{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 8 - Iterative solutions of linear equations"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example No. 8_01 Page No. 254"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x1 = (5 - x2 - x3)/2 \n",
"x2 = (15 - 3x1 - 2x3)/5 \n",
"x3 = (8 - 2x1 - x2)/4\n",
"\n",
" Iteration Number : 1\n",
"\n",
" \n",
" x1 = 2.500000\n",
" x2 = 3.000000\n",
" x3 = 2.000000\n",
"\n",
"\n",
" Iteration Number : 2\n",
"\n",
" \n",
" x1 = 0.000000\n",
" x2 = 0.700000\n",
" x3 = 0.000000\n",
"\n",
"\n",
" Iteration Number : 3\n",
"\n",
" \n",
" x1 = 2.150000\n",
" x2 = 3.000000\n",
" x3 = 1.825000\n",
"\n",
"\n",
" Iteration Number : 4\n",
"\n",
" \n",
" x1 = 0.087500\n",
" x2 = 0.980000\n",
" x3 = 0.175000\n",
"\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"#Gauss Jacobi\n",
"\n",
"A = mat([[ 2, 1 , 1] ,[3, 5, 2],[2, 1, 4]])\n",
"B = mat([[ 5],[ 15],[ 8]])\n",
"x1old = 0 ;x2old = 0 ; x3old = 0 #intial assumption of x1,x2 & x3\n",
"\n",
"print 'x1 = (5 - x2 - x3)/2 '\n",
"print 'x2 = (15 - 3x1 - 2x3)/5 '\n",
"print 'x3 = (8 - 2x1 - x2)/4'\n",
"\n",
"for i in range(1,5):\n",
" print '\\n Iteration Number : %d\\n'%(i)\n",
" \n",
" x1 = (5 - x2old - x3old)/2 #\n",
" x2 = (15 - 3*x1old - 2*x3old)/5 # \n",
" x3 = (8 - 2*x1old - x2old)/4 #\n",
" \n",
" print ' \\n x1 = %f\\n x2 = %f\\n x3 = %f\\n'%(x1,x2,x3)\n",
" \n",
" x1old = x1#\n",
" x2old = x2#\n",
" x3old = x3#\n",
" \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example No. 8_02 Page No.261"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(x1 = 5 - x2 - x3)/2 \n",
"(x2 = 15 - 3x1 - 2x3)/5 \n",
"(x3 = 8 - 2x1 - x2)/4\n",
"\n",
" Iteration Number : 1\n",
" \n",
" x1 = 2.500000\n",
" x2 = 1.500000\n",
" x3 = 0.375000\n",
"\n",
"\n",
" Iteration Number : 2\n",
" \n",
" x1 = 1.562500\n",
" x2 = 1.912500\n",
" x3 = 0.740625\n",
"\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"#Gauss Seidel\n",
"\n",
"A = mat([[ 2, 1 , 1] ,[3, 5, 2],[2, 1, 4]])\n",
"B = mat([[ 5],[ 15],[ 8]])\n",
"\n",
"x1old = 0 ;x2old = 0 ; x3old = 0 #intial assumption\n",
"\n",
"print '(x1 = 5 - x2 - x3)/2 '\n",
"print '(x2 = 15 - 3x1 - 2x3)/5 '\n",
"print '(x3 = 8 - 2x1 - x2)/4'\n",
" \n",
"for i in range(1,3):\n",
" \n",
" print '\\n Iteration Number : %d'%(i)\n",
" \n",
" x1 = (5 - x2old - x3old)/2 #\n",
" x1old = x1# \n",
" x2 = (15 - 3*x1old - 2*x3old)/5 #\n",
" x2old = x2# \n",
" x3 = (8 - 2*x1old - x2old)/4 #\n",
" x3old = x3#\n",
" \n",
" print ' \\n x1 = %f\\n x2 = %f\\n x3 = %f\\n'%(x1,x2,x3)\n",
" \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example No. 8_03 page no. 269"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Using a matrix to display the results after each iteration, first row represents initial assumption\n",
"x1 = (5-x2)/3\n",
"x2 = (x1 - 5)/3\n",
