From ab10f57b557550223932647a67094add6f517c74 Mon Sep 17 00:00:00 2001 From: zeptodoctor <44736852+zeptodoctor@users.noreply.github.com> Date: Fri, 13 Dec 2024 14:52:38 +0000 Subject: [PATCH] build based on 0c5cc86 --- dev/dev/index.html | 2 +- dev/generated/annulus.html | 4 ++-- dev/generated/annulus/index.html | 2 +- dev/generated/automaticdifferentiation/index.html | 2 +- dev/generated/chebyshev/index.html | 2 +- dev/generated/disk/index.html | 2 +- dev/generated/dunklxu.html | 4 ++-- dev/generated/halfrange.html | 4 ++-- dev/generated/halfrange/index.html | 2 +- dev/generated/nonlocaldiffusion/index.html | 2 +- dev/generated/padua/index.html | 2 +- dev/generated/proriol.html | 4 ++-- dev/generated/semiclassical.html | 4 ++-- dev/generated/semiclassical/index.html | 2 +- dev/generated/semiclassical1.html | 4 ++-- dev/generated/sphere/index.html | 2 +- dev/generated/sphere1.html | 4 ++-- dev/generated/sphere2.html | 4 ++-- dev/generated/sphere3.html | 4 ++-- dev/generated/spinweighted/index.html | 2 +- dev/generated/subspaceangles/index.html | 2 +- dev/generated/triangle/index.html | 2 +- dev/generated/zernike.html | 4 ++-- dev/index.html | 2 +- dev/search/index.html | 2 +- 25 files changed, 35 insertions(+), 35 deletions(-) diff --git a/dev/dev/index.html b/dev/dev/index.html index 75f8b29..024d929 100644 --- a/dev/dev/index.html +++ b/dev/dev/index.html @@ -34,4 +34,4 @@

Then we wait for the friendly folks at JuliaPackaging to merge the pull request to Yggdrasil, triggering a new release of the FastTransforms_jll.jl meta package that stores all precompiled binaries. With this release, we update the FastTransforms.jl Project.toml to point to the latest release and register the new version.

Since development of Yggdrasil is quite rapid, a fork may easily become stale. Git permits the developer to forcibly make a master branch on a fork even with upstream master:

git fetch upstream
 git checkout master
 git reset --hard upstream/master
-git push origin master --force
+git push origin master --force diff --git a/dev/generated/annulus.html b/dev/generated/annulus.html index 9bb414c..0294c52 100644 --- a/dev/generated/annulus.html +++ b/dev/generated/annulus.html @@ -1,7 +1,7 @@ -
+
+
-
+
+
-
+
-
+
+
+
+
+
Edit on GitHub

FastTransforms.jl Documentation

Introduction

FastTransforms.jl allows the user to conveniently work with orthogonal polynomials with degrees well into the millions.

This package provides a Julia wrapper for the C library of the same name. Additionally, all three types of nonuniform fast Fourier transforms available, as well as the Padua transform.

Fast orthogonal polynomial transforms

For this documentation, please see the documentation for FastTransforms. Most transforms have separate forward and inverse plans. In some instances, however, the inverse is in the sense of least-squares, and therefore only the forward transform is planned.

Fast Cholesky factorization of the Gram matrix

FastTransforms.GramMatrix — Type
GramMatrix(W::AbstractMatrix, X::AbstractMatrix)

Construct a symmetric positive-definite Gram matrix with data stored in $W$. Given a family of orthogonal polynomials ${\bf P}(x) = \{p_0(x), p_1(x),\ldots\}$ and a continuous inner product $\langle f, g\rangle$, the Gram matrix is defined by:

\[W_{i,j} = \langle p_{i-1}, p_{j-1}\rangle.\]

Moreover, given $X$, the transposed Jacobi matrix that satisfies $x {\bf P}(x) = {\bf P}(x) X$, the Gram matrix satisfies the skew-symmetric rank-2 displacement equation ($X = X_{1:n, 1:n}$):

\[X^\top W - WX = GJG^\top,\]

where $J = \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix}$ and where:

\[G_{:, 1} = e_n,\quad{\rm and}\quad G_{:, 2} = W_{n-1, :}X_{n-1, n} - X^\top W_{:, n}.\]

Fast ($O(n^2)$) Cholesky factorization of the Gram matrix returns the connection coefficients between ${\bf P}(x)$ and the polynomials ${\bf Q}(x)$ orthogonal in the modified inner product, ${\bf P}(x) = {\bf Q}(x) R$.

source
FastTransforms.ChebyshevGramMatrix — Type
ChebyshevGramMatrix(μ::AbstractVector)

Construct a Chebyshev–Gram matrix of size (length(μ)+1)÷2 with entries:

\[W_{i,j} = \frac{\mu_{|i-j|+1} +\mu_{i+j-1}}{2}.\]

Due to the linearization of a product of two first-kind Chebyshev polynomials, the Chebyshev–Gram matrix can be constructed from modified Chebyshev moments:

\[\mu_{n} = \langle T_{n-1}, 1\rangle.\]

Specialized construction and Cholesky factorization is given for this type.

