-also_15","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/workflow/#as_ordered_factors_1","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/workflow/#as_ordered_terms_1","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/workflow/#examples_56","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/workflow/#as_poly_1","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/workflow/#explanation_7","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/workflow/#as_powers_dict_1","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/workflow/#see-also_16","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/workflow/#as_real_imag_1","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/workflow/#as_set_1","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/workflow/#examples_57","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/workflow/#as_terms_1","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/workflow/#aseries_1","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/workflow/#parameters_6","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_58","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_3","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/workflow/#notes_8","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/workflow/#references_2","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/workflow/#see-also_17","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/workflow/#atoms_1","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/workflow/#examples_59","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/workflow/#cancel_1","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/workflow/#coeff_1","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/workflow/#explanation_8","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_60","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_18","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#collect_1","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/workflow/#combsimp_1","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/workflow/#compare_1","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/workflow/#examples_61","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/workflow/#compute_leading_term_1","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/workflow/#conjugate_1","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/workflow/#copy_1","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/workflow/#could_extract_minus_sign_1","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/workflow/#examples_62","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/workflow/#count_1","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/workflow/#count_ops_1","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/workflow/#diff_1","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/workflow/#dir_1","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/workflow/#doit_1","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/workflow/#dummy_eq_1","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/workflow/#examples_63","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/workflow/#equals_1","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/workflow/#explanation_9","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/workflow/#evalf_1","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/workflow/#parameters_7","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_9","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#expand_1","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/workflow/#extract_additively_1","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/workflow/#examples_64","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_19","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/workflow/#extract_branch_factor_1","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/workflow/#extract_multiplicatively_1","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/workflow/#examples_65","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/workflow/#factor_1","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/workflow/#find_1","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/workflow/#find_field_2","text":"def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference.","title":"find_field"},{"location":"reference/dewret/workflow/#fourier_series_1","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/workflow/#fps_1","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/workflow/#gammasimp_1","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/workflow/#geto_1","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/workflow/#getn_1","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/workflow/#explanation_10","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_66","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/workflow/#has_1","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/workflow/#examples_67","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/workflow/#has_free_1","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/workflow/#examples_68","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/workflow/#has_xfree_1","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/workflow/#examples_69","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/workflow/#integrate_1","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/workflow/#invert_1","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/workflow/#see-also_20","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/workflow/#is_algebraic_expr_1","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/workflow/#examples_70","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_21","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/workflow/#references_3","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/workflow/#is_constant_1","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/workflow/#explanation_11","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_71","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/workflow/#is_hypergeometric_1","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/workflow/#is_meromorphic_1","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/workflow/#examples_72","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/workflow/#is_polynomial_1","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/workflow/#examples_73","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/workflow/#is_rational_function_1","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/workflow/#examples_74","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/workflow/#is_same_1","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/workflow/#examples_75","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/workflow/#leadterm_1","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/workflow/#examples_76","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/workflow/#limit_1","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/workflow/#lseries_1","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/workflow/#match_1","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/workflow/#examples_77","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/workflow/#matches_1","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/workflow/#examples_78","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/workflow/#n_1","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/workflow/#parameters_8","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_10","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#normal_1","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/workflow/#see-also_22","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/workflow/#nseries_1","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/workflow/#examples_79","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/workflow/#nsimplify_1","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/workflow/#powsimp_1","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/workflow/#primitive_1","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/workflow/#examples_80","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/workflow/#radsimp_1","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/workflow/#ratsimp_1","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/workflow/#rcall_1","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/workflow/#refine_1","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/workflow/#removeo_1","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/workflow/#replace_1","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/workflow/#examples_81","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_23","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/workflow/#rewrite_1","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/workflow/#parameters_9","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_82","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/workflow/#round_1","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/workflow/#examples_83","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/workflow/#notes_11","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/workflow/#separate_1","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/workflow/#series_1","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/workflow/#parameters_10","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_84","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_4","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/workflow/#raises_1","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/workflow/#simplify_1","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/workflow/#sort_key_1","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/workflow/#examples_85","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/workflow/#subs_1","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/workflow/#examples_86","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_24","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/workflow/#taylor_term_1","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/workflow/#to_nnf_1","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/workflow/#together_1","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/workflow/#transpose_1","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/workflow/#trigsimp_1","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/workflow/#xreplace_1","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/workflow/#parameters_11","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/workflow/#returns_5","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/workflow/#examples_87","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_25","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/workflow/#lazy","text":"class Lazy ( * args , ** kwargs ) Requirements for a lazy-evaluatable function.","title":"Lazy"},{"location":"reference/dewret/workflow/#ancestors-in-mro_5","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#descendants_2","text":"dewret.workflow.LazyEvaluation","title":"Descendants"},{"location":"reference/dewret/workflow/#lazyevaluation","text":"class LazyEvaluation ( fn : 'Callable[..., RetType]' ) Tracks a single evaluation of a lazy function.","title":"LazyEvaluation"},{"location":"reference/dewret/workflow/#ancestors-in-mro_6","text":"dewret.workflow.Lazy typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#nestedstep","text":"class NestedStep ( workflow : 'Workflow' , name : 'str' , subworkflow : 'Workflow' , arguments : 'Mapping[str, Basic | Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Calling out to a subworkflow. Type of BaseStep to call a subworkflow, which holds a reference to it.","title":"NestedStep"},{"location":"reference/dewret/workflow/#ancestors-in-mro_7","text":"dewret.workflow.BaseStep dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#class-variables_4","text":"positional_args","title":"Class variables"},{"location":"reference/dewret/workflow/#instance-variables_5","text":"id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value. subworkflow Subworkflow that is wrapped.","title":"Instance variables"},{"location":"reference/dewret/workflow/#methods_5","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#make_reference_2","text":"def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None","title":"make_reference"},{"location":"reference/dewret/workflow/#set_workflow_2","text":"def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None","title":"set_workflow"},{"location":"reference/dewret/workflow/#parameter","text":"class Parameter ( name : 'str' , default : 'T | UnsetType[T]' , tethered : 'Literal[False] | None | Step | Workflow' = None , autoname : 'bool' = False , typ : 'type[T] | Unset' = < dewret . utils . Unset object at 0x7f1e70dddb90 > ) Global parameter. Independent parameter that will be used when a task is spotted reaching outside its scope. This wraps the variable it uses. To allow for potential arithmetic operations, etc. it is a Sympy symbol.","title":"Parameter"},{"location":"reference/dewret/workflow/#attributes_2","text":"Name Type Description Default name None name of the parameter. None default None captured default value from the original value. None","title":"Attributes"},{"location":"reference/dewret/workflow/#ancestors-in-mro_8","text":"typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#class-variables_5","text":"autoname default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/workflow/#static-methods_2","text":"","title":"Static methods"},{"location":"reference/dewret/workflow/#class_key_2","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/workflow/#fromiter_2","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/workflow/#examples_88","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/workflow/#is_loopable","text":"def is_loopable ( typ : 'type' ) -> 'bool' Checks if this type can be looped over. In particular, checks if this is an iterable that is NOT a str or bytes, possibly disguised behind an Annotated. Parameters: Name Type Description Default typ None type to check. None","title":"is_loopable"},{"location":"reference/dewret/workflow/#instance-variables_6","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/workflow/#examples_89","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/workflow/#notes_12","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/workflow/#examples_90","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} default Retrieve default value for this parameter, or an unset token. expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/workflow/#examples_91","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Extended name, suitable for rendering. This attempts to create a unique name by tying the parameter to a step if the user has not explicitly provided a name, ideally the one where we discovered it.","title":"Examples"},{"location":"reference/dewret/workflow/#methods_6","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#adjoint_2","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/workflow/#apart_2","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/workflow/#args_cnc_2","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/workflow/#explanation_12","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_92","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/workflow/#as_base_exp_2","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/workflow/#as_coeff_add_4","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/workflow/#as_coeff_mul_4","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/workflow/#as_coeff_add_5","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/workflow/#as_coeff_exponent_2","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/workflow/#as_coeff_mul_5","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/workflow/#as_coefficient_2","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/workflow/#examples_93","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_26","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#as_coefficients_dict_2","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/workflow/#examples_94","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/workflow/#as_content_primitive_2","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/workflow/#examples_95","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/workflow/#as_dummy_2","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/workflow/#examples_96","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/workflow/#notes_13","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/workflow/#as_expr_2","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/workflow/#examples_97","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/workflow/#as_independent_2","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/workflow/#examples_98","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_27","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/workflow/#as_leading_term_2","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/workflow/#examples_99","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/workflow/#as_numer_denom_2","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/workflow/#see-also_28","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/workflow/#as_ordered_factors_2","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/workflow/#as_ordered_terms_2","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/workflow/#examples_100","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/workflow/#as_poly_2","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/workflow/#explanation_13","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/workflow/#as_powers_dict_2","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/workflow/#see-also_29","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/workflow/#as_real_imag_2","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/workflow/#as_set_2","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/workflow/#examples_101","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/workflow/#as_terms_2","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/workflow/#aseries_2","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/workflow/#parameters_12","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_102","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_6","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/workflow/#notes_14","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/workflow/#references_4","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/workflow/#see-also_30","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/workflow/#atoms_2","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/workflow/#examples_103","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/workflow/#cancel_2","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/workflow/#coeff_2","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/workflow/#explanation_14","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_104","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_31","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#collect_2","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/workflow/#combsimp_2","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/workflow/#compare_2","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/workflow/#examples_105","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/workflow/#compute_leading_term_2","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/workflow/#conjugate_2","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/workflow/#copy_2","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/workflow/#could_extract_minus_sign_2","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/workflow/#examples_106","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/workflow/#count_2","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/workflow/#count_ops_2","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/workflow/#diff_2","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/workflow/#dir_2","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/workflow/#doit_2","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/workflow/#dummy_eq_2","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/workflow/#examples_107","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/workflow/#equals_2","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/workflow/#explanation_15","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/workflow/#evalf_2","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/workflow/#parameters_13","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_15","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#expand_2","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/workflow/#extract_additively_2","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/workflow/#examples_108","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_32","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/workflow/#extract_branch_factor_2","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/workflow/#extract_multiplicatively_2","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/workflow/#examples_109","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/workflow/#factor_2","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/workflow/#find_2","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/workflow/#fourier_series_2","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/workflow/#fps_2","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/workflow/#gammasimp_2","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/workflow/#geto_2","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/workflow/#getn_2","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/workflow/#explanation_16","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_110","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/workflow/#has_2","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/workflow/#examples_111","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/workflow/#has_free_2","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/workflow/#examples_112","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/workflow/#has_xfree_2","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/workflow/#examples_113","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/workflow/#integrate_2","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/workflow/#invert_2","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/workflow/#see-also_33","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/workflow/#is_algebraic_expr_2","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/workflow/#examples_114","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_34","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/workflow/#references_5","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/workflow/#is_constant_2","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/workflow/#explanation_17","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_115","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/workflow/#is_hypergeometric_2","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/workflow/#is_meromorphic_2","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/workflow/#examples_116","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/workflow/#is_polynomial_2","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/workflow/#examples_117","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/workflow/#is_rational_function_2","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/workflow/#examples_118","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/workflow/#is_same_2","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/workflow/#examples_119","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/workflow/#leadterm_2","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/workflow/#examples_120","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/workflow/#limit_2","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/workflow/#lseries_2","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/workflow/#make_reference_3","text":"def