LU分解求线性方程组_2011.1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1 -3  7]\n",
      " [ 2  4 -3]\n",
      " [-3  7  2]]\n"
     ]
    }
   ],
   "source": [
    "#输入一个矩阵(3阶)\n",
    "A = numpy.array([[1, -3 , 7],[2, 4, -3],[-3, 7, 2]])\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2]\n",
      " [-1]\n",
      " [ 3]]\n"
     ]
    }
   ],
   "source": [
    "b = numpy.array([[2], [-1] , [3]])\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L1:\n",
      "[[ 1.  0.  0.]\n",
      " [ 2.  1.  0.]\n",
      " [-3.  0.  1.]]\n",
      "L1_:\n",
      "[[ 1.  0.  0.]\n",
      " [-2.  1.  0.]\n",
      " [ 3.  0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "#构造矩阵L1\n",
    "L1 = numpy.array([[1, 0 , 0],[A[1,0]/A[0,0], 1, 0],[A[2,0]/A[0,0], 0, 1]])\n",
    "print(\"L1:\")\n",
    "print(L1)\n",
    "L1_ = numpy.array([[1, 0 , 0],[A[1,0]/-A[0,0], 1, 0],[A[2,0]/-A[0,0], 0, 1]])\n",
    "print(\"L1_:\")\n",
    "print(L1_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A1\n",
      "[[  1.  -3.   7.]\n",
      " [  0.  10. -17.]\n",
      " [  0.  -2.  23.]]\n"
     ]
    }
   ],
   "source": [
    "A1 = numpy.dot(L1_, A)\n",
    "print(\"A1\")\n",
    "print(A1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L2:\n",
      "[[ 1.   0.   0. ]\n",
      " [ 0.   1.   0. ]\n",
      " [ 0.  -0.2  1. ]]\n",
      "L2_:\n",
      "[[1.  0.  0. ]\n",
      " [0.  1.  0. ]\n",
      " [0.  0.2 1. ]]\n"
     ]
    }
   ],
   "source": [
    "#构造矩阵L2\n",
    "L2 = numpy.array([[1, 0 , 0],[0, 1, 0],[0, A1[2,1]/A1[1,1], 1]])\n",
    "print(\"L2:\")\n",
    "print(L2)\n",
    "L2_ = numpy.array([[1, 0 , 0],[0, 1, 0],[0, -A1[2,1]/A1[1,1], 1]])\n",
    "print(\"L2_:\")\n",
    "print(L2_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U\n",
      "[[  1.   -3.    7. ]\n",
      " [  0.   10.  -17. ]\n",
      " [  0.    0.   19.6]]\n"
     ]
    }
   ],
   "source": [
    "U = numpy.dot(L2_, A1)\n",
    "print(\"U\")\n",
    "print(U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L\n",
      "[[ 1.   0.   0. ]\n",
      " [ 2.   1.   0. ]\n",
      " [-3.  -0.2  1. ]]\n"
     ]
    }
   ],
   "source": [
    "L = numpy.dot(L1, L2)\n",
    "print(\"L\")\n",
    "print(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1. -3.  7.]\n",
      " [ 2.  4. -3.]\n",
      " [-3.  7.  2.]]\n"
     ]
    }
   ],
   "source": [
    "A = numpy.dot(L,U)\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.00000000e+00 -9.25185854e-18 -0.00000000e+00]\n",
      " [-2.00000000e+00  1.00000000e+00  0.00000000e+00]\n",
      " [ 2.60000000e+00  2.00000000e-01  1.00000000e+00]]\n",
      "[[ 1.00000000e+00 -9.25185854e-18  0.00000000e+00]\n",
      " [ 0.00000000e+00  1.00000000e+00  0.00000000e+00]\n",
      " [ 0.00000000e+00 -2.77555756e-17  1.00000000e+00]]\n",
      "[[ 2.]\n",
      " [-5.]\n",
      " [ 8.]]\n"
     ]
    }
   ],
   "source": [
    "L_ = numpy.linalg.inv(L)\n",
    "print(L_)\n",
    "print(numpy.dot(L_, L))\n",
    "y = numpy.dot(L_, b)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.          0.3        -0.09693878]\n",
      " [ 0.          0.1         0.08673469]\n",
      " [ 0.          0.          0.05102041]]\n",
      "[[ 1.00000000e+00  4.44089210e-16 -6.04051956e-16]\n",
      " [ 0.00000000e+00  1.00000000e+00 -3.43036257e-16]\n",
      " [ 0.00000000e+00  0.00000000e+00  1.00000000e+00]]\n",
      "[[-0.2755102 ]\n",
      " [ 0.19387755]\n",
      " [ 0.40816327]]\n"
     ]
    }
   ],
   "source": [
    "U_ = numpy.linalg.inv(U)\n",
    "print(U_)\n",
    "print(numpy.dot(U_, U))\n",
    "x = numpy.dot(U_, y)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.]\n",
      " [-1.]\n",
      " [ 3.]]\n"
     ]
    }
   ],
   "source": [
    "print(numpy.dot(L,y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.]\n",
      " [-5.]\n",
      " [ 8.]]\n"
     ]
    }
   ],
   "source": [
    "print(numpy.dot(U,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.]\n",
      " [-1.]\n",
      " [ 3.]]\n"
     ]
    }
   ],
   "source": [
    "print(numpy.dot(A,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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