II.10.6(p).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1) Сначала построим Matrix A и column f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "import numpy as np\n",
    "\n",
    "a = [1,5,1]*48\n",
    "b = [5,1] + [0]*48\n",
    "c = [[0]*(x/3) + a[x:x+3] + [0]*(47 - x/3) for x in xrange(0, 48*3, 3)]\n",
    "d = [1]*50\n",
    "A = np.vstack([b, c,d])\n",
    "#A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "f = range(1, 51)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2) Вот как они выглядят"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5 1 0 ... 0 0 0]\n",
      " [1 5 1 ... 0 0 0]\n",
      " [0 1 5 ... 0 0 0]\n",
      " ...\n",
      " [0 0 0 ... 5 1 0]\n",
      " [0 0 0 ... 1 5 1]\n",
      " [1 1 1 ... 1 1 1]] \n",
      "\n",
      "[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] \n",
      "\n"
     ]
    }
   ],
   "source": [
    "print A, '\\n\\n', f, '\\n'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3)Метод Зейделя"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def seidel(A, b, eps):\n",
    "    \n",
    "    n = len(A)\n",
    "    x = [.0 for i in range(n)]\n",
    "\n",
    "    converge = False\n",
    "    \n",
    "    while not converge:\n",
    "        x_new = np.copy(x)\n",
    "        for i in range(n):\n",
    "            s1 = sum(A[i][j] * x_new[j] for j in range(i))\n",
    "            s2 = sum(A[i][j] * x[j] for j in range(i + 1, n))\n",
    "            x_new[i] = (b[i] - s1 - s2) / A[i][i]\n",
    "#Критерий останова!!!\n",
    "        converge = sqrt(sum((x_new[i] - x[i]) ** 2 for i in range(n))) <= eps\n",
    "# Вычисление и печать невязки\n",
    "        residue = b - np.dot(A, x_new)\n",
    "        print 'The norm of the residue(third norm):'\n",
    "        print np.linalg.norm(residue, ord=3)\n",
    "        \n",
    "        x = x_new\n",
    "\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4) Метод Гаусса"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Данная функция не работает для решения произвольных систем методом гаусса, т.к. здесь реализован выбор первого элемента в столбце, а не наибольшего по модулю"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(A, b):\n",
    "    n = len(A)\n",
    "    A = np.array(A)\n",
    "    A = A.astype(float)\n",
    "    \n",
    "    \n",
    "    A = np.column_stack((A, b))\n",
    "    \n",
    "    for j in range(0, n):\n",
    "        for i in range(j+1,n):\n",
    "            if A[i][j] != 0:\n",
    "                A[i] = A[i] - A[i][j]/A[j][j] * A[j]\n",
    "                       \n",
    "    x = np.array([.0 for i in range(n)])\n",
    "    \n",
    "    for i in range(n-1, -1, -1):\n",
    "        x[i] = A[i][n]/A[i][i] - sum(A[i][k]/A[i][i]*x[k] for k in range(i+1, n))\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5) Теперь посчитаем результат для двух методов и сравним его с тем что должно было получиться"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "\n",
      "result is\n",
      "[ 1.42857331e-01  2.85714459e-01  4.28571621e-01  5.71428777e-01\n",
      "  7.14285934e-01  8.57143091e-01  1.00000025e+00  1.14285741e+00\n",
      "  1.28571456e+00  1.42857172e+00  1.57142888e+00  1.71428603e+00\n",
      "  1.85714319e+00  2.00000035e+00  2.14285750e+00  2.28571466e+00\n",
      "  2.42857182e+00  2.57142898e+00  2.71428613e+00  2.85714329e+00\n",
      "  3.00000045e+00  3.14285760e+00  3.28571476e+00  3.42857192e+00\n",