make_reference ( self , ** kwargs : 'Any' ) -> \"'ParameterReference[T]'\" Creates a new reference for the parameter. The kwargs will be passed to the constructor, but the Parameters: Name Type Description Default typ None type of the new reference's target. None **kwargs None arguments to pass to the constructor. None","title":"make_reference"},{"location":"reference/dewret/workflow/#match_2","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/workflow/#examples_121","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/workflow/#matches_2","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/workflow/#examples_122","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/workflow/#n_2","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/workflow/#parameters_14","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_16","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#normal_2","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/workflow/#see-also_35","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/workflow/#nseries_2","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/workflow/#examples_123","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/workflow/#nsimplify_2","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/workflow/#powsimp_2","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/workflow/#primitive_2","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/workflow/#examples_124","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/workflow/#radsimp_2","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/workflow/#ratsimp_2","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/workflow/#rcall_2","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/workflow/#refine_2","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/workflow/#register_caller","text":"def register_caller ( self , caller : 'BaseStep' ) -> 'None' Capture a step that uses this parameter. Gathers together the steps using this parameter. The first found will be recorded as the tethered step, and used for forming the name.","title":"register_caller"},{"location":"reference/dewret/workflow/#removeo_2","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/workflow/#replace_2","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/workflow/#examples_125","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_36","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/workflow/#rewrite_2","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/workflow/#parameters_15","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_126","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/workflow/#round_2","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/workflow/#examples_127","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/workflow/#notes_17","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/workflow/#separate_2","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/workflow/#series_2","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/workflow/#parameters_16","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_128","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_7","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/workflow/#raises_2","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/workflow/#simplify_2","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/workflow/#sort_key_2","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/workflow/#examples_129","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/workflow/#subs_2","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/workflow/#examples_130","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_37","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/workflow/#taylor_term_2","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/workflow/#to_nnf_2","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/workflow/#together_2","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/workflow/#transpose_2","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/workflow/#trigsimp_2","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/workflow/#xreplace_2","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/workflow/#parameters_17","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/workflow/#returns_8","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/workflow/#examples_131","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_38","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/workflow/#parameterreference","text":"class ParameterReference ( parameter : 'Parameter[U]' , * args : 'Any' , typ : 'type[U] | None' = None , ** kwargs : 'Any' ) Reference to an individual Parameter . Allows us to refer to the outputs of a Parameter in subsequent Parameter arguments.","title":"ParameterReference"},{"location":"reference/dewret/workflow/#attributes_3","text":"Name Type Description Default parameter None Parameter referred to. None workflow None Related workflow. In this case, as Parameters are generic but ParameterReferences are specific, this carries the actual workflow reference. None","title":"Attributes"},{"location":"reference/dewret/workflow/#ancestors-in-mro_9","text":"dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#descendants_3","text":"dewret.workflow.IterableParameterReference","title":"Descendants"},{"location":"reference/dewret/workflow/#class-variables_6","text":"ParameterReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/workflow/#static-methods_3","text":"","title":"Static methods"},{"location":"reference/dewret/workflow/#class_key_3","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/workflow/#fromiter_3","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/workflow/#examples_132","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/workflow/#instance-variables_7","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/workflow/#examples_133","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/workflow/#notes_18","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/workflow/#examples_134","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/workflow/#examples_135","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference.","title":"Examples"},{"location":"reference/dewret/workflow/#methods_7","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#adjoint_3","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/workflow/#apart_3","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/workflow/#args_cnc_3","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/workflow/#explanation_18","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_136","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/workflow/#as_base_exp_3","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/workflow/#as_coeff_add_6","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/workflow/#as_coeff_mul_6","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/workflow/#as_coeff_add_7","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/workflow/#as_coeff_exponent_3","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/workflow/#as_coeff_mul_7","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/workflow/#as_coefficient_3","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/workflow/#examples_137","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_39","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#as_coefficients_dict_3","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/workflow/#examples_138","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/workflow/#as_content_primitive_3","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/workflow/#examples_139","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/workflow/#as_dummy_3","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/workflow/#examples_140","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/workflow/#notes_19","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/workflow/#as_expr_3","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/workflow/#examples_141","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/workflow/#as_independent_3","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/workflow/#examples_142","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_40","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/workflow/#as_leading_term_3","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/workflow/#examples_143","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/workflow/#as_numer_denom_3","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/workflow/#see-also_41","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/workflow/#as_ordered_factors_3","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/workflow/#as_ordered_terms_3","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/workflow/#examples_144","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/workflow/#as_poly_3","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/workflow/#explanation_19","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/workflow/#as_powers_dict_3","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/workflow/#see-also_42","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/workflow/#as_real_imag_3","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/workflow/#as_set_3","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/workflow/#examples_145","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/workflow/#as_terms_3","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/workflow/#aseries_3","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/workflow/#parameters_18","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_146","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_9","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/workflow/#notes_20","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/workflow/#references_6","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/workflow/#see-also_43","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/workflow/#atoms_3","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/workflow/#examples_147","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/workflow/#cancel_3","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/workflow/#coeff_3","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/workflow/#explanation_20","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_148","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_44","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#collect_3","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/workflow/#combsimp_3","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/workflow/#compare_3","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/workflow/#examples_149","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/workflow/#compute_leading_term_3","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/workflow/#conjugate_3","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/workflow/#copy_3","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/workflow/#could_extract_minus_sign_3","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/workflow/#examples_150","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/workflow/#count_3","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/workflow/#count_ops_3","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/workflow/#diff_3","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/workflow/#dir_3","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/workflow/#doit_3","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/workflow/#dummy_eq_3","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/workflow/#examples_151","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/workflow/#equals_3","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/workflow/#explanation_21","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/workflow/#evalf_3","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/workflow/#parameters_19","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_21","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#expand_3","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/workflow/#extract_additively_3","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/workflow/#examples_152","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_45","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/workflow/#extract_branch_factor_3","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/workflow/#extract_multiplicatively_3","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/workflow/#examples_153","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/workflow/#factor_3","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/workflow/#find_3","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/workflow/#find_field_3","text":"def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference.","title":"find_field"},{"location":"reference/dewret/workflow/#fourier_series_3","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/workflow/#fps_3","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/workflow/#gammasimp_3","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/workflow/#geto_3","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/workflow/#getn_3","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/workflow/#explanation_22","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_154","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/workflow/#has_3","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/workflow/#examples_155","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/workflow/#has_free_3","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/workflow/#examples_156","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/workflow/#has_xfree_3","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/workflow/#examples_157","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/workflow/#integrate_3","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/workflow/#invert_3","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/workflow/#see-also_46","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/workflow/#is_algebraic_expr_3","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/workflow/#examples_158","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_47","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/workflow/#references_7","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/workflow/#is_constant_3","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/workflow/#explanation_23","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_159","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/workflow/#is_hypergeometric_3","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/workflow/#is_meromorphic_3","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/workflow/#examples_160","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/workflow/#is_polynomial_3","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/workflow/#examples_161","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/workflow/#is_rational_function_3","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/workflow/#examples_162","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/workflow/#is_same_3","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/workflow/#examples_163","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/workflow/#leadterm_3","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/workflow/#examples_164","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/workflow/#limit_3","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/workflow/#lseries_3","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/workflow/#match_3","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/workflow/#examples_165","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/workflow/#matches_3","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/workflow/#examples_166","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/workflow/#n_3","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/workflow/#parameters_20","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_22","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#normal_3","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/workflow/#see-also_48","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/workflow/#nseries_3","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/workflow/#examples_167","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/workflow/#nsimplify_3","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/workflow/#powsimp_3","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/workflow/#primitive_3","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/workflow/#examples_168","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/workflow/#radsimp_3","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/workflow/#ratsimp_3","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/workflow/#rcall_3","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/workflow/#refine_3","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/workflow/#removeo_3","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/workflow/#replace_3","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/workflow/#examples_169","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_49","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/workflow/#rewrite_3","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/workflow/#parameters_21","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_170","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/workflow/#round_3","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/workflow/#examples_171","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/workflow/#notes_23","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/workflow/#separate_3","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/workflow/#series_3","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/workflow/#parameters_22","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_172","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_10","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/workflow/#raises_3","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/workflow/#simplify_3","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/workflow/#sort_key_3","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/workflow/#examples_173","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/workflow/#subs_3","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/workflow/#examples_174","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_50","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/workflow/#taylor_term_3","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/workflow/#to_nnf_3","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/workflow/#together_3","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/workflow/#transpose_3","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/workflow/#trigsimp_3","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/workflow/#xreplace_3","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/workflow/#parameters_23","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/workflow/#returns_11","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/workflow/#examples_175","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_51","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/workflow/#step","text":"class