{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 7 Computations with Matrices"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:7.4.1 Pg: 238"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"intial v & w:\n",
"[[ 0.]\n",
" [ 0.]] \n",
"\n",
"k= 0\n",
"v(k+1) & w(k+1)= [[ 1.]\n",
" [ 1.]] \n",
"\n",
"k= 1\n",
"v(k+1) & w(k+1)= [[ 1.5]\n",
" [ 1.5]] \n",
"\n",
"k= 2\n",
"v(k+1) & w(k+1)= [[ 1.75]\n",
" [ 1.75]] \n",
"\n",
"k= 3\n",
"v(k+1) & w(k+1)= [[ 1.875]\n",
" [ 1.875]] \n",
"\n",
"k= 4\n",
"v(k+1) & w(k+1)= [[ 1.9375]\n",
" [ 1.9375]] \n",
"\n",
"k= 5\n",
"v(k+1) & w(k+1)= [[ 1.96875]\n",
" [ 1.96875]] \n",
"\n",
"k= 6\n",
"v(k+1) & w(k+1)= [[ 1.984375]\n",
" [ 1.984375]] \n",
"\n",
"k= 7\n",
"v(k+1) & w(k+1)= [[ 1.9921875]\n",
" [ 1.9921875]] \n",
"\n",
"k= 8\n",
"v(k+1) & w(k+1)= [[ 1.99609375]\n",
" [ 1.99609375]] \n",
"\n",
"k= 9\n",
"v(k+1) & w(k+1)= [[ 1.99804688]\n",
" [ 1.99804688]] \n",
"\n",
"k= 10\n",
"v(k+1) & w(k+1)= [[ 1.99902344]\n",
" [ 1.99902344]] \n",
"\n",
"k= 11\n",
"v(k+1) & w(k+1)= [[ 1.99951172]\n",
" [ 1.99951172]] \n",
"\n",
"k= 12\n",
"v(k+1) & w(k+1)= [[ 1.99975586]\n",
" [ 1.99975586]] \n",
"\n",
"k= 13\n",
"v(k+1) & w(k+1)= [[ 1.99987793]\n",
" [ 1.99987793]] \n",
"\n",
"k= 14\n",
"v(k+1) & w(k+1)= [[ 1.99993896]\n",
" [ 1.99993896]] \n",
"\n",
"k= 15\n",
"v(k+1) & w(k+1)= [[ 1.99996948]\n",
" [ 1.99996948]] \n",
"\n",
"k= 16\n",
"v(k+1) & w(k+1)= [[ 1.99998474]\n",
" [ 1.99998474]] \n",
"\n",
"k= 17\n",
"v(k+1) & w(k+1)= [[ 1.99999237]\n",
" [ 1.99999237]] \n",
"\n",
"k= 18\n",
"v(k+1) & w(k+1)= [[ 1.99999619]\n",
" [ 1.99999619]] \n",
"\n",
"k= 19\n",
"v(k+1) & w(k+1)= [[ 1.99999809]\n",
" [ 1.99999809]] \n",
"\n",
"k= 20\n",
"v(k+1) & w(k+1)= [[ 1.99999905]\n",
" [ 1.99999905]] \n",
"\n",
"k= 21\n",
"v(k+1) & w(k+1)= [[ 1.99999952]\n",
" [ 1.99999952]] \n",
"\n",
"k= 22\n",
"v(k+1) & w(k+1)= [[ 1.99999976]\n",
" [ 1.99999976]] \n",
"\n",
"k= 23\n",
"v(k+1) & w(k+1)= [[ 1.99999988]\n",
" [ 1.99999988]] \n",
"\n",
"k= 24\n",
"v(k+1) & w(k+1)= [[ 1.99999994]\n",
" [ 1.99999994]] \n",
"\n",
"k= 25\n",
"v(k+1) & w(k+1)= [[ 1.99999997]\n",
" [ 1.99999997]] \n",
"\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat,transpose,zeros\n",
"A=mat([[2 ,-1],[-1, 2]])\n",
"S=mat([[2, 0],[0, 2]])\n",
"T=mat([[0, 1],[1, 0]])\n",
"p=S**-1*T\n",
"b=transpose(mat([2 ,2]))\n",
"x=zeros([2,1])\n",
"print 'intial v & w:\\n',x,'\\n'\n",
"x_1=zeros([1,2])\n",
"for k in range(0,26):\n",
" x_1=p*x+(S**-1)*b\n",
" x=x_1\n",
" print 'k=',k\n",
" print 'v(k+1) & w(k+1)=',x_1,'\\n'\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:7.4.2 Pg: 238"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"intial v & w: [[ 0.]\n",
" [ 0.]]\n",
"k=\n",
"0\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.10887353]\n",
" [ 0.53877511]]\n",
"k=\n",
"1\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.37826108]\n",
" [ 0.67346888]]\n",
"k=\n",
"2\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.44560797]\n",
" [ 0.70714233]]\n",
"k=\n",
"3\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46244469]\n",
" [ 0.71556069]]\n",
"k=\n",
"4\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46665387]\n",
" [ 0.71766528]]\n",
"k=\n",
"5\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46770617]\n",
" [ 0.71819143]]\n",
"k=\n",
"6\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46796924]\n",
" [ 0.71832296]]\n",
"k=\n",
"7\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46803501]\n",
" [ 0.71835585]]\n",
"k=\n",
"8\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805145]\n",
" [ 0.71836407]]\n",
"k=\n",
"9\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805556]\n",
" [ 0.71836612]]\n",
"k=\n",
"10\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805659]\n",
" [ 0.71836664]]\n",
"k=\n",
"11\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805685]\n",
" [ 0.71836676]]\n",
"k=\n",
"12\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805691]\n",
" [ 0.7183668 ]]\n",
"k=\n",
"13\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"14\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"15\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"16\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"17\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"18\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"19\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"20\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"21\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"22\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"23\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"24\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n",
"k=\n",
"25\n",
"v(k+1) & w(k+1)=\n",
"[[ 0.46805693]\n",
" [ 0.71836681]]\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat,transpose,zeros\n",
"from numpy.random import rand\n",
"from numpy.linalg import inv\n",
"A=mat([[2, -1],[-1, 2]])\n",
"S=mat([[2, 0],[-1, 2]])\n",
"T=mat([[0, 1],[0, 0]])\n",
"b=rand(2,1)\n",
"p=inv(S)*T\n",
"x=zeros([2,1])\n",
"print 'intial v & w:',x\n",
"x_1=zeros([1,2])\n",
"for k in range(0,26):\n",
" x_1=p*x+inv(S)*b\n",
" x=x_1\n",
" print 'k=\\n',k\n",
" print 'v(k+1) & w(k+1)=\\n',x_1"
]
}
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