"The system converges to the solution ( 1.999949 , -1.000017 ) in 4 iterations\n",
"\n",
"Final Result is as : \n",
"0.0000000000 \t0.0000000000 \t1.9999491947 \t1.00001693509\n",
"-1.1111111111 \t1.6666666667 \t0.3332825281 \t0.111094176023\n",
"-0.9876543210 \t2.0370370370 \t0.0370878423 \t0.0123626141002\n",
"-1.0013717421 \t1.9958847737 \t0.0040644211 \t0.00135480702467\n",
"-0.9998475842 \t2.0004572474 \t0.0005080526 \t0.000169350878084\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import array,zeros,ones,nditer,vstack,hstack\n",
"\n",
"#Gauss Seidel\n",
"\n",
"\n",
"A = array([[ 3, 1],[ 1 ,-3]])\n",
"B = array([[ 5],[5 ]])\n",
"\n",
"X=zeros([6,2])\n",
"print ('Using a matrix to display the results after each iteration, first row represents initial assumption')\n",
"X[0,0] = 0; X[0,1] = 0 ;#initial assumption\n",
"\n",
"maxit = 1000;#Maximum number of iterations\n",
"err = 0.0003 ;\n",
"\n",
"print('x1 = (5-x2)/3');\n",
"print('x2 = (x1 - 5)/3');\n",
"\n",
"for i in range(1,maxit):\n",
" \n",
" X[i,0] = (5 - X[i-1,1])/3 ;\n",
" X[i,1] = (X[i,0] - 5)/3 ;\n",
" \n",
" #Error Calculations\n",
" err1 =abs((X[i,0] - X[i-1,0])/X[i,0]) \n",
" err2 =abs((X[i,1]- X[i-1,1])/X[i,1])\n",
" \n",
" #Terminating Condition \n",
" if err >= err1 and err >= err2:\n",
" print 'The system converges to the solution ( %f , %f ) in %d iterations\\n'%(X[i,0],X[i,1],i-1) \n",
" break\n",
" \n",
" \n",
"\n",
"#calcution of true error i.e. difference between final result and results from each iteration\n",
"trueerr1 = abs(X[:,0] - X[i,0]*ones([i+1,1])) ;\n",
"trueerr2 = abs(X[:,1] - X[i,1]*ones([i+1,1])) ;\n",
"\n",
"#displaying final results\n",
"print 'Final Result is as : '\n",
"for i in range(0,5):\n",
" print '%.10f'%X[i,:][1],'\\t%.10f'%X[i,:][0],'\\t%.10f'%trueerr1[0,i],'\\t',trueerr2[0,i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example No. 8_04 Page No.261"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x1 = 5 + 3*x2 \n",
"x2 = 5 - 3*x1 \n",
"\n",
" Iteration : 1 x1 = 5 and x2 = -10\n",
"\n",
"\n",
" Iteration : 2 x1 = -25 and x2 = 80\n",
"\n",
"\n",
" Iteration : 3 x1 = 245 and x2 = -730\n",
"\n",
"It is clear that the process do not converge towards the solution, rather it diverges.\n"
]
}
],
"source": [
"from numpy import mat\n",
"#Gauss Seidel\n",
"\n",
"A = mat([[ 1, -3],[3, 1 ]])\n",
"B = mat([[ 5],[5] ])\n",
"x1old = 0 #intial assumption\n",
"x2old = 0 #intial assumption\n",
"\n",
"print 'x1 = 5 + 3*x2 '\n",
"print 'x2 = 5 - 3*x1 '\n",
" \n",
"for i in range(1,4):\n",
" x1 = 5 + 3*x2old \n",
" x1old = x1\n",
" x2 = 5 - 3*x1old \n",
" x2old = x2\n",
" \n",
" print '\\n Iteration : %d x1 = %d and x2 = %d\\n'%(i,x1,x2)\n",
" \n",
"print 'It is clear that the process do not converge towards the solution, rather it diverges.'"
]
}
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