Step ( workflow : 'Workflow' , task : 'Task | Workflow' , arguments : 'Mapping[str, Reference[Any] | Raw]' , raw_as_parameter : 'bool' = False ) Regular step.","title":"Step"},{"location":"reference/dewret/workflow/#ancestors-in-mro_10","text":"dewret.workflow.BaseStep dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#descendants_4","text":"dewret.workflow.FactoryCall","title":"Descendants"},{"location":"reference/dewret/workflow/#class-variables_7","text":"positional_args","title":"Class variables"},{"location":"reference/dewret/workflow/#instance-variables_8","text":"id Consistent ID based on the value. name Name for this step. May be remapped by the workflow to something nicer than the ID. return_type Take the type of the wrapped function from the target. Unwraps and inspects the signature, meaning that the original wrapped function must have a typehint for the return value.","title":"Instance variables"},{"location":"reference/dewret/workflow/#methods_8","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#make_reference_4","text":"def make_reference ( self , ** kwargs : 'Any' ) -> \"'StepReference[T]'\" Create a reference to this step. Builds a reference to the (result of) this step, which will be iterable if appropriate. Parameters: Name Type Description Default **kwargs None arguments for reference constructor, which will be supplemented appropriately. None","title":"make_reference"},{"location":"reference/dewret/workflow/#set_workflow_3","text":"def set_workflow ( self , workflow : 'Workflow' , with_arguments : 'bool' = True ) -> 'None' Move the step reference to another workflow. This method is primarily intended to be called by a step, allowing it to switch to a new workflow. It also updates the workflow reference for any arguments that are steps themselves, if specified. Parameters: Name Type Description Default workflow None The new target workflow to which the step should be moved. None with_arguments None If True, also update the workflow reference for the step's arguments. None","title":"set_workflow"},{"location":"reference/dewret/workflow/#stepreference","text":"class StepReference ( step : 'BaseStep' , * args : 'Any' , typ : 'type[U] | None' = None , ** kwargs : 'Any' ) Reference to an individual Step . Allows us to refer to the outputs of a Step in subsequent Step arguments.","title":"StepReference"},{"location":"reference/dewret/workflow/#attributes_4","text":"Name Type Description Default _ None metadata wrapping the Step referred to. None","title":"Attributes"},{"location":"reference/dewret/workflow/#ancestors-in-mro_11","text":"dewret.workflow.FieldableMixin dewret.core.Reference typing.Generic sympy.core.symbol.Symbol sympy.core.expr.AtomicExpr sympy.core.basic.Atom sympy.core.expr.Expr sympy.logic.boolalg.Boolean sympy.core.basic.Basic sympy.printing.defaults.Printable sympy.core.evalf.EvalfMixin dewret.core.WorkflowComponent","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#descendants_5","text":"dewret.workflow.IterableStepReference","title":"Descendants"},{"location":"reference/dewret/workflow/#class-variables_8","text":"StepReferenceMetadata default_assumptions is_Add is_AlgebraicNumber is_Atom is_Boolean is_Derivative is_Dummy is_Equality is_Float is_Function is_Indexed is_Integer is_MatAdd is_MatMul is_Matrix is_Mul is_Not is_Number is_NumberSymbol is_Order is_Piecewise is_Point is_Poly is_Pow is_Rational is_Relational is_Symbol is_Vector is_Wild is_comparable is_number is_scalar is_symbol","title":"Class variables"},{"location":"reference/dewret/workflow/#static-methods_4","text":"","title":"Static methods"},{"location":"reference/dewret/workflow/#class_key_4","text":"def class_key ( ) Nice order of classes.","title":"class_key"},{"location":"reference/dewret/workflow/#fromiter_4","text":"def fromiter ( args , ** assumptions ) Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.","title":"fromiter"},{"location":"reference/dewret/workflow/#examples_176","text":"from sympy import Tuple Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4)","title":"Examples"},{"location":"reference/dewret/workflow/#instance-variables_9","text":"args Returns a tuple of arguments of 'self'.","title":"Instance variables"},{"location":"reference/dewret/workflow/#examples_177","text":"from sympy import cot from sympy.abc import x, y cot(x).args (x,) cot(x).args[0] x (x*y).args (x, y) (x*y).args[1] y","title":"Examples"},{"location":"reference/dewret/workflow/#notes_24","text":"Never use self._args, always use self.args. Only use _args in new when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). assumptions0 binary_symbols canonical_variables Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.","title":"Notes"},{"location":"reference/dewret/workflow/#examples_178","text":"from sympy import Lambda from sympy.abc import x Lambda(x, 2*x).canonical_variables {x: _0} expr_free_symbols free_symbols func The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args)","title":"Examples"},{"location":"reference/dewret/workflow/#examples_179","text":"from sympy.abc import x a = 2 x a.func a.args (2, x) a.func( a.args) 2 x a == a.func( a.args) True is_algebraic is_antihermitian is_commutative is_complex is_composite is_even is_extended_negative is_extended_nonnegative is_extended_nonpositive is_extended_nonzero is_extended_positive is_extended_real is_finite is_hermitian is_imaginary is_infinite is_integer is_irrational is_negative is_noninteger is_nonnegative is_nonpositive is_nonzero is_odd is_polar is_positive is_prime is_rational is_real is_transcendental is_zero kind name Printable name of the reference.","title":"Examples"},{"location":"reference/dewret/workflow/#methods_9","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#adjoint_4","text":"def adjoint ( self )","title":"adjoint"},{"location":"reference/dewret/workflow/#apart_4","text":"def apart ( self , x = None , ** args ) See the apart function in sympy.polys","title":"apart"},{"location":"reference/dewret/workflow/#args_cnc_4","text":"def args_cnc ( self , cset = False , warn = True , split_1 = True ) Return [commutative factors, non-commutative factors] of self.","title":"args_cnc"},{"location":"reference/dewret/workflow/#explanation_24","text":"self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False. Note: -1 is always separated from a Number unless split_1 is False.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_180","text":"from sympy import symbols, oo A, B = symbols('A B', commutative=0) x, y = symbols('x y') (-2 x y).args_cnc() [[-1, 2, x, y], []] (-2.5 x).args_cnc() [[-1, 2.5, x], []] (-2 x A B y).args_cnc() [[-1, 2, x, y], [A, B]] (-2 x A B y).args_cnc(split_1=False) [[-2, x, y], [A, B]] (-2 x*y).args_cnc(cset=True) [{-1, 2, x, y}, []] The arg is always treated as a Mul: (-2 + x + A).args_cnc() [[], [x - 2 + A]] (-oo).args_cnc() # -oo is a singleton [[-1, oo], []]","title":"Examples"},{"location":"reference/dewret/workflow/#as_base_exp_4","text":"def as_base_exp ( self ) -> 'tuple[Expr, Expr]'","title":"as_base_exp"},{"location":"reference/dewret/workflow/#as_coeff_add_8","text":"def as_coeff_Add ( self , rational = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a summation.","title":"as_coeff_Add"},{"location":"reference/dewret/workflow/#as_coeff_mul_8","text":"def as_coeff_Mul ( self , rational : 'bool' = False ) -> \"tuple['Number', Expr]\" Efficiently extract the coefficient of a product.","title":"as_coeff_Mul"},{"location":"reference/dewret/workflow/#as_coeff_add_9","text":"def as_coeff_add ( self , * deps ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as an Add, a . c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of a ; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. if you know self is an Add and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_add() (3, ()) (3 + x).as_coeff_add() (3, (x,)) (3 + x + y).as_coeff_add(x) (y + 3, (x,)) (3 + y).as_coeff_add(x) (y + 3, ())","title":"as_coeff_add"},{"location":"reference/dewret/workflow/#as_coeff_exponent_4","text":"def as_coeff_exponent ( self , x ) -> 'tuple[Expr, Expr]' c*x**e -> c,e where x can be any symbolic expression.","title":"as_coeff_exponent"},{"location":"reference/dewret/workflow/#as_coeff_mul_9","text":"def as_coeff_mul ( self , * deps , ** kwargs ) -> 'tuple[Expr, tuple[Expr, ...]]' Return the tuple (c, args) where self is written as a Mul, m . c should be a Rational multiplied by any factors of the Mul that are independent of deps. args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. if you know self is a Mul and want only the head, use self.args[0]; if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; if you want to split self into an independent and dependent parts use self.as_independent(*deps) from sympy import S from sympy.abc import x, y (S(3)).as_coeff_mul() (3, ()) (3 x y).as_coeff_mul() (3, (x, y)) (3 x y).as_coeff_mul(x) (3 y, (x,)) (3 y).as_coeff_mul(x) (3*y, ())","title":"as_coeff_mul"},{"location":"reference/dewret/workflow/#as_coefficient_4","text":"def as_coefficient ( self , expr ) Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into the product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None.","title":"as_coefficient"},{"location":"reference/dewret/workflow/#examples_181","text":"from sympy import E, pi, sin, I, Poly from sympy.abc import x E.as_coefficient(E) 1 (2 E).as_coefficient(E) 2 (2 sin(E)*E).as_coefficient(E) Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.) (2 E + x E).as_coefficient(E) x + 2 _.args[0] # just want the exact match 2 p = Poly(2 E + x E); p Poly(x E + 2 E, x, E, domain='ZZ') p.coeff_monomial(E) 2 p.nth(0, 1) 2 Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.) (2 E x + x).as_coefficient(E) (2 E x + x).coeff(E) 2*x (E*(x + 1) + x).as_coefficient(E) (2 pi I).as_coefficient(pi I) 2 (2 I).as_coefficient(pi*I)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_52","text":"coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#as_coefficients_dict_4","text":"def as_coefficients_dict ( self , * syms ) Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.","title":"as_coefficients_dict"},{"location":"reference/dewret/workflow/#examples_182","text":"from sympy.abc import a, x, y (3 x + a x + 4).as_coefficients_dict() {1: 4, x: 3, a x: 1} _[a] 0 (3 a x).as_coefficients_dict() {a x: 3} (3 a x).as_coefficients_dict(x) {x: 3 a} (3 a x).as_coefficients_dict(y) {1: 3 a*x}","title":"Examples"},{"location":"reference/dewret/workflow/#as_content_primitive_4","text":"def as_content_primitive ( self , radical = False , clear = True ) This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo . The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).","title":"as_content_primitive"},{"location":"reference/dewret/workflow/#examples_183","text":"from sympy import sqrt from sympy.abc import x, y, z eq = 2 + 2 x + 2 y (3 + 3 y) The as_content_primitive function is recursive and retains structure: eq.as_content_primitive() (2, x + 3 y (y + 1) + 1) Integer powers will have Rationals extracted from the base: ((2 + 6 x) 2).as_content_primitive() (4, (3 x + 1) 2) ((2 + 6*x) (2 y)).as_content_primitive() (1, (2 (3 x + 1)) (2 y)) Terms may end up joining once their as_content_primitives are added: ((5 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (11, x (y + 1)) ((3 (x (1 + y)) + 2 x (3 + 3 y))).as_content_primitive() (9, x (y + 1)) ((3 (z (1 + y)) + 2.0 x (3 + 3 y))).as_content_primitive() (1, 6.0 x (y + 1) + 3 z (y + 1)) ((5 (x (1 + y)) + 2 x (3 + 3 y)) 2).as_content_primitive() (121, x 2 (y + 1) 2) ((x (1 + y) + 0.4 x (3 + 3 y)) 2).as_content_primitive() (1, 4.84 x 2*(y + 1) 2) Radical content can also be factored out of the primitive: (2 sqrt(2) + 4 sqrt(10)).as_content_primitive(radical=True) (2, sqrt(2) (1 + 2 sqrt(5))) If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients. (x/2 + y).as_content_primitive() (1/2, x + 2*y) (x/2 + y).as_content_primitive(clear=False) (1, x/2 + y)","title":"Examples"},{"location":"reference/dewret/workflow/#as_dummy_4","text":"def as_dummy ( self ) Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.","title":"as_dummy"},{"location":"reference/dewret/workflow/#examples_184","text":"from sympy import Integral, Symbol from sympy.abc import x r = Symbol('r', real=True) Integral(r, (r, x)).as_dummy() Integral( 0, (_0, x)) .variables[0].is_real is None True r.as_dummy() _r","title":"Examples"},{"location":"reference/dewret/workflow/#notes_25","text":"Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.","title":"Notes"},{"location":"reference/dewret/workflow/#as_expr_4","text":"def as_expr ( self , * gens ) Convert a polynomial to a SymPy expression.","title":"as_expr"},{"location":"reference/dewret/workflow/#examples_185","text":"from sympy import sin from sympy.abc import x, y f = (x 2 + x*y).as_poly(x, y) f.as_expr() x 2 + x*y sin(x).as_expr() sin(x)","title":"Examples"},{"location":"reference/dewret/workflow/#as_independent_4","text":"def as_independent ( self , * deps , ** hint ) -> 'tuple[Expr, Expr]' A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: separatevars() to change Mul, Add and Pow (including exp) into Mul .expand(mul=True) to change Add or Mul into Add .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints. For nonzero self , the returned tuple (i, d) has the following interpretation: i will has no variable that appears in deps d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul) if self is an Add then self = i + d if self is a Mul then self = i*d otherwise (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True","title":"as_independent"},{"location":"reference/dewret/workflow/#examples_186","text":"-- self is an Add from sympy import sin, cos, exp from sympy.abc import x, y, z (x + x y).as_independent(x) (0, x y + x) (x + x y).as_independent(y) (x, x y) (2 x sin(x) + y + x + z).as_independent(x) (y + z, 2 x sin(x) + x) (2 x sin(x) + y + x + z).as_independent(x, y) (z, 2 x sin(x) + x + y) -- self is a Mul (x sin(x) cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul from sympy import symbols n1, n2, n3 = symbols('n1 n2 n3', commutative=False) (n1 + n1 n2).as_independent(n2) (n1, n1 n2) (n2 n1 + n1 n2).as_independent(n2) (0, n1 n2 + n2 n1) (n1 n2 n3).as_independent(n1) (1, n1 n2 n3) (n1 n2 n3).as_independent(n2) (n1, n2 n3) ((x-n1) (x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: (sin(x)).as_independent(x) (1, sin(x)) (sin(x)).as_independent(y) (sin(x), 1) exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: (3 x).as_independent(x, as_Add=True) (0, 3 x) -- force self to be treated as a Mul: (3+x).as_independent(x, as_Add=False) (1, x + 3) (-3+x).as_independent(x, as_Add=False) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. (y*(-3+x)).as_independent(x) (y, x - 3) -- use .as_independent() for true independence testing instead of .has(). The former considers only symbols in the free symbols while the latter considers all symbols from sympy import Integral I = Integral(x, (x, 1, 2)) I.has(x) True x in I.free_symbols False I.as_independent(x) == (I, 1) True (I + x).as_independent(x) == (I, x) True Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values from sympy import separatevars, log separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) (x + x y).as_independent(y) (x, x y) separatevars(x + x y).as_independent(y) (x, y + 1) (x (1 + y)).as_independent(y) (x, y + 1) (x (1 + y)).expand(mul=True).as_independent(y) (x, x y) a, b=symbols('a b', positive=True) (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b))","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_53","text":"separatevars expand_log sympy.core.add.Add.as_two_terms sympy.core.mul.Mul.as_two_terms as_coeff_mul","title":"See Also"},{"location":"reference/dewret/workflow/#as_leading_term_4","text":"def as_leading_term ( self , * symbols , logx = None , cdir = 0 ) Returns the leading (nonzero) term of the series expansion of self. The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.","title":"as_leading_term"},{"location":"reference/dewret/workflow/#examples_187","text":"from sympy.abc import x (1 + x + x 2).as_leading_term(x) 1 (1/x 2 + x + x 2).as_leading_term(x) x (-2)","title":"Examples"},{"location":"reference/dewret/workflow/#as_numer_denom_4","text":"def as_numer_denom ( self ) Return the numerator and the denominator of an expression. expression -> a/b -> a, b This is just a stub that should be defined by an object's class methods to get anything else.","title":"as_numer_denom"},{"location":"reference/dewret/workflow/#see-also_54","text":"normal: return a/b instead of (a, b)","title":"See Also"},{"location":"reference/dewret/workflow/#as_ordered_factors_4","text":"def as_ordered_factors ( self , order = None ) Return list of ordered factors (if Mul) else [self].","title":"as_ordered_factors"},{"location":"reference/dewret/workflow/#as_ordered_terms_4","text":"def as_ordered_terms ( self , order = None , data = False ) Transform an expression to an ordered list of terms.","title":"as_ordered_terms"},{"location":"reference/dewret/workflow/#examples_188","text":"from sympy import sin, cos from sympy.abc import x (sin(x) 2*cos(x) + sin(x) 2 + 1).as_ordered_terms() [sin(x) 2*cos(x), sin(x) 2, 1]","title":"Examples"},{"location":"reference/dewret/workflow/#as_poly_4","text":"def as_poly ( self , * gens , ** args ) Converts self to a polynomial or returns None .","title":"as_poly"},{"location":"reference/dewret/workflow/#explanation_25","text":"from sympy import sin from sympy.abc import x, y print((x 2 + x*y).as_poly()) Poly(x 2 + x*y, x, y, domain='ZZ') print((x 2 + x*y).as_poly(x, y)) Poly(x 2 + x*y, x, y, domain='ZZ') print((x**2 + sin(y)).as_poly(x, y)) None","title":"Explanation"},{"location":"reference/dewret/workflow/#as_powers_dict_4","text":"def as_powers_dict ( self ) Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.","title":"as_powers_dict"},{"location":"reference/dewret/workflow/#see-also_55","text":"as_ordered_factors: An alternative for noncommutative applications, returning an ordered list of factors. args_cnc: Similar to as_ordered_factors, but guarantees separation of commutative and noncommutative factors.","title":"See Also"},{"location":"reference/dewret/workflow/#as_real_imag_4","text":"def as_real_imag ( self , deep = True , ** hints ) Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. from sympy import symbols, I x, y = symbols('x,y', real=True) (x + y*I).as_real_imag() (x, y) from sympy.abc import z, w (z + w*I).as_real_imag() (re(z) - im(w), re(w) + im(z))","title":"as_real_imag"},{"location":"reference/dewret/workflow/#as_set_4","text":"def as_set ( self ) Rewrites Boolean expression in terms of real sets.","title":"as_set"},{"location":"reference/dewret/workflow/#examples_189","text":"from sympy import Symbol, Eq, Or, And x = Symbol('x', real=True) Eq(x, 0).as_set() {0} (x > 0).as_set() Interval.open(0, oo) And(-2 < x, x < 2).as_set() Interval.open(-2, 2) Or(x < -2, 2 < x).as_set() Union(Interval.open(-oo, -2), Interval.open(2, oo))","title":"Examples"},{"location":"reference/dewret/workflow/#as_terms_4","text":"def as_terms ( self ) Transform an expression to a list of terms.","title":"as_terms"},{"location":"reference/dewret/workflow/#aseries_4","text":"def aseries ( self , x = None , n = 6 , bound = 0 , hir = False ) Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n) .","title":"aseries"},{"location":"reference/dewret/workflow/#parameters_24","text":"self : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. hir : Boolean Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results. bound : Value, Integer Use the bound parameter to give limit on rewriting coefficients in its normalised form.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_190","text":"from sympy import sin, exp from sympy.abc import x e = sin(1/x + exp(-x)) - sin(1/x) e.aseries(x) (1/(24 x 4) - 1/(2 x 2) + 1 + O(x (-6), (x, oo)))*exp(-x) e.aseries(x, n=3, hir=True) -exp(-2 x) sin(1/x)/2 + exp(-x) cos(1/x) + O(exp(-3 x), (x, oo)) e = exp(exp(x)/(1 - 1/x)) e.aseries(x) exp(exp(x)/(1 - 1/x)) e.aseries(x, bound=3) # doctest: +SKIP exp(exp(x)/x 2) exp(exp(x)/x) exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x 2)*exp(exp(x)) For rational expressions this method may return original expression without the Order term. (1/x).aseries(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_12","text":"Expr Asymptotic series expansion of the expression.","title":"Returns"},{"location":"reference/dewret/workflow/#notes_26","text":"This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients. If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation. If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self .","title":"Notes"},{"location":"reference/dewret/workflow/#references_8","text":".. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series. In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993. pp. 239-244. .. [2] Gruntz thesis - p90 .. [3] https://en.wikipedia.org/wiki/Asymptotic_expansion","title":"References"},{"location":"reference/dewret/workflow/#see-also_56","text":"Expr.aseries: See the docstring of this function for complete details of this wrapper.","title":"See Also"},{"location":"reference/dewret/workflow/#atoms_4","text":"def atoms ( self , * types ) Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.","title":"atoms"},{"location":"reference/dewret/workflow/#examples_191","text":"from sympy import I, pi, sin from sympy.abc import x, y (1 + x + 2 sin(y + I pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. from sympy import Number, NumberSymbol, Symbol (1 + x + 2 sin(y + I pi)).atoms(Symbol) {x, y} (1 + x + 2 sin(y + I pi)).atoms(Number) {1, 2} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol) {1, 2, pi} (1 + x + 2 sin(y + I pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: (1 + x + 2 sin(y + I pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One , a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression: from sympy import S (1 + x + 2 sin(y + I pi)).atoms(S(1)) {1} (1 + x + 2 sin(y + I pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/ init .py) can be listed as an argument and those types of \"atoms\" as found in scanning the arguments of the expression recursively: from sympy import Function, Mul from sympy.core.function import AppliedUndef f = Function('f') (1 + f(x) + 2 sin(y + I pi)).atoms(Function) {f(x), sin(y + I pi)} (1 + f(x) + 2 sin(y + I*pi)).atoms(AppliedUndef) {f(x)} (1 + x + 2 sin(y + I pi)).atoms(Mul) {I pi, 2 sin(y + I*pi)}","title":"Examples"},{"location":"reference/dewret/workflow/#cancel_4","text":"def cancel ( self , * gens , ** args ) See the cancel function in sympy.polys","title":"cancel"},{"location":"reference/dewret/workflow/#coeff_4","text":"def coeff ( self , x , n = 1 , right = False , _first = True ) Returns the coefficient from the term(s) containing x**n . If n is zero then all terms independent of x will be returned.","title":"coeff"},{"location":"reference/dewret/workflow/#explanation_26","text":"When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_192","text":"from sympy import symbols from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: (-x + 2 y).coeff(-1) x (x - 2 y).coeff(-1) 2*y You can select terms with no Rational coefficient: (x + 2 y).coeff(1) x (3 + 2 x + 4 x *2).coeff(1) 0 You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None): (3 + 2 x + 4 x 2).coeff(x, 0) 3 eq = ((x + 1) 3).expand() + 1 eq x 3 + 3*x 2 + 3*x + 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 2] eq -= 2 [eq.coeff(x, i) for i in reversed(range(4))] [1, 3, 3, 0] You can select terms that have a numerical term in front of them: (-x - 2 y).coeff(2) -y from sympy import sqrt (x + sqrt(2) x).coeff(sqrt(2)) x The matching is exact: (3 + 2 x + 4 x 2).coeff(x) 2 (3 + 2 x + 4 x 2).coeff(x 2) 4 (3 + 2 x + 4 x 2).coeff(x 3) 0 (z*(x + y) 2).coeff((x + y) 2) z (z*(x + y) 2).coeff(x + y) 0 In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following: (x + z (x + x y)).coeff(x) 1 If such factoring is desired, factor_terms can be used first: from sympy import factor_terms factor_terms(x + z (x + x y)).coeff(x) z*(y + 1) + 1 n, m, o = symbols('n m o', commutative=False) n.coeff(n) 1 (3 n).coeff(n) 3 (n m + m n m).coeff(n) # = (1 + m) n m 1 + m (n m + m n m).coeff(n, right=True) # = (1 + m) n*m m If there is more than one possible coefficient 0 is returned: (n m + m n).coeff(n) 0 If there is only one possible coefficient, it is returned: (n m + x m n).coeff(m n) x (n m + x m n).coeff(m n, right=1) 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_57","text":"as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used","title":"See Also"},{"location":"reference/dewret/workflow/#collect_4","text":"def collect ( self , syms , func = None , evaluate = True , exact = False , distribute_order_term = True ) See the collect function in sympy.simplify","title":"collect"},{"location":"reference/dewret/workflow/#combsimp_4","text":"def combsimp ( self ) See the combsimp function in sympy.simplify","title":"combsimp"},{"location":"reference/dewret/workflow/#compare_4","text":"def compare ( self , other ) Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the ordering_of_classes then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned.","title":"compare"},{"location":"reference/dewret/workflow/#examples_193","text":"from sympy.abc import x, y x.compare(y) -1 x.compare(x) 0 y.compare(x) 1","title":"Examples"},{"location":"reference/dewret/workflow/#compute_leading_term_4","text":"def compute_leading_term ( self , x , logx = None ) Deprecated function to compute the leading term of a series. as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.","title":"compute_leading_term"},{"location":"reference/dewret/workflow/#conjugate_4","text":"def conjugate ( self ) Returns the complex conjugate of 'self'.","title":"conjugate"},{"location":"reference/dewret/workflow/#copy_4","text":"def copy ( self )","title":"copy"},{"location":"reference/dewret/workflow/#could_extract_minus_sign_4","text":"def could_extract_minus_sign ( self ) Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.","title":"could_extract_minus_sign"},{"location":"reference/dewret/workflow/#examples_194","text":"from sympy.abc import x, y e = x - y {i.could_extract_minus_sign() for i in (e, -e)} {False, True} Though the y - x is considered like -(x - y) , since it is in a product without a leading factor of -1, the result is false below: (x*(y - x)).could_extract_minus_sign() False To put something in canonical form wrt to sign, use signsimp : from sympy import signsimp signsimp(x (y - x)) -x (x - y) _.could_extract_minus_sign() True","title":"Examples"},{"location":"reference/dewret/workflow/#count_4","text":"def count ( self , query ) Count the number of matching subexpressions.","title":"count"},{"location":"reference/dewret/workflow/#count_ops_4","text":"def count_ops ( self , visual = None ) Wrapper for count_ops that returns the operation count.","title":"count_ops"},{"location":"reference/dewret/workflow/#diff_4","text":"def diff ( self , * symbols , ** assumptions )","title":"diff"},{"location":"reference/dewret/workflow/#dir_4","text":"def dir ( self , x , cdir )","title":"dir"},{"location":"reference/dewret/workflow/#doit_4","text":"def doit ( self , ** hints ) Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. from sympy import Integral from sympy.abc import x 2 Integral(x, x) 2 Integral(x, x) (2 Integral(x, x)).doit() x *2 (2 Integral(x, x)).doit(deep=False) 2 Integral(x, x)","title":"doit"},{"location":"reference/dewret/workflow/#dummy_eq_4","text":"def dummy_eq ( self , other , symbol = None ) Compare two expressions and handle dummy symbols.","title":"dummy_eq"},{"location":"reference/dewret/workflow/#examples_195","text":"from sympy import Dummy from sympy.abc import x, y u = Dummy('u') (u 2 + 1).dummy_eq(x 2 + 1) True (u 2 + 1) == (x 2 + 1) False (u 2 + y).dummy_eq(x 2 + y, x) True (u 2 + y).dummy_eq(x 2 + y, y) False","title":"Examples"},{"location":"reference/dewret/workflow/#equals_4","text":"def equals ( self , other , failing_expression = False ) Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.","title":"equals"},{"location":"reference/dewret/workflow/#explanation_27","text":"If self is a Number (or complex number) that is not zero, then the result is False. If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.","title":"Explanation"},{"location":"reference/dewret/workflow/#evalf_4","text":"def evalf ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"evalf"},{"location":"reference/dewret/workflow/#parameters_25","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_27","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#expand_4","text":"def expand ( self , deep = True , modulus = None , power_base = True , power_exp = True , mul = True , log = True , multinomial = True , basic = True , ** hints ) Expand an expression using hints. See the docstring of the expand() function in sympy.core.function for more information.","title":"expand"},{"location":"reference/dewret/workflow/#extract_additively_4","text":"def extract_additively ( self , c ) Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.","title":"extract_additively"},{"location":"reference/dewret/workflow/#examples_196","text":"from sympy.abc import x, y e = 2 x + 3 e.extract_additively(x + 1) x + 2 e.extract_additively(3 x) e.extract_additively(4) (y (x + 1)).extract_additively(x + 1) ((x + 1) (x + 2 y + 1) + 3).extract_additively(x + 1) (x + 1) (x + 2*y) + 3","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_58","text":"extract_multiplicatively coeff as_coefficient","title":"See Also"},{"location":"reference/dewret/workflow/#extract_branch_factor_4","text":"def extract_branch_factor ( self , allow_half = False ) Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n). from sympy import exp_polar, I, pi from sympy.abc import x, y exp_polar(I pi).extract_branch_factor() (exp_polar(I pi), 0) exp_polar(2 I pi).extract_branch_factor() (1, 1) exp_polar(-pi I).extract_branch_factor() (exp_polar(I pi), -1) exp_polar(3 pi I + x).extract_branch_factor() (exp_polar(x + I pi), 1) (y exp_polar(-5 pi I) exp_polar(3 pi I + 2 pi x)).extract_branch_factor() (y exp_polar(2 pi x), -1) exp_polar(-I pi/2).extract_branch_factor() (exp_polar(-I pi/2), 0) If allow_half is True, also extract exp_polar(I*pi): exp_polar(I pi).extract_branch_factor(allow_half=True) (1, 1/2) exp_polar(2 I pi).extract_branch_factor(allow_half=True) (1, 1) exp_polar(3 I pi).extract_branch_factor(allow_half=True) (1, 3/2) exp_polar(-I pi).extract_branch_factor(allow_half=True) (1, -1/2)","title":"extract_branch_factor"},{"location":"reference/dewret/workflow/#extract_multiplicatively_4","text":"def extract_multiplicatively ( self , c ) Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.","title":"extract_multiplicatively"},{"location":"reference/dewret/workflow/#examples_197","text":"from sympy import symbols, Rational x, y = symbols('x,y', real=True) ((x y) 3).extract_multiplicatively(x 2 * y) x y**2 ((x y) 3).extract_multiplicatively(x *4 * y) (2*x).extract_multiplicatively(2) x (2*x).extract_multiplicatively(3) (Rational(1, 2)*x).extract_multiplicatively(3) x/6","title":"Examples"},{"location":"reference/dewret/workflow/#factor_4","text":"def factor ( self , * gens , ** args ) See the factor() function in sympy.polys.polytools","title":"factor"},{"location":"reference/dewret/workflow/#find_4","text":"def find ( self , query , group = False ) Find all subexpressions matching a query.","title":"find"},{"location":"reference/dewret/workflow/#find_field_4","text":"def find_field ( self : 'FieldableProtocol' , field : 'str | int' , fallback_type : 'type | None' = None , ** init_kwargs : 'Any' ) -> 'Reference[Any]' Field within the reference, if possible. Parameters: Name Type Description Default field None the field to search for. None fallback_type None the type to use if we do not know a more specific one. None **init_kwargs None arguments to use for constructing a new reference (via __make_reference__ ). None Returns: Type Description None A field-specific version of this reference.","title":"find_field"},{"location":"reference/dewret/workflow/#fourier_series_4","text":"def fourier_series ( self , limits = None ) Compute fourier sine/cosine series of self. See the docstring of the :func: fourier_series in sympy.series.fourier for more information.","title":"fourier_series"},{"location":"reference/dewret/workflow/#fps_4","text":"def fps ( self , x = None , x0 = 0 , dir = 1 , hyper = True , order = 4 , rational = True , full = False ) Compute formal power power series of self. See the docstring of the :func: fps function in sympy.series.formal for more information.","title":"fps"},{"location":"reference/dewret/workflow/#gammasimp_4","text":"def gammasimp ( self ) See the gammasimp function in sympy.simplify","title":"gammasimp"},{"location":"reference/dewret/workflow/#geto_4","text":"def getO ( self ) Returns the additive O(..) symbol if there is one, else None.","title":"getO"},{"location":"reference/dewret/workflow/#getn_4","text":"def getn ( self ) Returns the order of the expression.","title":"getn"},{"location":"reference/dewret/workflow/#explanation_28","text":"The order is determined either from the O(...) term. If there is no O(...) term, it returns None.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_198","text":"from sympy import O from sympy.abc import x (1 + x + O(x**2)).getn() 2 (1 + x).getn()","title":"Examples"},{"location":"reference/dewret/workflow/#has_4","text":"def has ( self , * patterns ) Test whether any subexpression matches any of the patterns.","title":"has"},{"location":"reference/dewret/workflow/#examples_199","text":"from sympy import sin from sympy.abc import x, y, z (x 2 + sin(x*y)).has(z) False (x 2 + sin(x*y)).has(x, y, z) True x.has(x) True Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: from sympy import Interval i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) i.args (0, 5, True, False) i.has(4) # there is no \"4\" in the arguments False i.has(0) # there is a \"0\" in the arguments True Instead, use contains to determine whether a number is in the interval or not: i.contains(4) True i.contains(0) False Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns) . In particular, False is returned when the list of patterns is empty. x.has() False","title":"Examples"},{"location":"reference/dewret/workflow/#has_free_4","text":"def has_free ( self , * patterns ) Return True if self has object(s) x as a free expression else False.","title":"has_free"},{"location":"reference/dewret/workflow/#examples_200","text":"from sympy import Integral, Function from sympy.abc import x, y f = Function('f') g = Function('g') expr = Integral(f(x), (f(x), 1, g(y))) expr.free_symbols {y} expr.has_free(g(y)) True expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: expr.has_free(g) True (x + y + 1).has_free(y + 1) True","title":"Examples"},{"location":"reference/dewret/workflow/#has_xfree_4","text":"def has_xfree ( self , s : 'set[Basic]' ) Return True if self has any of the patterns in s as a free argument, else False. This is like Basic.has_free but this will only report exact argument matches.","title":"has_xfree"},{"location":"reference/dewret/workflow/#examples_201","text":"from sympy import Function from sympy.abc import x, y f = Function('f') f(x).has_xfree({f}) False f(x).has_xfree({f(x)}) True f(x + 1).has_xfree({x}) True f(x + 1).has_xfree({x + 1}) True f(x + y + 1).has_xfree({x + 1}) False","title":"Examples"},{"location":"reference/dewret/workflow/#integrate_4","text":"def integrate ( self , * args , ** kwargs ) See the integrate function in sympy.integrals","title":"integrate"},{"location":"reference/dewret/workflow/#invert_4","text":"def invert ( self , g , * gens , ** args ) Return the multiplicative inverse of self mod g where self (and g ) may be symbolic expressions).","title":"invert"},{"location":"reference/dewret/workflow/#see-also_59","text":"sympy.core.intfunc.mod_inverse, sympy.polys.polytools.invert","title":"See Also"},{"location":"reference/dewret/workflow/#is_algebraic_expr_4","text":"def is_algebraic_expr ( self , * syms ) This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"algebraic expressions\" with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.","title":"is_algebraic_expr"},{"location":"reference/dewret/workflow/#examples_202","text":"from sympy import Symbol, sqrt x = Symbol('x', real=True) sqrt(1 + x).is_rational_function() False sqrt(1 + x).is_algebraic_expr() True This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one. from sympy import exp, factor a = sqrt(exp(x)* 2 + 2 exp(x) + 1)/(exp(x) + 1) a.is_algebraic_expr(x) False factor(a).is_algebraic_expr() True","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_60","text":"is_rational_function","title":"See Also"},{"location":"reference/dewret/workflow/#references_9","text":".. [1] https://en.wikipedia.org/wiki/Algebraic_expression","title":"References"},{"location":"reference/dewret/workflow/#is_constant_4","text":"def is_constant ( self , * wrt , ** flags ) Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.","title":"is_constant"},{"location":"reference/dewret/workflow/#explanation_29","text":"If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried: 1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols. 2) differentiation with respect to variables in 'wrt' (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols. 3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant. If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True -- in that case the numerical value will be returned. If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.","title":"Explanation"},{"location":"reference/dewret/workflow/#examples_203","text":"from sympy import cos, sin, Sum, S, pi from sympy.abc import a, n, x, y x.is_constant() False S(2).is_constant() True Sum(x, (x, 1, 10)).is_constant() True Sum(x, (x, 1, n)).is_constant() False Sum(x, (x, 1, n)).is_constant(y) True Sum(x, (x, 1, n)).is_constant(n) False Sum(x, (x, 1, n)).is_constant(x) True eq = a cos(x) 2 + a sin(x)**2 - a eq.is_constant() True eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0 True (0 x).is_constant() False x.is_constant() False (x x).is_constant() False one = cos(x) 2 + sin(x) 2 one.is_constant() True ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1 True","title":"Examples"},{"location":"reference/dewret/workflow/#is_hypergeometric_4","text":"def is_hypergeometric ( self , k )","title":"is_hypergeometric"},{"location":"reference/dewret/workflow/#is_meromorphic_4","text":"def is_meromorphic ( self , x , a ) This tests whether an expression is meromorphic as a function of the given symbol x at the point a . This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.","title":"is_meromorphic"},{"location":"reference/dewret/workflow/#examples_204","text":"from sympy import zoo, log, sin, sqrt from sympy.abc import x f = 1/x 2 + 1 - 2*x 3 f.is_meromorphic(x, 0) True f.is_meromorphic(x, 1) True f.is_meromorphic(x, zoo) True g = x**log(3) g.is_meromorphic(x, 0) False g.is_meromorphic(x, 1) True g.is_meromorphic(x, zoo) False h = sin(1/x) x *2 h.is_meromorphic(x, 0) False h.is_meromorphic(x, 1) True h.is_meromorphic(x, zoo) True Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints. log(x).is_meromorphic(x, -1) True log(x).is_meromorphic(x, 0) False sqrt(x).is_meromorphic(x, -1) True sqrt(x).is_meromorphic(x, 0) False","title":"Examples"},{"location":"reference/dewret/workflow/#is_polynomial_4","text":"def is_polynomial ( self , * syms ) Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are \"polynomials\" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True).","title":"is_polynomial"},{"location":"reference/dewret/workflow/#examples_205","text":"from sympy import Symbol, Function x = Symbol('x') ((x 2 + 1) 4).is_polynomial(x) True ((x 2 + 1) 4).is_polynomial() True (2 x + 1).is_polynomial(x) False (2 x + 1).is_polynomial(2**x) True f = Function('f') (f(x) + 1).is_polynomial(x) False (f(x) + 1).is_polynomial(f(x)) True (1/f(x) + 1).is_polynomial(f(x)) False n = Symbol('n', nonnegative=True, integer=True) (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. from sympy import sqrt, factor, cancel y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1) a.is_polynomial(y) False factor(a) y + 1 factor(a).is_polynomial(y) True b = (y* 2 + 2 y + 1)/(y + 1) b.is_polynomial(y) False cancel(b) y + 1 cancel(b).is_polynomial(y) True See also .is_rational_function()","title":"Examples"},{"location":"reference/dewret/workflow/#is_rational_function_4","text":"def is_rational_function ( self , * syms ) Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are \"rational functions\" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True).","title":"is_rational_function"},{"location":"reference/dewret/workflow/#examples_206","text":"from sympy import Symbol, sin from sympy.abc import x, y (x/y).is_rational_function() True (x**2).is_rational_function() True (x/sin(y)).is_rational_function(y) False n = Symbol('n', integer=True) (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. from sympy import sqrt, factor y = Symbol('y', positive=True) a = sqrt(y* 2 + 2 y + 1)/y a.is_rational_function(y) False factor(a) (y + 1)/y factor(a).is_rational_function(y) True See also is_algebraic_expr().","title":"Examples"},{"location":"reference/dewret/workflow/#is_same_4","text":"def is_same ( a , b , approx = None ) Return True if a and b are structurally the same, else False. If approx is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5).","title":"is_same"},{"location":"reference/dewret/workflow/#examples_207","text":"In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: from sympy import S 2.0 == S(2) False 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. equal_valued will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. from sympy import Float from sympy.core.numbers import equal_valued (S.Half/4).is_same(Float(0.125, 1), equal_valued) True Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: import math Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. from sympy import eye, Basic eye(1) == S(eye(1)) # mutable vs immutable True Basic.is_same(eye(1), S(eye(1))) False","title":"Examples"},{"location":"reference/dewret/workflow/#leadterm_4","text":"def leadterm ( self , x , logx = None , cdir = 0 ) Returns the leading term a x *b as a tuple (a, b).","title":"leadterm"},{"location":"reference/dewret/workflow/#examples_208","text":"from sympy.abc import x (1+x+x 2).leadterm(x) (1, 0) (1/x 2+x+x**2).leadterm(x) (1, -2)","title":"Examples"},{"location":"reference/dewret/workflow/#limit_4","text":"def limit ( self , x , xlim , dir = '+' ) Compute limit x->xlim.","title":"limit"},{"location":"reference/dewret/workflow/#lseries_4","text":"def lseries ( self , x = None , x0 = 0 , dir = '+' , logx = None , cdir = 0 ) Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you do not know how many you should ask for in nseries() using the \"n\" parameter. See also nseries().","title":"lseries"},{"location":"reference/dewret/workflow/#match_4","text":"def match ( self , pattern , old = False ) Pattern matching. Wild symbols match all. Return None when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self","title":"match"},{"location":"reference/dewret/workflow/#examples_209","text":"from sympy import Wild, Sum from sympy.abc import x, y p = Wild(\"p\") q = Wild(\"q\") r = Wild(\"r\") e = (x+y) (x+y) e.match(p p) {p_: x + y} e.match(p q) {p_: x + y, q_: x + y} e = (2*x) 2 e.match(p q r) {p_: 4, q_: x, r_: 2} (p q r).xreplace(e.match(p*q r)) 4 x *2 Structurally bound symbols are