See also GramMatrix for the general case.

source

Nonuniform fast Fourier transforms

FastTransforms.nufft1 — Function

Computes a nonuniform fast Fourier transform of type I:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} \frac{j}{N} \omega_k},\quad{\rm for}\quad 0 \le j \le N-1.\]
source

Computes a 2D nonuniform fast Fourier transform of type I-I:

\[F_{i,j} = \sum_{k=0}^{M-1}\sum_{\ell=0}^{N-1} C_{k,\ell} e^{-2\pi{\rm i} (\frac{i}{M} \omega_k + \frac{j}{N} \pi_{\ell})},\quad{\rm for}\quad 0 \le i \le M-1,\quad 0 \le j \le N-1.\]
source
FastTransforms.nufft2 — Function

Computes a nonuniform fast Fourier transform of type II:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} x_j k},\quad{\rm for}\quad 0 \le j \le N-1.\]
source

Computes a 2D nonuniform fast Fourier transform of type II-II:

\[F_{i,j} = \sum_{k=0}^{M-1}\sum_{\ell=0}^{N-1} C_{k,\ell} e^{-2\pi{\rm i} (x_i k + y_j \ell)},\quad{\rm for}\quad 0 \le i \le M-1,\quad 0 \le j \le N-1.\]
source
FastTransforms.nufft3 — Function

Computes a nonuniform fast Fourier transform of type III:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} x_j \omega_k},\quad{\rm for}\quad 0 \le j \le N-1.\]
source

Other Exported Methods

FastTransforms.gaunt — Function

Calculates the Gaunt coefficients, defined by:

\[a(m,n,\mu,\nu,q) = \frac{2(n+\nu-2q)+1}{2} \frac{(n+\nu-2q-m-\mu)!}{(n+\nu-2q+m+\mu)!} \int_{-1}^{+1} P_n^m(x) P_\nu^\mu(x) P_{n+\nu-2q}^{m+\mu}(x) {\rm\,d}x.\]

or defined by:

\[P_n^m(x) P_\nu^\mu(x) = \sum_{q=0}^{q_{\rm max}} a(m,n,\mu,\nu,q) P_{n+\nu-2q}^{m+\mu}(x)\]

This is a Julia implementation of the stable recurrence described in:

Y.-l. Xu, Fast evaluation of Gaunt coefficients: recursive approach, J. Comp. Appl. Math., 85:53–65, 1997.

source

Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.

source
FastTransforms.sphevaluate — Function

Pointwise evaluation of real orthonormal spherical harmonic:

\[Y_\ell^m(\theta,\varphi) = (-1)^{|m|}\sqrt{(\ell+\frac{1}{2})\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_\ell^{|m|}(\cos\theta) \sqrt{\frac{2-\delta_{m,0}}{2\pi}} \left\{\begin{array}{ccc} \cos m\varphi & {\rm for} & m \ge 0,\\ \sin(-m\varphi) & {\rm for} & m < 0.\end{array}\right.\]
source

Internal Methods

Miscellaneous Special Functions

FastTransforms.δ — Function

The Kronecker $\delta$ function:

\[\delta_{k,j} = \left\{\begin{array}{ccc} 1 & {\rm for} & k = j,\\ 0 & {\rm for} & k \ne j.\end{array}\right.\]
source
FastTransforms.Λ — Function

The Lambda function $\Lambda(z) = \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}$ for the ratio of gamma functions.

source

For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for $\tau$ in Appendix B of

I. Bogaert and B. Michiels and J. Fostier, 𝒪(1) computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing, SIAM J. Sci. Comput., 34:C83–C101, 2012.

source

The Lambda function $\Lambda(z,λ₁,λ₂) = \frac{\Gamma(z+\lambda_1)}{Γ(z+\lambda_2)}$ for the ratio of gamma functions.

source
FastTransforms.lambertw — Function

The principal branch of the Lambert-W function, defined by $x = W_0(x) e^{W_0(x)}$, computed using Halley's method for $x \in [-e^{-1},\infty)$.

source

Modified Chebyshev Moment-Based Quadrature

FastTransforms.chebyshevlogmoments1 — Function

Modified Chebyshev moments of the first kind with respect to the logarithmic weight:

\[ \int_{-1}^{+1} T_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
source
FastTransforms.chebyshevlogmoments2 — Function

Modified Chebyshev moments of the second kind with respect to the logarithmic weight:

\[ \int_{-1}^{+1} U_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
source

Elliptic

FastTransforms.Elliptic — Module

FastTransforms submodule for the computation of some elliptic integrals and functions.