      "  3.57142907e+00  3.71428623e+00  3.85714339e+00  4.00000055e+00\n",
      "  4.14285770e+00  4.28571486e+00  4.42857202e+00  4.57142917e+00\n",
      "  4.71428633e+00  4.85714349e+00  5.00000064e+00  5.14285780e+00\n",
      "  5.28571496e+00  5.42857212e+00  5.57142927e+00  5.71428643e+00\n",
      "  5.85714359e+00  5.99958118e+00  6.14548559e+00  6.27259040e+00\n",
      "  6.49178810e+00  6.26836588e+00  8.16642360e+00 -1.00496513e-01\n",
      "  4.03360634e+01 -1.52579820e+02]\n",
      "\n",
      "erf is:\n",
      "[ 1.88230627e-07  1.72921642e-07  1.92095813e-07  2.05422138e-07\n",
      "  2.19742702e-07  2.33893278e-07  2.48073246e-07  2.62248024e-07\n",
      "  2.76423752e-07  2.90599297e-07  3.04774879e-07  3.18950454e-07\n",
      "  3.33126030e-07  3.47301605e-07  3.61477182e-07  3.75652757e-07\n",
      "  3.89828333e-07  4.04003909e-07  4.18179484e-07  4.32355060e-07\n",
      "  4.46530636e-07  4.60706211e-07  4.74881787e-07  4.89057363e-07\n",
      "  5.03232937e-07  5.17408523e-07  5.31584055e-07  5.45759837e-07\n",
      "  5.59934422e-07  5.74114748e-07  5.88267566e-07  6.02552173e-07\n",
      "  6.16205352e-07  6.32883883e-07  6.35067075e-07  7.06701615e-07\n",
      "  4.45574755e-07  1.77880354e-06 -4.52698450e-06  2.57679560e-05\n",
      " -1.19301729e-04  1.56283578e-04 -1.27058933e-04  7.85178079e-05\n",
      " -3.98424774e-05  1.74877421e-05 -6.47908379e-06  2.25939349e-06\n",
      " -4.32898375e-07  2.41062537e-07]\n"
     ]
    }
   ],
   "source": [
    "x = seidel(A, f, 10**-3)\n",
    "print '\\nresult is'\n",
    "print x\n",
    "print '\\nerf is:'\n",
    "print x - np.linalg.solve(A,f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "result is\n",
      "[ 1.42857143e-01  2.85714286e-01  4.28571429e-01  5.71428571e-01\n",
      "  7.14285714e-01  8.57142857e-01  1.00000000e+00  1.14285714e+00\n",
      "  1.28571429e+00  1.42857143e+00  1.57142857e+00  1.71428571e+00\n",
      "  1.85714286e+00  2.00000000e+00  2.14285714e+00  2.28571429e+00\n",
      "  2.42857143e+00  2.57142857e+00  2.71428571e+00  2.85714286e+00\n",
      "  3.00000000e+00  3.14285714e+00  3.28571429e+00  3.42857143e+00\n",
      "  3.57142857e+00  3.71428571e+00  3.85714286e+00  4.00000000e+00\n",
      "  4.14285714e+00  4.28571429e+00  4.42857143e+00  4.57142857e+00\n",
      "  4.71428571e+00  4.85714286e+00  5.00000001e+00  5.14285710e+00\n",
      "  5.28571451e+00  5.42857034e+00  5.57143380e+00  5.71426066e+00\n",
      "  5.85726289e+00  5.99942489e+00  6.14561264e+00  6.27251188e+00\n",
      "  6.49182794e+00  6.26834840e+00  8.16643008e+00 -1.00498772e-01\n",
      "  4.03360638e+01 -1.52579820e+02]\n",
      "\n",
      "erf is:\n",
      "[ 2.77555756e-17 -5.55111512e-17 -5.55111512e-17  0.00000000e+00\n",
      "  0.00000000e+00 -1.11022302e-16  0.00000000e+00  0.00000000e+00\n",
      "  0.00000000e+00  0.00000000e+00 -2.22044605e-16  0.00000000e+00\n",
      " -2.22044605e-16  0.00000000e+00 -4.44089210e-16  0.00000000e+00\n",
      "  4.44089210e-16  0.00000000e+00  4.44089210e-16  0.00000000e+00\n",
      " -4.44089210e-16 -4.44089210e-16 -4.44089210e-16  4.44089210e-16\n",
      " -4.44089210e-16  4.44089210e-16  8.88178420e-16  0.00000000e+00\n",
      "  0.00000000e+00  8.88178420e-16 -8.88178420e-16 -8.88178420e-16\n",
      "  0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n",
      "  0.00000000e+00 -8.88178420e-16  0.00000000e+00  0.00000000e+00\n",
      "  0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n",