ignored during matching: Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p))) {p_: 2} But they can be identified if desired: Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The old flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless old=True : (x - 2).match(p - x, old=True) {p_: 2 x - 2} (2/x).match(p x, old=True) {p_: 2/x**2}","title":"Examples"},{"location":"reference/dewret/workflow/#matches_4","text":"def matches ( self , expr , repl_dict = None , old = False ) Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.","title":"matches"},{"location":"reference/dewret/workflow/#examples_210","text":"from sympy import symbols, Wild, Basic a, b, c = symbols('a b c') x = Wild('x') Basic(a + x, x).matches(Basic(a + b, c)) is None True Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c}","title":"Examples"},{"location":"reference/dewret/workflow/#n_4","text":"def n ( self , n = 15 , subs = None , maxn = 100 , chop = False , strict = False , quad = None , verbose = False ) Evaluate the given formula to an accuracy of n digits.","title":"n"},{"location":"reference/dewret/workflow/#parameters_26","text":"subs : dict, optional Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi} . The substitutions must be given as a dictionary. maxn : int, optional Allow a maximum temporary working precision of maxn digits. chop : bool or number, optional Specifies how to replace tiny real or imaginary parts in subresults by exact zeros. When ``True`` the chop value defaults to standard precision. Otherwise the chop value is used to determine the magnitude of \"small\" for purposes of chopping. >>> from sympy import N >>> x = 1e-4 >>> N(x, chop=True) 0.000100000000000000 >>> N(x, chop=1e-5) 0.000100000000000000 >>> N(x, chop=1e-4) 0 strict : bool, optional Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec. quad : str, optional Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc' . verbose : bool, optional Print debug information.","title":"Parameters"},{"location":"reference/dewret/workflow/#notes_28","text":"When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following: from sympy.abc import x, y, z values = {x: 1e16, y: 1, z: 1e16} (x + y - z).subs(values) 0 Using the subs argument for evalf is the accurate way to evaluate such an expression: (x + y - z).evalf(subs=values) 1.00000000000000","title":"Notes"},{"location":"reference/dewret/workflow/#normal_4","text":"def normal ( self ) Return the expression as a fraction. expression -> a/b","title":"normal"},{"location":"reference/dewret/workflow/#see-also_61","text":"as_numer_denom: return (a, b) instead of a/b","title":"See Also"},{"location":"reference/dewret/workflow/#nseries_4","text":"def nseries ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates \"n\" terms in the innermost expressions and then builds up the final series just by \"cross-multiplying\" everything out. The optional logx parameter can be used to replace any log(x) in the returned series with a symbolic value to avoid evaluating log(x) at 0. A symbol to use in place of log(x) should be provided. Advantage -- it's fast, because we do not have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries().","title":"nseries"},{"location":"reference/dewret/workflow/#examples_211","text":"from sympy import sin, log, Symbol from sympy.abc import x, y sin(x).nseries(x, 0, 6) x - x 3/6 + x 5/120 + O(x 6) log(x+1).nseries(x, 0, 5) x - x 2/2 + x 3/3 - x 4/4 + O(x**5) Handling of the logx parameter --- in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0): e = sin(log(x)) e.nseries(x, 0, 6) Traceback (most recent call last): ... PoleError: ... ... logx = Symbol('logx') e.nseries(x, 0, 6, logx=logx) sin(logx) In the following example, the expansion works but only returns self unless the logx parameter is used: e = x y e.nseries(x, 0, 2) x y e.nseries(x, 0, 2, logx=logx) exp(logx*y)","title":"Examples"},{"location":"reference/dewret/workflow/#nsimplify_4","text":"def nsimplify ( self , constants = (), tolerance = None , full = False ) See the nsimplify function in sympy.simplify","title":"nsimplify"},{"location":"reference/dewret/workflow/#powsimp_4","text":"def powsimp ( self , * args , ** kwargs ) See the powsimp function in sympy.simplify","title":"powsimp"},{"location":"reference/dewret/workflow/#primitive_4","text":"def primitive ( self ) Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).","title":"primitive"},{"location":"reference/dewret/workflow/#examples_212","text":"from sympy.abc import x (3 (x + 1) 2).primitive() (3, (x + 1) 2) a = (6 x + 2); a.primitive() (2, 3 x + 1) b = (x/2 + 3); b.primitive() (1/2, x + 6) (a b).primitive() == (1, a*b) True","title":"Examples"},{"location":"reference/dewret/workflow/#radsimp_4","text":"def radsimp ( self , ** kwargs ) See the radsimp function in sympy.simplify","title":"radsimp"},{"location":"reference/dewret/workflow/#ratsimp_4","text":"def ratsimp ( self ) See the ratsimp function in sympy.simplify","title":"ratsimp"},{"location":"reference/dewret/workflow/#rcall_4","text":"def rcall ( self , * args ) Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: (x+Lambda(y, 2*y))(z) == x+2*z , however, you can use: from sympy import Lambda from sympy.abc import x, y, z (x + Lambda(y, 2 y)).rcall(z) x + 2 z","title":"rcall"},{"location":"reference/dewret/workflow/#refine_4","text":"def refine ( self , assumption = True ) See the refine function in sympy.assumptions","title":"refine"},{"location":"reference/dewret/workflow/#removeo_4","text":"def removeO ( self ) Removes the additive O(..) symbol if there is one","title":"removeO"},{"location":"reference/dewret/workflow/#replace_4","text":"def replace ( self , query , value , map = False , simultaneous = True , exact = None ) Replace matching subexpressions of self with value . If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())) . Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below:","title":"replace"},{"location":"reference/dewret/workflow/#examples_213","text":"Initial setup from sympy import log, sin, cos, tan, Wild, Mul, Add from sympy.abc import x, y f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired: Here, we want exact=False : from sympy import Function f = Function('f') e = f(1) + f(0) q = f(a), lambda a: f(a + 1) e.replace( q, exact=False) f(1) + f(2) e.replace( q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: e = x (1 + y) (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=False) x (x (1 + y)).replace(x (1 + a), lambda a: x -a, exact=True) x (-x - y + 1) (x y).replace(x (1 + a), lambda a: x -a, exact=False) x (x y).replace(x (1 + a), lambda a: x -a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: (1 + x (1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base (1 - (x.exp - 1))) ... x**(1 - y) + 1","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_62","text":"subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules","title":"See Also"},{"location":"reference/dewret/workflow/#rewrite_4","text":"def rewrite ( self , * args , deep = True , ** hints ) Rewrite self using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.","title":"rewrite"},{"location":"reference/dewret/workflow/#parameters_27","text":"args : Expr A rule , or pattern and rule . - pattern is a type or an iterable of types. - rule can be any object. deep : bool, optional If True , subexpressions are recursively transformed. Default is True .","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_214","text":"If pattern is unspecified, all possible expressions are transformed. from sympy import cos, sin, exp, I from sympy.abc import x expr = cos(x) + I sin(x) expr.rewrite(exp) exp(I x) Pattern can be a type or an iterable of types. expr.rewrite(sin, exp) exp(I x)/2 + cos(x) - exp(-I x)/2 expr.rewrite([cos,], exp) exp(I x)/2 + I sin(x) + exp(-I x)/2 expr.rewrite([cos, sin], exp) exp(I x) Rewriting behavior can be implemented by defining _eval_rewrite() method. from sympy import Expr, sqrt, pi class MySin(Expr): ... def _eval_rewrite(self, rule, args, hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x) 2) MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. class MySin(Expr): ... def _eval_rewrite_as_cos(self, args, *hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) MySin(x).rewrite(cos) cos(-x + pi/2)","title":"Examples"},{"location":"reference/dewret/workflow/#round_4","text":"def round ( self , n = None ) Return x rounded to the given decimal place. If a complex number would results, apply round to the real and imaginary components of the number.","title":"round"},{"location":"reference/dewret/workflow/#examples_215","text":"from sympy import pi, E, I, S, Number pi.round() 3 pi.round(2) 3.14 (2 pi + E I).round() 6 + 3*I The round method has a chopping effect: (2 pi + I/10).round() 6 (pi/10 + 2 I).round() 2 I (pi/10 + E I).round(2) 0.31 + 2.72*I","title":"Examples"},{"location":"reference/dewret/workflow/#notes_29","text":"The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int): isinstance(round(S(123), -2), Number) True","title":"Notes"},{"location":"reference/dewret/workflow/#separate_4","text":"def separate ( self , deep = False , force = False ) See the separate function in sympy.simplify","title":"separate"},{"location":"reference/dewret/workflow/#series_4","text":"def series ( self , x = None , x0 = 0 , n = 6 , dir = '+' , logx = None , cdir = 0 ) Series expansion of \"self\" around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Returns the series expansion of \"self\" around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6). If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.","title":"series"},{"location":"reference/dewret/workflow/#parameters_28","text":"expr : Expression The expression whose series is to be expanded. x : Symbol It is the variable of the expression to be calculated. x0 : Value The value around which x is calculated. Can be any value from -oo to oo . n : Value The value used to represent the order in terms of x**n , up to which the series is to be expanded. dir : String, optional The series-expansion can be bi-directional. If dir=\"+\" , then (x->x0+). If dir=\"-\", then (x->x0-). For infinite x0 ( oo or -oo ), the dir argument is determined from the direction of the infinity (i.e., dir=\"-\" for oo``). logx : optional It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value. cdir : optional It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.","title":"Parameters"},{"location":"reference/dewret/workflow/#examples_216","text":"from sympy import cos, exp, tan from sympy.abc import x, y cos(x).series() 1 - x 2/2 + x 4/24 + O(x 6) cos(x).series(n=4) 1 - x 2/2 + O(x 4) cos(x).series(x, x0=1, n=2) cos(1) - (x - 1)*sin(1) + O((x - 1) 2, (x, 1)) e = cos(x + exp(y)) e.series(y, n=2) cos(x + 1) - y sin(x + 1) + O(y 2) e.series(x, n=2) cos(exp(y)) - x sin(exp(y)) + O(x**2) If n=None then a generator of the series terms will be returned. term=cos(x).series(n=None) [next(term) for i in range(2)] [1, -x**2/2] For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results. abs(x).series(dir=\"+\") x abs(x).series(dir=\"-\") -x f = tan(x) f.series(x, 2, 6, \"+\") tan(2) + (1 + tan(2) 2)*(x - 2) + (x - 2) 2 (tan(2) 3 + tan(2)) + (x - 2) 3 (1/3 + 4 tan(2) 2/3 + tan(2) 4) + (x - 2) 4 (tan(2) 5 + 5*tan(2) 3/3 + 2 tan(2)/3) + (x - 2) 5 (2/15 + 17 tan(2) 2/15 + 2 tan(2) 4 + tan(2) 6) + O((x - 2)**6, (x, 2)) f.series(x, 2, 3, \"-\") tan(2) + (2 - x) (-tan(2) 2 - 1) + (2 - x) 2 (tan(2) 3 + tan(2)) + O((x - 2) 3, (x, 2)) For rational expressions this method may return original expression without the Order term. (1/x).series(x, n=8) 1/x","title":"Examples"},{"location":"reference/dewret/workflow/#returns_13","text":"Expr : Expression Series expansion of the expression about x0","title":"Returns"},{"location":"reference/dewret/workflow/#raises_4","text":"TypeError If \"n\" and \"x0\" are infinity objects PoleError If \"x0\" is an infinity object","title":"Raises"},{"location":"reference/dewret/workflow/#simplify_4","text":"def simplify ( self , ** kwargs ) See the simplify function in sympy.simplify","title":"simplify"},{"location":"reference/dewret/workflow/#sort_key_4","text":"def sort_key ( self , order = None ) Return a sort key.","title":"sort_key"},{"location":"reference/dewret/workflow/#examples_217","text":"from sympy import S, I sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] S(\"[x, 1/x, 1/x 2, x 2, x (1/2), x (1/4), x (3/2)]\") [x, 1/x, x (-2), x 2, sqrt(x), x (1/4), x (3/2)] sorted(_, key=lambda x: x.sort_key()) [x (-2), 1/x, x (1/4), sqrt(x), x, x (3/2), x**2]","title":"Examples"},{"location":"reference/dewret/workflow/#subs_4","text":"def subs ( self , * args , ** kwargs ) Substitutes old for new in an expression after sympifying args. args is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.","title":"subs"},{"location":"reference/dewret/workflow/#examples_218","text":"from sympy import pi, exp, limit, oo from sympy.abc import x, y (1 + x y).subs(x, pi) pi y + 1 (1 + x y).subs({x:pi, y:2}) 1 + 2 pi (1 + x y).subs([(x, pi), (y, 2)]) 1 + 2 pi reps = [(y, x 2), (x, 2)] (x + y).subs(reps) 6 (x + y).subs(reversed(reps)) x 2 + 2 (x 2 + x 4).subs(x 2, y) y 2 + y To replace only the x 2 but not the x 4, use xreplace: (x 2 + x 4).xreplace({x 2: y}) x 4 + y To delay evaluation until all substitutions have been made, set the keyword simultaneous to True: (x/y).subs([(x, 0), (y, 0)]) 0 (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: ((x + y)/y).subs({x + y: y, y: x + y}) 1 ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. from sympy import sqrt, sin, cos from sympy.abc import a, b, c, d, e A = (sqrt(sin(2 x)), a) B = (sin(2 x), b) C = (cos(2*x), c) D = (x, d) E = (exp(x), e) expr = sqrt(sin(2 x)) sin(exp(x) x) cos(2 x) + sin(2 x) expr.subs(dict([A, B, C, D, E])) a c sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: (x* 3 - 3 x).subs({x: oo}) nan limit(x* 3 - 3 x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained.","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_63","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision","title":"See Also"},{"location":"reference/dewret/workflow/#taylor_term_4","text":"def taylor_term ( self , n , x , * previous_terms ) General method for the taylor term. This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the \"previous_terms\".","title":"taylor_term"},{"location":"reference/dewret/workflow/#to_nnf_4","text":"def to_nnf ( self , simplify = True )","title":"to_nnf"},{"location":"reference/dewret/workflow/#together_4","text":"def together ( self , * args , ** kwargs ) See the together function in sympy.polys","title":"together"},{"location":"reference/dewret/workflow/#transpose_4","text":"def transpose ( self )","title":"transpose"},{"location":"reference/dewret/workflow/#trigsimp_4","text":"def trigsimp ( self , ** args ) See the trigsimp function in sympy.simplify","title":"trigsimp"},{"location":"reference/dewret/workflow/#xreplace_4","text":"def xreplace ( self , rule , hack2 = False ) Replace occurrences of objects within the expression.","title":"xreplace"},{"location":"reference/dewret/workflow/#parameters_29","text":"rule : dict-like Expresses a replacement rule","title":"Parameters"},{"location":"reference/dewret/workflow/#returns_14","text":"xreplace : the result of the replacement","title":"Returns"},{"location":"reference/dewret/workflow/#examples_219","text":"from sympy import symbols, pi, exp x, y, z = symbols('x y z') (1 + x y).xreplace({x: pi}) pi y + 1 (1 + x y).xreplace({x: pi, y: 2}) 1 + 2 pi Replacements occur only if an entire node in the expression tree is matched: (x y + z).xreplace({x y: pi}) z + pi (x y z).xreplace({x y: pi}) x y z (2 x).xreplace({2 x: y, x: z}) y (2 2 x).xreplace({2 x: y, x: z}) 4*z (x + y + 2).xreplace({x + y: 2}) x + y + 2 (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: from sympy import Integral Integral(x, (x, 1, 2 x)).xreplace({x: y}) Integral(y, (y, 1, 2 y)) Trying to replace x with an expression raises an error: Integral(x, (x, 1, 2 x)).xreplace({x: 2 y}) # doctest: +SKIP ValueError: Invalid limits given: ((2 y, 1, 4 y),)","title":"Examples"},{"location":"reference/dewret/workflow/#see-also_64","text":"replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves.","title":"See Also"},{"location":"reference/dewret/workflow/#task","text":"class Task ( name : 'str' , target : 'Lazy' ) Named wrapper of a lazy-evaluatable function. Wraps a lazy-evaluatable function ( dewret.workflow.Lazy ) with any metadata needed to render it later. At present, this is the name.","title":"Task"},{"location":"reference/dewret/workflow/#attributes_5","text":"Name Type Description Default name None Name of the lazy function as it will appear in the output workflow text. None target None Callable that is wrapped. None","title":"Attributes"},{"location":"reference/dewret/workflow/#unsettype","text":"class UnsetType ( raw_type : 'type[T]' ) Unset variable with a specific type.","title":"UnsetType"},{"location":"reference/dewret/workflow/#attributes_6","text":"Name Type Description Default type None type of the variable. None","title":"Attributes"},{"location":"reference/dewret/workflow/#ancestors-in-mro_12","text":"dewret.utils.Unset typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/workflow/#workflow","text":"class Workflow ( name : 'str | None' = None ) Overarching workflow concept. Represents a whole workflow, as a singleton maintaining all state information needed ahead of rendering. It is built up as the lazy-evaluations are finally evaluated.","title":"Workflow"},{"location":"reference/dewret/workflow/#attributes_7","text":"Name Type Description Default steps None the sequence of calls to lazy-evaluable functions, built as they are evaluated. None tasks None the mapping of names used in the steps to the actual Task wrappers they represent. None result None target reference to evaluate, if yet present. None","title":"Attributes"},{"location":"reference/dewret/workflow/#static-methods_5","text":"","title":"Static methods"},{"location":"reference/dewret/workflow/#assimilate","text":"def assimilate ( * workflow_args : 'Workflow' ) -> \"'Workflow'\" Combine two Workflows into one Workflow. Takes two workflows and unifies them by combining steps and tasks. If it sees mismatched identifiers for the same component, it will error. This could happen if the hashing function is flawed or some Python magic to do with Targets being passed. Parameters: Name Type Description Default workflow_args None workflows to use as base None","title":"assimilate"},{"location":"reference/dewret/workflow/#from_result","text":"def from_result ( result : 'StepReference[Any] | list[StepReference[Any]] | tuple[StepReference[Any], ...]' , simplify_ids : 'bool' = False , nested : 'bool' = True ) -> 'Workflow' Create from a desired result. Starts from a result, and builds a workflow to output it.","title":"from_result"},{"location":"reference/dewret/workflow/#instance-variables_10","text":"has_result Confirms whether this workflow has a non-empty result. Either None or an empty list/tuple are considered empty for this purpose. Returns: True if the workflow has a result, False otherwise. id Consistent ID based off the step IDs. indexed_steps Steps mapped by ID. Forces generation of IDs. Note that this effectively freezes the steps, so it should not be used until we are confident the steps are all ready to be hashed. name Get the name of the workflow. result_type Overall return type of this workflow. steps Get deduplicated steps. Returns: steps for looping over without duplicates.","title":"Instance variables"},{"location":"reference/dewret/workflow/#methods_10","text":"","title":"Methods"},{"location":"reference/dewret/workflow/#add_nested_step","text":"def add_nested_step ( self , name : 'str' , subworkflow : 'Workflow' , return_type : 'type | None' , kwargs : 'dict[str, Any]' , positional_args : 'dict[str, bool] | None' = None ) -> 'StepReference[Any]' Append a nested step. Calls a subworkflow. Parameters: Name Type Description Default name None name of the subworkflow. None subworkflow None the subworkflow itself. None return_type None a forced type for the return, or None. None kwargs None any key-value arguments to pass in the call. None positional_args None a mapping of arguments to bools, True if the argument is positional or otherwise False. None","title":"add_nested_step"},{"location":"reference/dewret/workflow/#add_step","text":"def add_step ( self , fn : 'Lazy' , kwargs : 'dict[str, Raw | Reference[Any]]' , raw_as_parameter : 'bool' = False , is_factory : 'bool' = False , positional_args : 'dict[str, bool] | None' = None ) -> 'StepReference[Any]' Append a step. Adds a step, for running a target with key-value arguments, to the workflow. Parameters: Name Type Description Default fn None the target function to turn into a step. None kwargs None any key-value arguments to pass in the call. None raw_as_parameter None whether to turn any discovered raw arguments into workflow parameters. None is_factory None whether this step should be a Factory. None positional_args None a mapping of arguments to bools, True if the argument is positional or otherwise False. None","title":"add_step"},{"location":"reference/dewret/workflow/#find_factories","text":"def find_factories ( self ) -> 'dict[str, FactoryCall]' Steps that are factory calls.","title":"find_factories"},{"location":"reference/dewret/workflow/#find_parameters","text":"def find_parameters ( self , include_factory_calls : 'bool' = True ) -> 'set[Parameter[Any]]' Crawl steps for parameter references. As the workflow does not hold its own list of parameters, this dynamically finds them. Returns: Type Description None Set of all references to parameters across the steps.","title":"find_parameters"},{"location":"reference/dewret/workflow/#register_task","text":"def register_task ( self , fn : 'Lazy' ) -> 'Task' Note the existence of a