Complete elliptic integrals of the first and second kinds:

\[K(k) = \int_0^{\frac{\pi}{2}} \frac{{\rm d}\theta}{\sqrt{1-k^2\sin^2\theta}},\quad{\rm and},\]
\[E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2\theta} {\rm\,d}\theta.\]

Jacobian elliptic functions:

\[x = \int_0^{\operatorname{sn}(x,k)} \frac{{\rm d}t}{\sqrt{(1-t^2)(1-k^2t^2)}},\]
\[x = \int_{\operatorname{cn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)[1-k^2(1-t^2)]}},\]
\[x = \int_{\operatorname{dn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)(t^2-1+k^2)}},\]

and the remaining nine are defined by:

\[\operatorname{pq}(x,k) = \frac{\operatorname{pr}(x,k)}{\operatorname{qr}(x,k)} = \frac{1}{\operatorname{qp}(x,k)}.\]
source
+Home · FastTransforms.jl
Edit on GitHub

FastTransforms.jl Documentation

Introduction

FastTransforms.jl allows the user to conveniently work with orthogonal polynomials with degrees well into the millions.

This package provides a Julia wrapper for the C library of the same name. Additionally, all three types of nonuniform fast Fourier transforms available, as well as the Padua transform.

Fast orthogonal polynomial transforms

For this documentation, please see the documentation for FastTransforms. Most transforms have separate forward and inverse plans. In some instances, however, the inverse is in the sense of least-squares, and therefore only the forward transform is planned.

Fast Cholesky factorization of the Gram matrix

FastTransforms.GramMatrix — Type
GramMatrix(W::AbstractMatrix, X::AbstractMatrix)

Construct a symmetric positive-definite Gram matrix with data stored in $W$. Given a family of orthogonal polynomials ${\bf P}(x) = \{p_0(x), p_1(x),\ldots\}$ and a continuous inner product $\langle f, g\rangle$, the Gram matrix is defined by:

\[W_{i,j} = \langle p_{i-1}, p_{j-1}\rangle.\]

Moreover, given $X$, the transposed Jacobi matrix that satisfies $x {\bf P}(x) = {\bf P}(x) X$, the Gram matrix satisfies the skew-symmetric rank-2 displacement equation ($X = X_{1:n, 1:n}$):

\[X^\top W - WX = GJG^\top,\]

where $J = \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix}$ and where:

\[G_{:, 1} = e_n,\quad{\rm and}\quad G_{:, 2} = W_{n-1, :}X_{n-1, n} - X^\top W_{:, n}.\]

Fast ($O(n^2)$) Cholesky factorization of the Gram matrix returns the connection coefficients between ${\bf P}(x)$ and the polynomials ${\bf Q}(x)$ orthogonal in the modified inner product, ${\bf P}(x) = {\bf Q}(x) R$.

source
FastTransforms.ChebyshevGramMatrix — Type
ChebyshevGramMatrix(μ::AbstractVector)

Construct a Chebyshev–Gram matrix of size (length(μ)+1)÷2 with entries:

\[W_{i,j} = \frac{\mu_{|i-j|+1} +\mu_{i+j-1}}{2}.\]

Due to the linearization of a product of two first-kind Chebyshev polynomials, the Chebyshev–Gram matrix can be constructed from modified Chebyshev moments:

\[\mu_{n} = \langle T_{n-1}, 1\rangle.\]

Specialized construction and Cholesky factorization is given for this type.

See also GramMatrix for the general case.

source

Nonuniform fast Fourier transforms

FastTransforms.nufft1 — Function

Computes a nonuniform fast Fourier transform of type I:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} \frac{j}{N} \omega_k},\quad{\rm for}\quad 0 \le j \le N-1.\]
source

Computes a 2D nonuniform fast Fourier transform of type I-I:

\[F_{i,j} = \sum_{k=0}^{M-1}\sum_{\ell=0}^{N-1} C_{k,\ell} e^{-2\pi{\rm i} (\frac{i}{M} \omega_k + \frac{j}{N} \pi_{\ell})},\quad{\rm for}\quad 0 \le i \le M-1,\quad 0 \le j \le N-1.\]
source
FastTransforms.nufft2 — Function

Computes a nonuniform fast Fourier transform of type II:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} x_j k},\quad{\rm for}\quad 0 \le j \le N-1.\]
source

