Gauss-Seidel Algorithm.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def gauss_seidel(A, b, x0, tol, K):\n",
    "    \n",
    "    x_iter = []\n",
    "    n = A.shape[0]\n",
    "    err = tol + 1\n",
    "    x = x0.copy()\n",
    "    k = 0\n",
    "    \n",
    "    while err > tol and k < K:\n",
    "        \n",
    "        xold = x.copy()\n",
    "        x = gauss_iter(A, b, x)\n",
    "        print(x)\n",
    "        x_iter.append(x.copy())\n",
    "        err = np.linalg.norm(x-xold, np.inf)/np.linalg.norm(x, np.inf)\n",
    "        k = k+1\n",
    "    \n",
    "    return x, x_iter\n",
    "\n",
    "def gauss_iter(A, b, x):\n",
    "    \n",
    "    n = A.shape[0]\n",
    "    \n",
    "    for i in range(n):\n",
    "        s = 0\n",
    "        for j in range(n):\n",
    "            if j!=i:\n",
    "                s += A[i,j]*x[j]\n",
    "        x[i] = (-s +b[i])/A[i,i]\n",
    "        \n",
    "    return x\n",
    "        \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.5        -0.33333333  0.38888889  1.18055556]\n",
      "[ 0.66666667 -0.83796296  0.86651235  1.0476466 ]\n",
      "[ 0.91898148 -0.96186986  0.84286051  0.9797802 ]\n",
      "[ 0.98093493 -0.98055002  0.82335174  0.96556292]\n",
      "[ 0.99027501 -0.98203765  0.81886058  0.96369632]\n",
      "[ 0.99101882 -0.98191132  0.81821733  0.96359867]\n",
      "[ 0.99095566 -0.98183666  0.81817233  0.96362475]\n",
      "[ 0.99095566 -0.98183666  0.81817233  0.96362475]\n",
      "[array([ 0.5       , -0.33333333,  0.38888889,  1.18055556]), array([ 0.66666667, -0.83796296,  0.86651235,  1.0476466 ]), array([ 0.91898148, -0.96186986,  0.84286051,  0.9797802 ]), array([ 0.98093493, -0.98055002,  0.82335174,  0.96556292]), array([ 0.99027501, -0.98203765,  0.81886058,  0.96369632]), array([ 0.99101882, -0.98191132,  0.81821733,  0.96359867]), array([ 0.99095566, -0.98183666,  0.81817233,  0.96362475])]\n"
     ]
    }
   ],
   "source": [
    "A = np. array([[2., 1., 0., 0.],\n",
    "               [2., 3.,  0., 1.],\n",
    "               [0., -1., -6., 2.],\n",
    "               [0., 2., 1., -4.]])\n",
    "\n",
    "b = np. array([1., 0., -2., -5 ]) \n",
    "\n",
    "tol = 0.0001\n",
    "K = 1000\n",
    "x0 = np.zeros(4)\n",
    "x, x_iter = gauss_seidel(A,b, x0, tol, K)\n",
    "print(x)\n",
    "print(x_iter)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def error(x_iter):\n",
    "    errors = []\n",
    "    for i in range(len(x_iter)):\n",
    "        err = np.linalg.norm(A@x_iter[i] - b)\n",
    "        errors.append(err)\n",
    "    return errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "err = error(x_iter)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10b543e50>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.plot([i for i in range(len(x_iter))],err)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}