lazy-evaluatable function, and wrap it as a Task . Parameters: Name Type Description Default fn None the wrapped function. None Returns: Type Description None A new Task that wraps the function, and is retained in the Workflow.tasks dict.","title":"register_task"},{"location":"reference/dewret/workflow/#remap","text":"def remap ( self , step_id : 'str' ) -> 'str' Apply name simplification if requested. Parameters: Name Type Description Default step_id None step to check. None Returns: Type Description None Same ID or a remapped name.","title":"remap"},{"location":"reference/dewret/workflow/#set_result","text":"def set_result ( self , result : 'Basic | list[Basic] | tuple[Basic]' ) -> 'None' Choose the result step. Sets a step as being the result for the entire workflow. When we evaluate a dynamic workflow, the engine (e.g. dask) creates a graph to realize the result of a single collection. Similarly, in the static case, we need to have a result that drives the calculation. Parameters: Name Type Description Default result None reference to the chosen step. None","title":"set_result"},{"location":"reference/dewret/workflow/#simplify_ids","text":"def simplify_ids ( self , infix : 'list[str] | None' = None ) -> 'None' Work out mapping to simple ints from hashes. Goes through and numbers each step by the order of use of its task.","title":"simplify_ids"},{"location":"reference/dewret/workflow/#workflowlinkedcomponent","text":"class WorkflowLinkedComponent ( * args , ** kwargs ) Protocol for objects dynamically tied to a Workflow .","title":"WorkflowLinkedComponent"},{"location":"reference/dewret/workflow/#ancestors-in-mro_13","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/backends/","text":"Module dewret.backends Backends for supplying lazy-evaluation. By default, we use dask, but it is possible to extend this functionality as the behaviour is encapsulated within these modules. Sub-modules dewret.backends.backend_dask","title":"Index"},{"location":"reference/dewret/backends/#module-dewretbackends","text":"Backends for supplying lazy-evaluation. By default, we use dask, but it is possible to extend this functionality as the behaviour is encapsulated within these modules.","title":"Module dewret.backends"},{"location":"reference/dewret/backends/#sub-modules","text":"dewret.backends.backend_dask","title":"Sub-modules"},{"location":"reference/dewret/backends/backend_dask/","text":"Module dewret.backends.backend_dask Dask backend. Lazy-evaluation via dask.delayed . Variables config Functions is_lazy def is_lazy ( task : Any ) -> bool Checks if a task is really a lazy-evaluated function for this backend. Parameters: Name Type Description Default task None suspected lazy-evaluated function. None Returns: Type Description None True if so, False otherwise. lazy def lazy ( obj = '__no__default__' , name = None , pure = None , nout = None , traverse = True ) Wraps a function or object to produce a Delayed . Delayed objects act as proxies for the object they wrap, but all operations on them are done lazily by building up a dask graph internally. Parameters: Name Type Description Default obj object The function or object to wrap None name Dask key The key to use in the underlying graph for the wrapped object. Defaults to hashing content. Note that this only affects the name of the object wrapped by this call to delayed, and not the output of delayed function calls - for that use dask_key_name= as described below. .. note:: Because this name is used as the key in task graphs, you should ensure that it uniquely identifies obj . If you'd like to provide a descriptive name that is still unique, combine the descriptive name with :func: dask.base.tokenize of the array_like . See :ref: graphs for more. s pure bool Indicates whether calling the resulting Delayed object is a pure operation. If True, arguments to the call are hashed to produce deterministic keys. If not provided, the default is to check the global delayed_pure setting, and fallback to False if unset. to nout int The number of outputs returned from calling the resulting Delayed object. If provided, the Delayed output of the call can be iterated into nout objects, allowing for unpacking of results. By default iteration over Delayed objects will error. Note, that nout=1 expects obj to return a tuple of length 1, and consequently for nout=0 , obj should return an empty tuple. iteration traverse bool By default dask traverses builtin python collections looking for dask objects passed to delayed . For large collections this can be expensive. If obj doesn't contain any dask objects, set traverse=False to avoid doing this traversal. dask run def run ( workflow : dewret . workflow . Workflow | None , task : dewret . workflow . Lazy | list [ dewret . workflow . Lazy ] | tuple [ dewret . workflow . Lazy ], thread_pool : concurrent . futures . thread . ThreadPoolExecutor | None = None , ** kwargs : Any ) -> Any Execute a task as the output of a workflow. Runs a task with dask. Parameters: Name Type Description Default workflow None Workflow in which to record the execution. None task None dask.delayed function, wrapped by dewret, that we wish to compute. None thread_pool None thread pool for executing the workflows, to allow initialization of configuration contextvars. None **kwargs None any configuration arguments for this backend. None unwrap def unwrap ( task : dewret . workflow . Lazy ) -> collections . abc . Callable [ ... , typing . Any ] Unwraps a lazy-evaluated function to get the function. In recent dask (>=2024.3) this works with inspect.wraps, but earlier versions do not have the __wrapped__ property. Parameters: Name Type Description Default task None task to be unwrapped. None Returns: Type Description None Original target. Raises: Type Description RuntimeError if the task is not a wrapped function. Classes Delayed class Delayed ( * args , ** kwargs ) Description of a dask delayed . Since dask.delayed does not have a hintable type, this stands in its place, making sure that all the features of a dask.delayed are available. More info: https://github.com/dask/dask/issues/7779 Ancestors (in MRO) typing.Protocol typing.Generic Methods compute def compute ( self , __workflow__ : dewret . workflow . Workflow | None ) -> dewret . workflow . StepReference [ typing . Any ] Evaluate this dask.delayed . Evaluate a delayed (dask lazy-evaluated) function. dewret will have replaced it with a wrapper that expects a Workflow and all arguments will already be known to the wrapped delayed so the signature here is simple. Parameters: Name Type Description Default workflow None Workflow that this is tied to, if applicable. None Returns: Type Description None Reference to the final output step.","title":"Backend Dask"},{"location":"reference/dewret/backends/backend_dask/#module-dewretbackendsbackend_dask","text":"Dask backend. Lazy-evaluation via dask.delayed .","title":"Module dewret.backends.backend_dask"},{"location":"reference/dewret/backends/backend_dask/#variables","text":"config","title":"Variables"},{"location":"reference/dewret/backends/backend_dask/#functions","text":"","title":"Functions"},{"location":"reference/dewret/backends/backend_dask/#is_lazy","text":"def is_lazy ( task : Any ) -> bool Checks if a task is really a lazy-evaluated function for this backend. Parameters: Name Type Description Default task None suspected lazy-evaluated function. None Returns: Type Description None True if so, False otherwise.","title":"is_lazy"},{"location":"reference/dewret/backends/backend_dask/#lazy","text":"def lazy ( obj = '__no__default__' , name = None , pure = None , nout = None , traverse = True ) Wraps a function or object to produce a Delayed . Delayed objects act as proxies for the object they wrap, but all operations on them are done lazily by building up a dask graph internally. Parameters: Name Type Description Default obj object The function or object to wrap None name Dask key The key to use in the underlying graph for the wrapped object. Defaults to hashing content. Note that this only affects the name of the object wrapped by this call to delayed, and not the output of delayed function calls - for that use dask_key_name= as described below. .. note:: Because this name is used as the key in task graphs, you should ensure that it uniquely identifies obj . If you'd like to provide a descriptive name that is still unique, combine the descriptive name with :func: dask.base.tokenize of the array_like . See :ref: graphs for more. s pure bool Indicates whether calling the resulting Delayed object is a pure operation. If True, arguments to the call are hashed to produce deterministic keys. If not provided, the default is to check the global delayed_pure setting, and fallback to False if unset. to nout int The number of outputs returned from calling the resulting Delayed object. If provided, the Delayed output of the call can be iterated into nout objects, allowing for unpacking of results. By default iteration over Delayed objects will error. Note, that nout=1 expects obj to return a tuple of length 1, and consequently for nout=0 , obj should return an empty tuple. iteration traverse bool By default dask traverses builtin python collections looking for dask objects passed to delayed . For large collections this can be expensive. If obj doesn't contain any dask objects, set traverse=False to avoid doing this traversal. dask","title":"lazy"},{"location":"reference/dewret/backends/backend_dask/#run","text":"def run ( workflow : dewret . workflow . Workflow | None , task : dewret . workflow . Lazy | list [ dewret . workflow . Lazy ] | tuple [ dewret . workflow . Lazy ], thread_pool : concurrent . futures . thread . ThreadPoolExecutor | None = None , ** kwargs : Any ) -> Any Execute a task as the output of a workflow. Runs a task with dask. Parameters: Name Type Description Default workflow None Workflow in which to record the execution. None task None dask.delayed function, wrapped by dewret, that we wish to compute. None thread_pool None thread pool for executing the workflows, to allow initialization of configuration contextvars. None **kwargs None any configuration arguments for this backend. None","title":"run"},{"location":"reference/dewret/backends/backend_dask/#unwrap","text":"def unwrap ( task : dewret . workflow . Lazy ) -> collections . abc . Callable [ ... , typing . Any ] Unwraps a lazy-evaluated function to get the function. In recent dask (>=2024.3) this works with inspect.wraps, but earlier versions do not have the __wrapped__ property. Parameters: Name Type Description Default task None task to be unwrapped. None Returns: Type Description None Original target. Raises: Type Description RuntimeError if the task is not a wrapped function.","title":"unwrap"},{"location":"reference/dewret/backends/backend_dask/#classes","text":"","title":"Classes"},{"location":"reference/dewret/backends/backend_dask/#delayed","text":"class Delayed ( * args , ** kwargs ) Description of a dask delayed . Since dask.delayed does not have a hintable type, this stands in its place, making sure that all the features of a dask.delayed are available. More info: https://github.com/dask/dask/issues/7779","title":"Delayed"},{"location":"reference/dewret/backends/backend_dask/#ancestors-in-mro","text":"typing.Protocol typing.Generic","title":"Ancestors (in MRO)"},{"location":"reference/dewret/backends/backend_dask/#methods","text":"","title":"Methods"},{"location":"reference/dewret/backends/backend_dask/#compute","text":"def compute ( self , __workflow__ : dewret . workflow . Workflow | None ) -> dewret . workflow . StepReference [ typing . Any ] Evaluate this dask.delayed . Evaluate a delayed (dask lazy-evaluated) function. dewret will have replaced it with a wrapper that expects a Workflow and all arguments will already be known to the wrapped delayed so the signature here is simple. Parameters: Name Type Description Default workflow None Workflow that this is tied to, if applicable. None Returns: Type Description None Reference to the final output step.","title":"compute"},{"location":"reference/dewret/renderers/","text":"Module dewret.renderers Renderers for outputting static workflow representations. Take a Workflow and output a static representation in the desired format. Sub-modules dewret.renderers.cwl dewret.renderers.snakemake","title":"Index"},{"location":"reference/dewret/renderers/#module-dewretrenderers","text":"Renderers for outputting static workflow representations. Take a Workflow and output a static representation in the desired format.","title":"Module dewret.renderers"},{"location":"reference/dewret/renderers/#sub-modules","text":"dewret.renderers.cwl dewret.renderers.snakemake","title":"Sub-modules"},{"location":"reference/dewret/renderers/cwl/","text":"Module dewret.renderers.cwl CWL Renderer. Outputs a Common Workflow Language representation of the current workflow. Variables InputSchemaType Functions cwl_type_from_value def cwl_type_from_value ( label : str , val : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], dewret . utils . Unset ] ) -> dewret . renderers . cwl . CommandInputSchema Find a CWL type for a given (possibly Unset) value. Parameters: Name Type Description Default label None the label for the variable being checked to prefill the input def and improve debugging info. None val None a raw Python variable or an unset variable. None Returns: Type Description None Input schema type. default_renderer_config def default_renderer_config ( ) -> dewret . renderers . cwl . CWLRendererConfiguration Default configuration for this renderer. This is a hook-like call to give a configuration dict that this renderer will respect, and sets any necessary default values. Returns: a dict with (preferably) raw type structures to enable easy setting from YAML/JSON. raw_to_command_input_schema def raw_to_command_input_schema ( label : str , value : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], dewret . utils . Unset ] ) -> Union [ str , dewret . renderers . cwl . CommandInputSchema , list [ str ], list [ 'InputSchemaType' ], dict [ str , 'str | InputSchemaType' ]] Infer the CWL input structure for this value. Inspects the value, to work out an appropriate structure describing it in CWL. Parameters: Name Type Description Default label None name of the variable. None value None basic-typed variable from which to build structure. None Returns: Type Description None Structure used to define (possibly compound) basic types for input. render def render ( workflow : dewret . workflow . Workflow , ** kwargs : *< class ' dewret . renderers . cwl . CWLRendererConfiguration '> ) -> dict [ str , dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]]] Render to a dict-like structure. Parameters: Name Type Description Default workflow None workflow to evaluate result. None **kwargs None additional configuration arguments - these should match CWLRendererConfiguration. None Returns: Type Description None Reduced form as a native Python dict structure for serialization. render_expression def render_expression ( ref : Any ) -> 'ReferenceDefinition' Turn a rich (sympy) expression into a CWL JS expression. Parameters: Name Type Description Default ref None a structure whose elements are all string-renderable or sympy Basic. None to_cwl_type def to_cwl_type ( label : str , typ : type ) -> dewret . renderers . cwl . CommandInputSchema Map Python types to CWL types. Parameters: Name Type Description Default label None the label for the variable being checked to prefill the input def and improve debugging info. None typ None a Python basic type. None Returns: Type Description None CWL specification type dict. to_name def to_name ( result : dewret . core . Reference [ typing . Any ] ) -> str Take a reference and get a name representing it. The primary purpose of this method is to deal with the case where a reference is to the whole result, as we always put this into an imagined out field for CWL consistency. Returns: the name of the reference, including any field portion, appending an \"out\" fieldname if none. to_output_schema def to_output_schema ( label : str , typ : type [ typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], attr . AttrsInstance , dewret . utils . DataclassProtocol ]], output_source : str | None = None ) -> dewret . renderers . cwl . CommandOutputSchema Turn a step's output into an output schema. Takes a source, type and label and provides a description for CWL. Parameters: Name Type Description Default label None name of this field. None typ None either a basic type, compound of basic types, or a TypedDict representing a pre-defined result structure. None output_source None if provided, a CWL step result reference to input here. None Returns: Type Description None CWL CommandOutputSchema-like structure for embedding into an outputs block with_field def with_field ( result : Any ) -> str Get a string representing any 'field' suffix of a value. This only makes sense in the context of a Reference, which can represent a deep reference with a known variable (parameter or step result, say) using its __field__ attribute. Defaults to \"out\" as this produces compliant CWL where every output has a \"fieldname\". Returns: a string representation of the field portion of the passed value or \"out\" . with_type def with_type ( result : Any ) -> type | typing . Any Get a Python type from a value. Does so either by using its __type__ field (for example, for References) or if unavailable, using type() . Returns: a Python type. Classes CWLRendererConfiguration class CWLRendererConfiguration ( / , * args , ** kwargs ) Configuration for the renderer. Attributes Name Type Description Default allow_complex_types None can input/output types be other than raw? None factories_as_params None should factories be treated as input or steps? None Ancestors (in MRO) builtins.dict Methods clear def clear ( ... ) D.clear() -> None. Remove all items from D. copy def copy ( ... ) D.copy() -> a shallow copy of D fromkeys def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value. get def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default. items def items ( ... ) D.items() -> a set-like object providing a view on D's items keys def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys pop def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError. popitem def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty. setdefault def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default. update def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k] values def values ( ... ) D.values() -> an object providing a view on D's values CommandInputSchema class CommandInputSchema ( / , * args , ** kwargs ) Structure for referring to a raw type in CWL. Encompasses several CWL types. In future, it may be best to use cwltool or another library for these basic structures. Attributes Name Type Description Default type None CWL type of this input. None label None name to show for this input. None fields None (for record ) individual fields in a dict-like structure. None items None (for array ) type that each field will have. None Ancestors (in MRO) builtins.dict Methods clear def clear ( ... ) D.clear() -> None. Remove all items from D. copy def copy ( ... ) D.copy() -> a shallow copy of D fromkeys def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value. get def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default. items def items ( ... ) D.items() -> a set-like object providing a view on D's items keys def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys pop def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError. popitem def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty. setdefault def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default. update def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k] values def values ( ... ) D.values() -> an object providing a view on D's values CommandOutputSchema class CommandOutputSchema ( / , * args , ** kwargs ) Structure for referring to an output in CWL. As a simplification, this is an input schema with an extra outputSource field. Attributes Name Type Description Default outputSource None step result to use for this output. None Ancestors (in MRO) builtins.dict Methods clear def clear ( ... ) D.clear() -> None. Remove all items from D. copy def copy ( ... ) D.copy() -> a shallow copy of D fromkeys def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value. get def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default. items def items ( ... ) D.items() -> a set-like object providing a view on D's items keys def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys pop def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError. popitem def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty. setdefault def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default. update def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k] values def values ( ... ) D.values() -> an object providing a view on D's values InputsDefinition class InputsDefinition ( inputs : dict [ str , 'CommandInputParameter' ] ) CWL-renderable representation of an input parameter block. Turns dewret results into a CWL input block. Attributes Name Type Description Default input None sequence of results from a workflow. None Class variables CommandInputParameter Static methods from_parameters def from_parameters ( parameters : list [ typing . Union [ dewret . workflow . ParameterReference [ typing . Any ], dewret . workflow . FactoryCall ]] ) -> 'InputsDefinition' Takes a list of parameters into a CWL structure. Uses the parameters to fill out the necessary input fields. Returns: Type Description None CWL-like structure representing all workflow outputs. Instance variables inputs Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization. OutputsDefinition class OutputsDefinition ( outputs : dict [ str , 'CommandOutputSchema' ] | list [ 'CommandOutputSchema' ] | dewret . renderers . cwl . CommandOutputSchema ) CWL-renderable set of workflow outputs. Turns dewret results into a CWL output block. Attributes Name Type Description Default outputs None sequence of results from a workflow. None Static methods from_results def from_results ( results : dict [ str , dewret . workflow . StepReference [ typing . Any ]] | list [ dewret . workflow . StepReference [ typing . Any ]] | tuple [ dewret . workflow . StepReference [ typing . Any ], ... ] ) -> 'OutputsDefinition' Takes a mapping of results into a CWL structure. Pulls the result type from the signature, ultimately, if possible. Returns: Type Description None CWL-like structure representing all workflow outputs. Instance variables outputs Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] | list [ typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization. ReferenceDefinition class ReferenceDefinition ( source : str | None , value_from : str | None ) CWL-renderable internal reference. Normally points to a value or a step. Static methods from_reference def from_reference ( ref : dewret . core . Reference [ typing . Any ] ) -> 'ReferenceDefinition' Build from a Reference . Converts a dewret.workflow.Reference into a CWL-rendering object. Parameters: Name Type Description Default ref None reference to convert. None Instance variables source value_from Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization. StepDefinition class StepDefinition ( name : str , run : str , out : dict [ str , 'CommandInputSchema' ] | list [ str ], in_ : collections . abc . Mapping [ str , dewret . renderers . cwl . ReferenceDefinition | dewret . core . Raw ] ) CWL-renderable step. Coerces the dewret structure of a step into that needed for valid CWL. Attributes Name Type Description Default name None identifier to call this step by. None run None task to execute for this step. None in_ None inputs from values or other steps. None Static methods from_step def from_step ( step : dewret . workflow . BaseStep ) -> 'StepDefinition' Build from a BaseStep . Converts a dewret.workflow.Step into a CWL-rendering object. Parameters: Name Type Description Default step None step to convert. None Instance variables in_ name out run Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization. WorkflowDefinition class WorkflowDefinition ( steps : list [ dewret . renderers . cwl . StepDefinition ], inputs : dewret . renderers . cwl . InputsDefinition , outputs : dewret . renderers . cwl . OutputsDefinition , name : None | str ) CWL-renderable workflow. Coerces the dewret structure of a workflow into that needed for valid CWL. Attributes Name Type Description Default steps None sequence of steps in the workflow. None Static methods from_workflow def from_workflow ( workflow : dewret . workflow . Workflow , name : None | str = None ) -> 'WorkflowDefinition' Build from a Workflow . Converts a dewret.workflow.Workflow into a CWL-rendering object. Parameters: Name Type Description Default workflow None workflow to convert. None name None name of this workflow, if it should have one. None Instance variables inputs name outputs steps Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"Cwl"},{"location":"reference/dewret/renderers/cwl/#module-dewretrendererscwl","text":"CWL Renderer. Outputs a Common Workflow Language representation of the current workflow.","title":"Module dewret.renderers.cwl"},{"location":"reference/dewret/renderers/cwl/#variables","text":"InputSchemaType","title":"Variables"},{"location":"reference/dewret/renderers/cwl/#functions","text":"","title":"Functions"},{"location":"reference/dewret/renderers/cwl/#cwl_type_from_value","text":"def cwl_type_from_value ( label : str , val : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], dewret . utils . Unset ] ) -> dewret . renderers . cwl . CommandInputSchema Find a CWL type for a given (possibly Unset) value. Parameters: Name Type Description Default label None the label for the variable being checked to prefill the input def and improve debugging info. None val None a raw Python variable or an unset variable. None Returns: Type Description None Input schema type.","title":"cwl_type_from_value"},{"location":"reference/dewret/renderers/cwl/#default_renderer_config","text":"def default_renderer_config ( ) -> dewret . renderers . cwl . CWLRendererConfiguration Default configuration for this renderer. This is a hook-like call to give a configuration dict that this renderer will respect, and sets any necessary default values. Returns: a dict with (preferably) raw type structures to enable easy setting from YAML/JSON.","title":"default_renderer_config"},{"location":"reference/dewret/renderers/cwl/#raw_to_command_input_schema","text":"def raw_to_command_input_schema ( label : str , value : Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], dewret . utils . Unset ] ) -> Union [ str , dewret . renderers . cwl . CommandInputSchema , list [ str ], list [ 'InputSchemaType' ], dict [ str , 'str | InputSchemaType' ]] Infer the CWL input structure for this value. Inspects the value, to work out an appropriate structure describing it in CWL. Parameters: Name Type Description Default label None name of the variable. None value None basic-typed variable from which to build structure. None Returns: Type Description None Structure used to define (possibly compound) basic types for input.","title":"raw_to_command_input_schema"},{"location":"reference/dewret/renderers/cwl/#render","text":"def render ( workflow : dewret . workflow . Workflow , ** kwargs : *< class ' dewret . renderers . cwl . CWLRendererConfiguration '> ) -> dict [ str , dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]]] Render to a dict-like structure. Parameters: Name Type Description Default workflow None workflow to evaluate result. None **kwargs None additional configuration arguments - these should match CWLRendererConfiguration. None Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"render"},{"location":"reference/dewret/renderers/cwl/#render_expression","text":"def render_expression ( ref : Any ) -> 'ReferenceDefinition' Turn a rich (sympy) expression into a CWL JS expression. Parameters: Name Type Description Default ref None a structure whose elements are all string-renderable or sympy Basic. None","title":"render_expression"},{"location":"reference/dewret/renderers/cwl/#to_cwl_type","text":"def to_cwl_type ( label : str , typ : type ) -> dewret . renderers . cwl . CommandInputSchema Map Python types to CWL types. Parameters: Name Type Description Default label None the label for the variable being checked to prefill the input def and improve debugging info. None typ None a Python basic type. None Returns: Type Description None CWL specification type dict.","title":"to_cwl_type"},{"location":"reference/dewret/renderers/cwl/#to_name","text":"def to_name ( result : dewret . core . Reference [ typing . Any ] ) -> str Take a reference and get a name representing it. The primary purpose of this method is to deal with the case where a reference is to the whole result, as we always put this into an imagined out field for CWL consistency. Returns: the name of the reference, including any field portion, appending an \"out\" fieldname if none.","title":"to_name"},{"location":"reference/dewret/renderers/cwl/#to_output_schema","text":"def to_output_schema ( label : str , typ : type [ typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ], attr . AttrsInstance , dewret . utils . DataclassProtocol ]], output_source : str | None = None ) -> dewret . renderers . cwl . CommandOutputSchema Turn a step's output into an output schema. Takes a source, type and label and provides a description for CWL. Parameters: Name Type Description Default label None name of this field. None typ None either a basic type, compound of basic types, or a TypedDict representing a pre-defined result structure. None output_source None if provided, a CWL step result reference to input here. None Returns: Type Description None CWL CommandOutputSchema-like structure for embedding into an outputs block","title":"to_output_schema"},{"location":"reference/dewret/renderers/cwl/#with_field","text":"def with_field ( result : Any ) -> str Get a string representing any 'field' suffix of a value. This only makes sense in the context of a Reference, which can represent a deep reference with a known variable (parameter or step result, say) using its __field__ attribute. Defaults to \"out\" as this produces compliant CWL where every output has a \"fieldname\". Returns: a string representation of the field portion of the passed value or \"out\" .","title":"with_field"},{"location":"reference/dewret/renderers/cwl/#with_type","text":"def with_type ( result : Any ) -> type | typing . Any Get a Python type from a value. Does so either by using its __type__ field (for example, for References) or if unavailable, using type() . Returns: a Python type.","title":"with_type"},{"location":"reference/dewret/renderers/cwl/#classes","text":"","title":"Classes"},{"location":"reference/dewret/renderers/cwl/#cwlrendererconfiguration","text":"class CWLRendererConfiguration ( / , * args , ** kwargs ) Configuration for the renderer.","title":"CWLRendererConfiguration"},{"location":"reference/dewret/renderers/cwl/#attributes","text":"Name Type Description Default allow_complex_types None can input/output types be other than raw? None factories_as_params None should factories be treated as input or steps? None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#ancestors-in-mro","text":"builtins.dict","title":"Ancestors (in MRO)"},{"location":"reference/dewret/renderers/cwl/#methods","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#clear","text":"def clear ( ... ) D.clear() -> None. Remove all items from D.","title":"clear"},{"location":"reference/dewret/renderers/cwl/#copy","text":"def copy ( ... ) D.copy() -> a shallow copy of D","title":"copy"},{"location":"reference/dewret/renderers/cwl/#fromkeys","text":"def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value.","title":"fromkeys"},{"location":"reference/dewret/renderers/cwl/#get","text":"def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default.","title":"get"},{"location":"reference/dewret/renderers/cwl/#items","text":"def items ( ... ) D.items() -> a set-like object providing a view on D's items","title":"items"},{"location":"reference/dewret/renderers/cwl/#keys","text":"def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys","title":"keys"},{"location":"reference/dewret/renderers/cwl/#pop","text":"def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError.","title":"pop"},{"location":"reference/dewret/renderers/cwl/#popitem","text":"def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty.","title":"popitem"},{"location":"reference/dewret/renderers/cwl/#setdefault","text":"def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default.","title":"setdefault"},{"location":"reference/dewret/renderers/cwl/#update","text":"def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k]","title":"update"},{"location":"reference/dewret/renderers/cwl/#values","text":"def values ( ... ) D.values() -> an object providing a view on D's values","title":"values"},{"location":"reference/dewret/renderers/cwl/#commandinputschema","text":"class CommandInputSchema ( / , * args , ** kwargs ) Structure for referring to a raw type in CWL. Encompasses several CWL types. In future, it may be best to use cwltool or another library for these basic structures.","title":"CommandInputSchema"},{"location":"reference/dewret/renderers/cwl/#attributes_1","text":"Name Type Description Default type None CWL type of this input. None label None name to show for this input. None fields None (for record ) individual fields in a dict-like structure. None items None (for array ) type that each field will have. None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#ancestors-in-mro_1","text":"builtins.dict","title":"Ancestors (in MRO)"},{"location":"reference/dewret/renderers/cwl/#methods_1","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#clear_1","text":"def clear ( ... ) D.clear() -> None. Remove all items from D.","title":"clear"},{"location":"reference/dewret/renderers/cwl/#copy_1","text":"def copy ( ... ) D.copy() -> a shallow copy of D","title":"copy"},{"location":"reference/dewret/renderers/cwl/#fromkeys_1","text":"def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value.","title":"fromkeys"},{"location":"reference/dewret/renderers/cwl/#get_1","text":"def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default.","title":"get"},{"location":"reference/dewret/renderers/cwl/#items_1","text":"def items ( ... ) D.items() -> a set-like object providing a view on D's items","title":"items"},{"location":"reference/dewret/renderers/cwl/#keys_1","text":"def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys","title":"keys"},{"location":"reference/dewret/renderers/cwl/#pop_1","text":"def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError.","title":"pop"},{"location":"reference/dewret/renderers/cwl/#popitem_1","text":"def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty.","title":"popitem"},{"location":"reference/dewret/renderers/cwl/#setdefault_1","text":"def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default.","title":"setdefault"},{"location":"reference/dewret/renderers/cwl/#update_1","text":"def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k]","title":"update"},{"location":"reference/dewret/renderers/cwl/#values_1","text":"def values ( ... ) D.values() -> an object providing a view on D's values","title":"values"},{"location":"reference/dewret/renderers/cwl/#commandoutputschema","text":"class CommandOutputSchema ( / , * args , ** kwargs ) Structure for referring to an output in CWL. As a simplification, this is an input schema with an extra outputSource field.","title":"CommandOutputSchema"},{"location":"reference/dewret/renderers/cwl/#attributes_2","text":"Name Type Description Default outputSource None step result to use for this output. None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#ancestors-in-mro_2","text":"builtins.dict","title":"Ancestors (in MRO)"},{"location":"reference/dewret/renderers/cwl/#methods_2","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#clear_2","text":"def clear ( ... ) D.clear() -> None. Remove all items from D.","title":"clear"},{"location":"reference/dewret/renderers/cwl/#copy_2","text":"def copy ( ... ) D.copy() -> a shallow copy of D","title":"copy"},{"location":"reference/dewret/renderers/cwl/#fromkeys_2","text":"def fromkeys ( iterable , value = None , / ) Create a new dictionary with keys from iterable and values set to value.","title":"fromkeys"},{"location":"reference/dewret/renderers/cwl/#get_2","text":"def get ( self , key , default = None , / ) Return the value for key if key is in the dictionary, else default.","title":"get"},{"location":"reference/dewret/renderers/cwl/#items_2","text":"def items ( ... ) D.items() -> a set-like object providing a view on D's items","title":"items"},{"location":"reference/dewret/renderers/cwl/#keys_2","text":"def keys ( ... ) D.keys() -> a set-like object providing a view on D's keys","title":"keys"},{"location":"reference/dewret/renderers/cwl/#pop_2","text":"def pop ( ... ) D.pop(k[,d]) -> v, remove specified key and return the corresponding value. If the key is not found, return the default if given; otherwise, raise a KeyError.","title":"pop"},{"location":"reference/dewret/renderers/cwl/#popitem_2","text":"def popitem ( self , / ) Remove and return a (key, value) pair as a 2-tuple. Pairs are returned in LIFO (last-in, first-out) order. Raises KeyError if the dict is empty.","title":"popitem"},{"location":"reference/dewret/renderers/cwl/#setdefault_2","text":"def setdefault ( self , key , default = None , / ) Insert key with a value of default if key is not in the dictionary. Return the value for key if key is in the dictionary, else default.","title":"setdefault"},{"location":"reference/dewret/renderers/cwl/#update_2","text":"def update ( ... ) D.update([E, ]**F) -> None. Update D from dict/iterable E and F. If E is present and has a .keys() method, then does: for k in E: D[k] = E[k] If E is present and lacks a .keys() method, then does: for k, v in E: D[k] = v In either case, this is followed by: for k in F: D[k] = F[k]","title":"update"},{"location":"reference/dewret/renderers/cwl/#values_2","text":"def values ( ... ) D.values() -> an object providing a view on D's values","title":"values"},{"location":"reference/dewret/renderers/cwl/#inputsdefinition","text":"class InputsDefinition ( inputs : dict [ str , 'CommandInputParameter' ] ) CWL-renderable representation of an input parameter block. Turns dewret results into a CWL input block.","title":"InputsDefinition"},{"location":"reference/dewret/renderers/cwl/#attributes_3","text":"Name Type Description Default input None sequence of results from a workflow. None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#class-variables","text":"CommandInputParameter","title":"Class variables"},{"location":"reference/dewret/renderers/cwl/#static-methods","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/cwl/#from_parameters","text":"def from_parameters ( parameters : list [ typing . Union [ dewret . workflow . ParameterReference [ typing . Any ], dewret . workflow . FactoryCall ]] ) -> 'InputsDefinition' Takes a list of parameters into a CWL structure. Uses the parameters to fill out the necessary input fields. Returns: Type Description None CWL-like structure representing all workflow outputs.","title":"from_parameters"},{"location":"reference/dewret/renderers/cwl/#instance-variables","text":"inputs","title":"Instance variables"},{"location":"reference/dewret/renderers/cwl/#methods_3","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#render_1","text":"def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"render"},{"location":"reference/dewret/renderers/cwl/#outputsdefinition","text":"class OutputsDefinition ( outputs : dict [ str , 'CommandOutputSchema' ] | list [ 'CommandOutputSchema' ] | dewret . renderers . cwl . CommandOutputSchema ) CWL-renderable set of workflow outputs. Turns dewret results into a CWL output block.","title":"OutputsDefinition"},{"location":"reference/dewret/renderers/cwl/#attributes_4","text":"Name Type Description Default outputs None sequence of results from a workflow. None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#static-methods_1","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/cwl/#from_results","text":"def from_results ( results : dict [ str , dewret . workflow . StepReference [ typing . Any ]] | list [ dewret . workflow . StepReference [ typing . Any ]] | tuple [ dewret . workflow . StepReference [ typing . Any ], ... ] ) -> 'OutputsDefinition' Takes a mapping of results into a CWL structure. Pulls the result type from the signature, ultimately, if possible. Returns: Type Description None CWL-like structure representing all workflow outputs.","title":"from_results"},{"location":"reference/dewret/renderers/cwl/#instance-variables_1","text":"outputs","title":"Instance variables"},{"location":"reference/dewret/renderers/cwl/#methods_4","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#render_2","text":"def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] | list [ typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"render"},{"location":"reference/dewret/renderers/cwl/#referencedefinition","text":"class ReferenceDefinition ( source : str | None , value_from : str | None ) CWL-renderable internal reference. Normally points to a value or a step.","title":"ReferenceDefinition"},{"location":"reference/dewret/renderers/cwl/#static-methods_2","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/cwl/#from_reference","text":"def from_reference ( ref : dewret . core . Reference [ typing . Any ] ) -> 'ReferenceDefinition' Build from a Reference . Converts a dewret.workflow.Reference into a CWL-rendering object. Parameters: Name Type Description Default ref None reference to convert. None","title":"from_reference"},{"location":"reference/dewret/renderers/cwl/#instance-variables_2","text":"source value_from","title":"Instance variables"},{"location":"reference/dewret/renderers/cwl/#methods_5","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#render_3","text":"def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"render"},{"location":"reference/dewret/renderers/cwl/#stepdefinition","text":"class StepDefinition ( name : str , run : str , out : dict [ str , 'CommandInputSchema' ] | list [ str ], in_ : collections . abc . Mapping [ str , dewret . renderers . cwl . ReferenceDefinition | dewret . core . Raw ] ) CWL-renderable step. Coerces the dewret structure of a step into that needed for valid CWL.","title":"StepDefinition"},{"location":"reference/dewret/renderers/cwl/#attributes_5","text":"Name Type Description Default name None identifier to call this step by. None run None task to execute for this step. None in_ None inputs from values or other steps. None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#static-methods_3","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/cwl/#from_step","text":"def from_step ( step : dewret . workflow . BaseStep ) -> 'StepDefinition' Build from a BaseStep . Converts a dewret.workflow.Step into a CWL-rendering object. Parameters: Name Type Description Default step None step to convert. None","title":"from_step"},{"location":"reference/dewret/renderers/cwl/#instance-variables_3","text":"in_ name out run","title":"Instance variables"},{"location":"reference/dewret/renderers/cwl/#methods_6","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#render_4","text":"def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"render"},{"location":"reference/dewret/renderers/cwl/#workflowdefinition","text":"class