      " -8.88178420e-16  8.88178420e-16 -1.77635684e-15  2.34534614e-15\n",
      " -7.10542736e-15  0.00000000e+00]\n"
     ]
    }
   ],
   "source": [
    "x = gauss(A, f)\n",
    "print '\\nresult is'\n",
    "print x\n",
    "print '\\nerf is:'\n",
    "print x - np.linalg.solve(A,f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Видно, что даже метод гаусса(прямой метод) не дает точного решения данной системы из-за неточного представления числа в памяти компьютера"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6) Найдем наибольшие и наименьшие собственные числа матрицы, определим число обусловленности"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "All eigenvalues\n",
      "[0.79793197 3.00397825 3.0157706  3.03570655 3.06283368 3.09864451\n",
      " 3.14044703 3.1917674  3.24738664 3.3135675  3.38196601 3.46208508\n",
      " 3.54206275 3.63494485 3.72515202 3.82939648 3.92834641 4.04235868\n",
      " 4.14844142 4.27046687 4.38196601 4.51012456 4.62523737 4.75755805\n",
      " 4.87441896 5.00887424 5.12558104 5.26012047 5.37476263 5.50734577\n",
      " 5.61803399 5.74666227 7.02359238 6.9842294  6.97523991 6.93716632\n",
      " 6.91370837 6.85955297 6.82142343 6.75261336 6.70039037 6.61803399\n",
      " 6.55262501 5.85155858 6.45793725 6.38048615 6.27484798 6.186695\n",
      " 5.97430589 6.07165359]\n",
      "Max eigenvalue:\n",
      "7.02359237578879\n",
      "Min eigenvalue:\n",
      "0.7979319694208489\n",
      "The condition number of a matrix(frobenius norm):\n",
      "87.85592866972526\n",
      "The condition number of a matrix(first norm):\n",
      "12.220201853215574\n",
      "The condition number of a matrix(second norm):\n",
      "17.46494635321572\n"
     ]
    }
   ],
   "source": [
    "n, v = np.linalg.eig(A)\n",
    "print 'All eigenvalues'\n",
    "print n\n",
    "\n",
    "print 'Max eigenvalue:'\n",
    "print max(n)\n",
    "print 'Min eigenvalue:'\n",
    "print min(n)\n",
    "\n",
    "print 'The condition number of a matrix(frobenius norm):'\n",
    "print np.linalg.cond(A, p = 'fro')\n",
    "print 'The condition number of a matrix(first norm):'\n",
    "print np.linalg.cond(A, p = 1)\n",
    "print 'The condition number of a matrix(second norm):'\n",
    "print np.linalg.cond(A, p = 2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Видно, что в зависимости от определения нормы мы получаем различные числа обусловленности для матрицы"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7)Давайте теперь проверим скорость исполнения методов расчета системы"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 1.4 ms per loop\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "10 loops, best of 3: 51.9 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit gauss(A, f)\n",
    "%timeit seidel(A, f, 10**-3)\n",
    "#timeit производит несколько запусков и вычисляет среднее время работы"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## При таком расчете метод гаусса дает 1,4 мс, а метод Зейделя 51.9 мс. Теперь выберем меньшую погрешность и сравним результаты."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.77555756e-17 -5.55111512e-17 -5.55111512e-17  0.00000000e+00\n",
      "  0.00000000e+00 -1.11022302e-16  0.00000000e+00  0.00000000e+00\n",
      "  0.00000000e+00  0.00000000e+00 -2.22044605e-16  0.00000000e+00\n",
      " -2.22044605e-16  0.00000000e+00 -4.44089210e-16  0.00000000e+00\n",
      "  4.44089210e-16  0.00000000e+00  4.44089210e-16  0.00000000e+00\n",