Computes a 2D nonuniform fast Fourier transform of type II-II:

\[F_{i,j} = \sum_{k=0}^{M-1}\sum_{\ell=0}^{N-1} C_{k,\ell} e^{-2\pi{\rm i} (x_i k + y_j \ell)},\quad{\rm for}\quad 0 \le i \le M-1,\quad 0 \le j \le N-1.\]
source
FastTransforms.nufft3 — Function

Computes a nonuniform fast Fourier transform of type III:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} x_j \omega_k},\quad{\rm for}\quad 0 \le j \le N-1.\]
source

Other Exported Methods

FastTransforms.gaunt — Function

Calculates the Gaunt coefficients, defined by:

\[a(m,n,\mu,\nu,q) = \frac{2(n+\nu-2q)+1}{2} \frac{(n+\nu-2q-m-\mu)!}{(n+\nu-2q+m+\mu)!} \int_{-1}^{+1} P_n^m(x) P_\nu^\mu(x) P_{n+\nu-2q}^{m+\mu}(x) {\rm\,d}x.\]

or defined by:

\[P_n^m(x) P_\nu^\mu(x) = \sum_{q=0}^{q_{\rm max}} a(m,n,\mu,\nu,q) P_{n+\nu-2q}^{m+\mu}(x)\]

This is a Julia implementation of the stable recurrence described in:

Y.-l. Xu, Fast evaluation of Gaunt coefficients: recursive approach, J. Comp. Appl. Math., 85:53–65, 1997.

source

Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.

source
FastTransforms.sphevaluate — Function

Pointwise evaluation of real orthonormal spherical harmonic:

\[Y_\ell^m(\theta,\varphi) = (-1)^{|m|}\sqrt{(\ell+\frac{1}{2})\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_\ell^{|m|}(\cos\theta) \sqrt{\frac{2-\delta_{m,0}}{2\pi}} \left\{\begin{array}{ccc} \cos m\varphi & {\rm for} & m \ge 0,\\ \sin(-m\varphi) & {\rm for} & m < 0.\end{array}\right.\]
source

Internal Methods

Miscellaneous Special Functions

FastTransforms.δ — Function

The Kronecker $\delta$ function:

\[\delta_{k,j} = \left\{\begin{array}{ccc} 1 & {\rm for} & k = j,\\ 0 & {\rm for} & k \ne j.\end{array}\right.\]
source
FastTransforms.Λ — Function

The Lambda function $\Lambda(z) = \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}$ for the ratio of gamma functions.

source

For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for $\tau$ in Appendix B of

I. Bogaert and B. Michiels and J. Fostier, 𝒪(1) computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing, SIAM J. Sci. Comput., 34:C83–C101, 2012.

source

The Lambda function $\Lambda(z,λ₁,λ₂) = \frac{\Gamma(z+\lambda_1)}{Γ(z+\lambda_2)}$ for the ratio of gamma functions.

source
FastTransforms.lambertw — Function

The principal branch of the Lambert-W function, defined by $x = W_0(x) e^{W_0(x)}$, computed using Halley's method for $x \in [-e^{-1},\infty)$.

source

Modified Chebyshev Moment-Based Quadrature

FastTransforms.chebyshevlogmoments1 — Function

Modified Chebyshev moments of the first kind with respect to the logarithmic weight:

\[ \int_{-1}^{+1} T_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
source
FastTransforms.chebyshevlogmoments2 — Function

Modified Chebyshev moments of the second kind with respect to the logarithmic weight:

\[ \int_{-1}^{+1} U_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
source

Elliptic

FastTransforms.Elliptic — Module

FastTransforms submodule for the computation of some elliptic integrals and functions.

Complete elliptic integrals of the first and second kinds:

\[K(k) = \int_0^{\frac{\pi}{2}} \frac{{\rm d}\theta}{\sqrt{1-k^2\sin^2\theta}},\quad{\rm and},\]
\[E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2\theta} {\rm\,d}\theta.\]

Jacobian elliptic functions:

\[x = \int_0^{\operatorname{sn}(x,k)} \frac{{\rm d}t}{\sqrt{(1-t^2)(1-k^2t^2)}},\]
\[x = \int_{\operatorname{cn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)[1-k^2(1-t^2)]}},\]
\[x = \int_{\operatorname{dn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)(t^2-1+k^2)}},\]

and the remaining nine are defined by:

\[\operatorname{pq}(x,k) = \frac{\operatorname{pr}(x,k)}{\operatorname{qr}(x,k)} = \frac{1}{\operatorname{qp}(x,k)}.\]
source
diff --git a/dev/search/index.html b/dev/search/index.html index b185962..6e0888a 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · FastTransforms.jl

Loading search...

    +Search · FastTransforms.jl

    Loading search...