WorkflowDefinition ( steps : list [ dewret . renderers . cwl . StepDefinition ], inputs : dewret . renderers . cwl . InputsDefinition , outputs : dewret . renderers . cwl . OutputsDefinition , name : None | str ) CWL-renderable workflow. Coerces the dewret structure of a workflow into that needed for valid CWL.","title":"WorkflowDefinition"},{"location":"reference/dewret/renderers/cwl/#attributes_6","text":"Name Type Description Default steps None sequence of steps in the workflow. None","title":"Attributes"},{"location":"reference/dewret/renderers/cwl/#static-methods_4","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/cwl/#from_workflow","text":"def from_workflow ( workflow : dewret . workflow . Workflow , name : None | str = None ) -> 'WorkflowDefinition' Build from a Workflow . Converts a dewret.workflow.Workflow into a CWL-rendering object. Parameters: Name Type Description Default workflow None workflow to convert. None name None name of this workflow, if it should have one. None","title":"from_workflow"},{"location":"reference/dewret/renderers/cwl/#instance-variables_4","text":"inputs name outputs steps","title":"Instance variables"},{"location":"reference/dewret/renderers/cwl/#methods_7","text":"","title":"Methods"},{"location":"reference/dewret/renderers/cwl/#render_5","text":"def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ 'RawType' ], dict [ str , 'RawType' ]]] Render to a dict-like structure. Returns: Type Description None Reduced form as a native Python dict structure for serialization.","title":"render"},{"location":"reference/dewret/renderers/snakemake/","text":"Module dewret.renderers.snakemake Snakemake Renderer. Outputs a Snakemake representation of the current workflow. Variables MainTypes Functions get_method_args def get_method_args ( func : dewret . workflow . Lazy ) -> inspect . Signature Retrieve the argument names and types of a lazy-evaluatable function. Parameters: Name Type Description Default func None A function that adheres to the Lazy protocol. None Returns: Type Description None An ItemsView object containing the argument names and their corresponding inspect.Parameter objects. get_method_rel_path def get_method_rel_path ( func : dewret . workflow . Lazy ) -> Any Get the relative path of the module containing the given function. Parameters: Name Type Description Default func Lazy The function for which the relative path is to be determined. None Returns: Type Description any The relative path of the module containing the function. raw_render def raw_render ( workflow : dewret . workflow . Workflow ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ str ], list [ 'MainTypes' ], dict [ str , 'MainTypes' ]]] Render the workflow as a Snakemake (SMK) string. This function converts a Workflow object into a object containing snakemake rules. Parameters: Name Type Description Default workflow Workflow The workflow to be rendered. None Returns: Type Description dict[str, MainTypes] A dictionary containing the components of the Workflow definition, for use in Snakemake workflows. render def render ( workflow : dewret . workflow . Workflow ) -> dict [ str , typing . Any ] Render the workflow as a Snakemake (SMK) string. This function converts a Workflow object into a Snakemake-compatible yaml. Parameters: Name Type Description Default workflow Workflow The workflow to be rendered. None Returns: Type Description str A Snakemake-compatible yaml representation of the workflow. to_snakemake_type def to_snakemake_type ( param : dewret . core . Raw ) -> str Convert a raw type to a corresponding Snakemake-compatible Python type. This function maps raw types to their corresponding Python types as used in Snakemake. Snakemake primarily uses Python types for its parameters, and this function ensures that the provided type is appropriately converted. Parameters: Name Type Description Default param None The raw type to be converted, which can be of any type. None Returns: Type Description None A string representing the corresponding Python type for Snakemake. Raises: Type Description TypeError If the parameter's type cannot be mapped to a known Python type. Classes InputDefinition class InputDefinition ( inputs : list [ str ], params : list [ str ] ) Represents input and parameter definitions block for a Snakemake-renderable workflow step. Attributes Name Type Description Default inputs List[str] A list of input definitions. None params List[str] A list of parameter definitions. None Static methods from_step def from_step ( step : dewret . workflow . BaseStep ) -> 'InputDefinition' Constructs an InputDefinition object from a Step. Parameters: Name Type Description Default step Step The Step object from which input and parameter block definitions are extracted. None Returns: Type Description InputDefinition An InputDefinition object. Instance variables inputs params Methods render def render ( self ) -> dict [ str , list [ str ]] Renders the input and parameter definitions as a dictionary. Returns: Type Description dict[str, list[MainTypes]] A dictionary containing the input and parameter definitions, for use in Snakemake Input and Params blocks. OutputDefinition class OutputDefinition ( output_file : str ) Represents the output definition block for a Snakemake-renderable workflow step. Attributes Name Type Description Default output_file str The output file definition. None Static methods from_step def from_step ( step : dewret . workflow . BaseStep ) -> 'OutputDefinition' Constructs an OutputDefinition object from a Step. Parameters: Name Type Description Default step Step The Step object from which the output file definition is extracted. None Returns: Type Description OutputDefinition An OutputDefinition object, for use in Snakemake Output block. Instance variables output_file Methods render def render ( self ) -> list [ str ] Renders the output definition as a list. Returns: Type Description list[str] A list containing the output file definition, for use in a Snakemake Output block. ReferenceDefinition class ReferenceDefinition ( source : str ) Represents a Snakemake-renderable internal reference. Attributes Name Type Description Default source str The source of the internal reference. None Static methods from_reference def from_reference ( ref : dewret . core . Reference [ typing . Any ] ) -> 'ReferenceDefinition' Build from a Reference . Converts a dewret.workflow.Reference into a Snakemake-rendering object. Parameters: Name Type Description Default ref None reference to convert. None Instance variables source Methods render def render ( self ) -> str Render the internal reference definition as a string. Returns: Type Description str internal reference. RunDefinition class RunDefinition ( method_name : str , rel_import : str , args : list [ str ] ) Represents a Snakemake-renderable run block for a dewret workflow step. Attributes Name Type Description Default method_name str The name of the method to be executed in the snakefile run block. None rel_import str The relative import path of the method. None args List[str] The arguments to be passed to the method. None Static methods from_task def from_task ( task : dewret . workflow . Task | dewret . workflow . Workflow ) -> 'RunDefinition' Constructs a RunDefinition object from a Task. Parameters: Name Type Description Default task Task The Task object from which method information and arguments are extracted. None Returns: Type Description RunDefinition A RunDefinition object containing the converted method information and arguments. Instance variables args method_name rel_import Methods render def render ( self ) -> list [ str ] Renders the run block as a list of strings. Returns: Type Description list[str] A list containing the import statement and the method call statement, for use in Snakemake run block. StepDefinition class StepDefinition ( name : str , run : list [ str ], params : list [ str ], output : list [ str ], input : list [ str ] ) Represents a Snakemake-renderable step definition in a dewret workflow. Attributes Name Type Description Default name str The name of the step. None run str The run block definition for the step. None params List[str] The parameter definitions for the step. None output (list[str] None The output definition for the step. None input List[str] The input definitions for the step. None Static methods from_step def from_step ( step : dewret . workflow . BaseStep ) -> 'StepDefinition' Constructs a StepDefinition object from a Step. Parameters: Name Type Description Default step Step The Step object from which step information and components are extracted. None Returns: Type Description StepDefinition A StepDefinition object containing the converted step information and components. Instance variables input name output params run Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ str ], list [ 'MainTypes' ], dict [ str , 'MainTypes' ]]] Renders the step definition as a dictionary. Returns: Type Description dict[str, MainTypes] A dictionary containing the components of the step definition, for use in Snakemake workflows. WorkflowDefinition class WorkflowDefinition ( steps : list [ dewret . renderers . snakemake . StepDefinition ] ) Represents a Snakemake-renderable workflow definition from a dewret workflow. Attributes Name Type Description Default steps List[StepDefinition] A list of StepDefinition objects representing the steps in the workflow. None Static methods from_workflow def from_workflow ( workflow : dewret . workflow . Workflow ) -> 'WorkflowDefinition' Creates a WorkflowDefinition object from a Workflow object. Parameters: Name Type Description Default workflow Workflow The workflow to be rendered. None Returns: Type Description str WorkflowDefinition with definited steps. Instance variables steps Methods render def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ str ], list [ 'MainTypes' ], dict [ str , 'MainTypes' ]]] Render the WorkflowDefinition. Returns: Type Description dict[str, MainTypes] A dictionary containing the components of the Workflow definition, for use in Snakemake workflows.","title":"Snakemake"},{"location":"reference/dewret/renderers/snakemake/#module-dewretrendererssnakemake","text":"Snakemake Renderer. Outputs a Snakemake representation of the current workflow.","title":"Module dewret.renderers.snakemake"},{"location":"reference/dewret/renderers/snakemake/#variables","text":"MainTypes","title":"Variables"},{"location":"reference/dewret/renderers/snakemake/#functions","text":"","title":"Functions"},{"location":"reference/dewret/renderers/snakemake/#get_method_args","text":"def get_method_args ( func : dewret . workflow . Lazy ) -> inspect . Signature Retrieve the argument names and types of a lazy-evaluatable function. Parameters: Name Type Description Default func None A function that adheres to the Lazy protocol. None Returns: Type Description None An ItemsView object containing the argument names and their corresponding inspect.Parameter objects.","title":"get_method_args"},{"location":"reference/dewret/renderers/snakemake/#get_method_rel_path","text":"def get_method_rel_path ( func : dewret . workflow . Lazy ) -> Any Get the relative path of the module containing the given function. Parameters: Name Type Description Default func Lazy The function for which the relative path is to be determined. None Returns: Type Description any The relative path of the module containing the function.","title":"get_method_rel_path"},{"location":"reference/dewret/renderers/snakemake/#raw_render","text":"def raw_render ( workflow : dewret . workflow . Workflow ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ str ], list [ 'MainTypes' ], dict [ str , 'MainTypes' ]]] Render the workflow as a Snakemake (SMK) string. This function converts a Workflow object into a object containing snakemake rules. Parameters: Name Type Description Default workflow Workflow The workflow to be rendered. None Returns: Type Description dict[str, MainTypes] A dictionary containing the components of the Workflow definition, for use in Snakemake workflows.","title":"raw_render"},{"location":"reference/dewret/renderers/snakemake/#render","text":"def render ( workflow : dewret . workflow . Workflow ) -> dict [ str , typing . Any ] Render the workflow as a Snakemake (SMK) string. This function converts a Workflow object into a Snakemake-compatible yaml. Parameters: Name Type Description Default workflow Workflow The workflow to be rendered. None Returns: Type Description str A Snakemake-compatible yaml representation of the workflow.","title":"render"},{"location":"reference/dewret/renderers/snakemake/#to_snakemake_type","text":"def to_snakemake_type ( param : dewret . core . Raw ) -> str Convert a raw type to a corresponding Snakemake-compatible Python type. This function maps raw types to their corresponding Python types as used in Snakemake. Snakemake primarily uses Python types for its parameters, and this function ensures that the provided type is appropriately converted. Parameters: Name Type Description Default param None The raw type to be converted, which can be of any type. None Returns: Type Description None A string representing the corresponding Python type for Snakemake. Raises: Type Description TypeError If the parameter's type cannot be mapped to a known Python type.","title":"to_snakemake_type"},{"location":"reference/dewret/renderers/snakemake/#classes","text":"","title":"Classes"},{"location":"reference/dewret/renderers/snakemake/#inputdefinition","text":"class InputDefinition ( inputs : list [ str ], params : list [ str ] ) Represents input and parameter definitions block for a Snakemake-renderable workflow step.","title":"InputDefinition"},{"location":"reference/dewret/renderers/snakemake/#attributes","text":"Name Type Description Default inputs List[str] A list of input definitions. None params List[str] A list of parameter definitions. None","title":"Attributes"},{"location":"reference/dewret/renderers/snakemake/#static-methods","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/snakemake/#from_step","text":"def from_step ( step : dewret . workflow . BaseStep ) -> 'InputDefinition' Constructs an InputDefinition object from a Step. Parameters: Name Type Description Default step Step The Step object from which input and parameter block definitions are extracted. None Returns: Type Description InputDefinition An InputDefinition object.","title":"from_step"},{"location":"reference/dewret/renderers/snakemake/#instance-variables","text":"inputs params","title":"Instance variables"},{"location":"reference/dewret/renderers/snakemake/#methods","text":"","title":"Methods"},{"location":"reference/dewret/renderers/snakemake/#render_1","text":"def render ( self ) -> dict [ str , list [ str ]] Renders the input and parameter definitions as a dictionary. Returns: Type Description dict[str, list[MainTypes]] A dictionary containing the input and parameter definitions, for use in Snakemake Input and Params blocks.","title":"render"},{"location":"reference/dewret/renderers/snakemake/#outputdefinition","text":"class OutputDefinition ( output_file : str ) Represents the output definition block for a Snakemake-renderable workflow step.","title":"OutputDefinition"},{"location":"reference/dewret/renderers/snakemake/#attributes_1","text":"Name Type Description Default output_file str The output file definition. None","title":"Attributes"},{"location":"reference/dewret/renderers/snakemake/#static-methods_1","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/snakemake/#from_step_1","text":"def from_step ( step : dewret . workflow . BaseStep ) -> 'OutputDefinition' Constructs an OutputDefinition object from a Step. Parameters: Name Type Description Default step Step The Step object from which the output file definition is extracted. None Returns: Type Description OutputDefinition An OutputDefinition object, for use in Snakemake Output block.","title":"from_step"},{"location":"reference/dewret/renderers/snakemake/#instance-variables_1","text":"output_file","title":"Instance variables"},{"location":"reference/dewret/renderers/snakemake/#methods_1","text":"","title":"Methods"},{"location":"reference/dewret/renderers/snakemake/#render_2","text":"def render ( self ) -> list [ str ] Renders the output definition as a list. Returns: Type Description list[str] A list containing the output file definition, for use in a Snakemake Output block.","title":"render"},{"location":"reference/dewret/renderers/snakemake/#referencedefinition","text":"class ReferenceDefinition ( source : str ) Represents a Snakemake-renderable internal reference.","title":"ReferenceDefinition"},{"location":"reference/dewret/renderers/snakemake/#attributes_2","text":"Name Type Description Default source str The source of the internal reference. None","title":"Attributes"},{"location":"reference/dewret/renderers/snakemake/#static-methods_2","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/snakemake/#from_reference","text":"def from_reference ( ref : dewret . core . Reference [ typing . Any ] ) -> 'ReferenceDefinition' Build from a Reference . Converts a dewret.workflow.Reference into a Snakemake-rendering object. Parameters: Name Type Description Default ref None reference to convert. None","title":"from_reference"},{"location":"reference/dewret/renderers/snakemake/#instance-variables_2","text":"source","title":"Instance variables"},{"location":"reference/dewret/renderers/snakemake/#methods_2","text":"","title":"Methods"},{"location":"reference/dewret/renderers/snakemake/#render_3","text":"def render ( self ) -> str Render the internal reference definition as a string. Returns: Type Description str internal reference.","title":"render"},{"location":"reference/dewret/renderers/snakemake/#rundefinition","text":"class RunDefinition ( method_name : str , rel_import : str , args : list [ str ] ) Represents a Snakemake-renderable run block for a dewret workflow step.","title":"RunDefinition"},{"location":"reference/dewret/renderers/snakemake/#attributes_3","text":"Name Type Description Default method_name str The name of the method to be executed in the snakefile run block. None rel_import str The relative import path of the method. None args List[str] The arguments to be passed to the method. None","title":"Attributes"},{"location":"reference/dewret/renderers/snakemake/#static-methods_3","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/snakemake/#from_task","text":"def from_task ( task : dewret . workflow . Task | dewret . workflow . Workflow ) -> 'RunDefinition' Constructs a RunDefinition object from a Task. Parameters: Name Type Description Default task Task The Task object from which method information and arguments are extracted. None Returns: Type Description RunDefinition A RunDefinition object containing the converted method information and arguments.","title":"from_task"},{"location":"reference/dewret/renderers/snakemake/#instance-variables_3","text":"args method_name rel_import","title":"Instance variables"},{"location":"reference/dewret/renderers/snakemake/#methods_3","text":"","title":"Methods"},{"location":"reference/dewret/renderers/snakemake/#render_4","text":"def render ( self ) -> list [ str ] Renders the run block as a list of strings. Returns: Type Description list[str] A list containing the import statement and the method call statement, for use in Snakemake run block.","title":"render"},{"location":"reference/dewret/renderers/snakemake/#stepdefinition","text":"class StepDefinition ( name : str , run : list [ str ], params : list [ str ], output : list [ str ], input : list [ str ] ) Represents a Snakemake-renderable step definition in a dewret workflow.","title":"StepDefinition"},{"location":"reference/dewret/renderers/snakemake/#attributes_4","text":"Name Type Description Default name str The name of the step. None run str The run block definition for the step. None params List[str] The parameter definitions for the step. None output (list[str] None The output definition for the step. None input List[str] The input definitions for the step. None","title":"Attributes"},{"location":"reference/dewret/renderers/snakemake/#static-methods_4","text":"","title":"Static methods"},{"location":"reference/dewret/renderers/snakemake/#from_step_2","text":"def from_step ( step : dewret . workflow . BaseStep ) -> 'StepDefinition' Constructs a StepDefinition object from a Step. Parameters: Name Type Description Default step Step The Step object from which step information and components are extracted. None Returns: Type Description StepDefinition A StepDefinition object containing the converted step information and components.","title":"from_step"},{"location":"reference/dewret/renderers/snakemake/#instance-variables_4","text":"input name output params run","title":"Instance variables"},{"location":"reference/dewret/renderers/snakemake/#methods_4","text":"","title":"Methods"},{"location":"reference/dewret/renderers/snakemake/#render_5","text":"def render ( self ) -> dict [ str , typing . Union [ str , float , bool , bytes , int , NoneType , list [ str ], list [ 'MainTypes' ], dict [ str , 'MainTypes' ]]] Renders the step definition as a dictionary. Returns: Type Description dict[str, MainTypes] A dictionary containing the components of the step definition, for use in Snakemake workflows.","title":"render"},{"location":"reference/dewret/renderers/snakemake/#workflowdefinition","text":"class WorkflowDefinition ( steps : list [ dewret . renderers . snakemake . StepDefinition ] ) Represents a Snakemake-renderable workflow definition from a dewret workflow.","title":"WorkflowDefinition"},{"location":"reference/dewret/renderers/snakemake/#attributes_5","text":"Name Type Description Default steps List[StepDefinition] A list of StepDefinition objects representing the steps in the workflow. 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