      " -4.44089210e-16 -4.44089210e-16 -4.44089210e-16  4.44089210e-16\n",
      " -4.44089210e-16  4.44089210e-16  8.88178420e-16  0.00000000e+00\n",
      "  0.00000000e+00  8.88178420e-16 -8.88178420e-16 -8.88178420e-16\n",
      "  0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n",
      "  0.00000000e+00 -8.88178420e-16  0.00000000e+00  0.00000000e+00\n",
      "  0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n",
      " -8.88178420e-16  8.88178420e-16 -1.77635684e-15  2.34534614e-15\n",
      " -7.10542736e-15  0.00000000e+00]\n",
      "CPU times: user 5.44 ms, sys: 57 µs, total: 5.5 ms\n",
      "Wall time: 4.55 ms\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "x = gauss(A, f)\n",
    "print x - np.linalg.solve(A,f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The norm of the residue(third norm):\n",
      "155.6293744226537\n",
      "The norm of the residue(third norm):\n",
      "31.401518443688527\n",
      "The norm of the residue(third norm):\n",
      "6.325537414841872\n",
      "The norm of the residue(third norm):\n",
      "1.283985028542123\n",
      "The norm of the residue(third norm):\n",
      "0.26642503432827086\n",
      "The norm of the residue(third norm):\n",
      "0.057085404149876055\n",
      "The norm of the residue(third norm):\n",
      "0.012650187445103392\n",
      "The norm of the residue(third norm):\n",
      "0.0028829778857805658\n",
      "The norm of the residue(third norm):\n",
      "0.000670382052818896\n",
      "The norm of the residue(third norm):\n",
      "0.00015801336279888775\n",
      "The norm of the residue(third norm):\n",
      "3.7589574512977734e-05\n",
      "The norm of the residue(third norm):\n",
      "9.000999867582166e-06\n",
      "The norm of the residue(third norm):\n",
      "2.1659950458933697e-06\n",
      "The norm of the residue(third norm):\n",
      "5.232516215538841e-07\n",
      "The norm of the residue(third norm):\n",
      "1.2680430565767658e-07\n",
      "The norm of the residue(third norm):\n",
      "3.081032157099129e-08\n",
      "[-1.15482623e-12 -1.01985087e-12 -1.10167431e-12 -1.14708243e-12\n",
      " -1.19859678e-12 -1.24888988e-12 -1.29984912e-12 -1.35047529e-12\n",
      " -1.40132350e-12 -1.45150558e-12 -1.50257584e-12 -1.55253588e-12\n",
      " -1.60316205e-12 -1.65423231e-12 -1.70485848e-12 -1.75504056e-12\n",
      " -1.80566673e-12 -1.85629290e-12 -1.90736316e-12 -1.95754524e-12\n",
      " -2.00861550e-12 -2.06012984e-12 -2.10986784e-12 -2.15960583e-12\n",
      " -2.21289653e-12 -2.25375274e-12 -2.34745556e-12 -2.19202434e-12\n",
      " -3.23385763e-12  1.46549439e-12 -2.13420392e-11  8.76383410e-11\n",
      " -4.34808634e-10  2.06809503e-09 -4.55387550e-09  6.38156195e-09\n",
      " -6.66925626e-09  5.64858382e-09 -4.07521039e-09  2.58312127e-09\n",
      " -1.47110768e-09  7.62026886e-10 -3.64305919e-10  1.59894320e-10\n",
      " -6.63824551e-11  2.41326958e-11 -9.40758582e-12  1.83340843e-12\n",
      " -1.58451030e-12 -7.95807864e-13]\n",
      "CPU times: user 119 ms, sys: 192 µs, total: 119 ms\n",
      "Wall time: 105 ms\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "x = seidel(A, f, 10**-7)\n",
    "print x - np.linalg.solve(A,f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## При таком расчете метод гаусса дает 5 мс, а метод Зейделя 110 мс. Результат медленнее для Зейделя, но зато стал гораздо точнее"
   ]
  }
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