{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n=3                                       #dimenze, tj. počet rovnic a neznámých\n",
    "A=np.array([[1,2,2],[1,1,1],[1,-1,1]])    #matice soustavy\n",
    "b=np.array([1,1,5])                       #vektor pravých stran"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Jacobiho iterace:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1. -1.  3.]\n",
      "[-3. -3.  3.]\n",
      "[1. 1. 5.]\n",
      "[-11.  -5.   5.]\n",
      "4\n"
     ]
    }
   ],
   "source": [
    "x=np.array([2,0,0])\n",
    "for k in range(4):\n",
    "    y=x\n",
    "    x=np.array([(b[i]-sum([A[i][j]*y[j] for j in range(i)])-sum([A[i][j]*y[j] for j in range(i+1,n)]))/A[i][i] for i in range(n)])\n",
    "    print(x)\n",
    "#    if np.linalg.norm(y-x,np.infty)<0.01:                #ukončovací podmínka: rozdíl po sobě jdoucích iterací\n",
    "#        break\n",
    "#    if np.linalg.norm(np.matmul(A,x)-b,np.infty)<0.01:   #ukončovací podmínka: reziduum\n",
    "#        break\n",
    "print(k+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gauss–Seidel:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0, 0.0, 4.0]\n",
      "[-7.0, 4.0, 16.0]\n",
      "[-39.0, 24.0, 68.0]\n",
      "[-183.0, 116.0, 304.0]\n",
      "4\n"
     ]
    }
   ],
   "source": [
    "x=[2,0,0]\n",
    "for k in range(4):\n",
    "    for i in range(n):\n",
    "        x[i]=(b[i]-sum([A[i][j]*x[j] for j in range(i)])-sum([A[i][j]*x[j] for j in range(i+1,n)]))/A[i][i]\n",
    "    print(x)\n",
    "#    if np.linalg.norm(y-x,np.infty)<0.01:                #ukončovací podmínka: rozdíl po sobě jdoucích iterací\n",
    "#        break\n",
    "#    if np.linalg.norm(np.matmul(A,x)-b,np.infty)<0.01:   #ukončovací podmínka: reziduum\n",
    "#        break\n",
    "print(k+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zkusíme zanalyzovat iterační matice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "B=\n",
      " [[ 0. -2. -2.]\n",
      " [-1.  0. -1.]\n",
      " [-1.  1.  0.]]\n",
      "c=\n",
      " [1. 1. 5.]\n"
     ]
    }
   ],
   "source": [
    "B=np.array([[-A[i][j]/A[i][i] for j in range(n)] for i in range(n)])\n",
    "for i in range(n):\n",
    "    B[i][i]=0\n",
    "#omega=1\n",
    "#B=omega*B+(1-omega)*np.identity(n)\n",
    "print(\"B=\\n\",B)\n",
    "c=np.array(b/np.diag(A))\n",
    "print(\"c=\\n\",c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=[2,0,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1. -1.  3.]\n"
     ]
    }
   ],
   "source": [
    "x=np.matmul(B,x)+c\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Spektrální poloměr: 1.732050807568877\n",
      "Řádková norma:      4.0\n",
      "Sloupcová norma:    3.0\n"
     ]
    }
   ],
   "source": [
    "print(\"Spektrální poloměr:\",max(np.abs(np.linalg.eigvals(B))))\n",
    "print(\"Řádková norma:     \",np.linalg.norm(B,np.infty))\n",
    "print(\"Sloupcová norma:   \",np.linalg.norm(B,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "B=\n",
      " [[-0. -2. -2.]\n",
      " [-0.  2.  1.]\n",
      " [-0.  4.  3.]]\n",
      "c=\n",
      " [1. 0. 4.]\n"
     ]
    }
   ],
   "source": [
    "B=np.linalg.inv(np.tril(A))\n",
    "c=np.matmul(B,b)\n",
    "B=-np.matmul(B,np.triu(A,1))\n",
    "#omega=1\n",
    "#B=omega*B+(1-omega)*np.identity(n)\n",
    "print(\"B=\\n\",B)\n",
    "print(\"c=\\n\",c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=[2,0,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1. 0. 4.]\n"
     ]
    }
   ],
   "source": [
    "x=np.matmul(B,x)+c\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Spektrální poloměr: 4.561552812808831\n",
      "Řádková norma:      7.0\n",
      "Sloupcová norma:    8.0\n"
     ]
    }
   ],
   "source": [
    "print(\"Spektrální poloměr:\",max(np.abs(np.linalg.eigvals(B))))\n",
    "print(\"Řádková norma:     \",np.linalg.norm(B,np.infty))\n",
    "print(\"Sloupcová norma:   \",np.linalg.norm(B,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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