Chapter 8 - Iterative solutions of linear equations

Example No. 8_01 Page No. 254

In [1]:
from __future__ import division
from numpy import mat
#Gauss Jacobi

A = mat([[ 2,  1 , 1] ,[3,  5,  2],[2,  1,  4]])
B = mat([[ 5],[ 15],[  8]])
x1old = 0 ;x2old = 0 ; x3old = 0 #intial assumption of x1,x2 & x3

print 'x1 = (5 - x2 - x3)/2 '
print 'x2 = (15 - 3x1 - 2x3)/5 '
print 'x3 = (8 - 2x1 - x2)/4'

for i in range(1,5):
    print '\n Iteration Number : %d\n'%(i)
    
    x1 = (5 - x2old - x3old)/2 #
    x2 = (15 - 3*x1old - 2*x3old)/5 # 
    x3 = (8 - 2*x1old - x2old)/4 #
    
    print ' \n x1 = %f\n x2 = %f\n x3 = %f\n'%(x1,x2,x3)
   
    x1old = x1#
    x2old = x2#
    x3old = x3#
    

x1 = (5 - x2 - x3)/2 
x2 = (15 - 3x1 - 2x3)/5 
x3 = (8 - 2x1 - x2)/4

 Iteration Number : 1

 
 x1 = 2.500000
 x2 = 3.000000
 x3 = 2.000000


 Iteration Number : 2

 
 x1 = 0.000000
 x2 = 0.700000
 x3 = 0.000000


 Iteration Number : 3

 
 x1 = 2.150000
 x2 = 3.000000
 x3 = 1.825000


 Iteration Number : 4

 
 x1 = 0.087500
 x2 = 0.980000
 x3 = 0.175000

Example No. 8_02 Page No.261

In [2]:
from __future__ import division
from numpy import mat
#Gauss Seidel

A = mat([[ 2,  1 , 1] ,[3,  5,  2],[2,  1,  4]])
B = mat([[ 5],[ 15],[  8]])

x1old = 0 ;x2old = 0 ; x3old = 0 #intial assumption

print '(x1 = 5 - x2 - x3)/2 '
print '(x2 = 15 - 3x1 - 2x3)/5 '
print '(x3 = 8 - 2x1 - x2)/4'
 
for i in range(1,3):
     
    print '\n Iteration Number : %d'%(i)
    
    x1 = (5 - x2old - x3old)/2 #
    x1old = x1#   
    x2 = (15 - 3*x1old - 2*x3old)/5 #
    x2old = x2# 
    x3 = (8 - 2*x1old - x2old)/4 #
    x3old = x3#
    
    print ' \n x1 = %f\n x2 = %f\n x3 = %f\n'%(x1,x2,x3)
     

(x1 = 5 - x2 - x3)/2 
(x2 = 15 - 3x1 - 2x3)/5 
(x3 = 8 - 2x1 - x2)/4

 Iteration Number : 1
 
 x1 = 2.500000
 x2 = 1.500000
 x3 = 0.375000


 Iteration Number : 2
 
 x1 = 1.562500
 x2 = 1.912500
 x3 = 0.740625

Example No. 8_03 page no. 269

In [3]:
from __future__ import division
from numpy import array,zeros,ones,nditer,vstack,hstack

#Gauss Seidel


A = array([[ 3, 1],[ 1 ,-3]])
B = array([[ 5],[5 ]])

X=zeros([6,2])
print ('Using a matrix to display the results after each iteration, first row represents initial assumption')
X[0,0] = 0; X[0,1] = 0 ;#initial assumption

maxit = 1000;#Maximum number of iterations
err = 0.0003 ;

print('x1 = (5-x2)/3');
print('x2 = (x1 - 5)/3');

for i in range(1,maxit):
    
    X[i,0] = (5 - X[i-1,1])/3 ;
    X[i,1] = (X[i,0] - 5)/3 ;
    
    #Error Calculations
    err1 =abs((X[i,0] - X[i-1,0])/X[i,0]) 
    err2 =abs((X[i,1]- X[i-1,1])/X[i,1])
    
    #Terminating Condition 
    if err >= err1 and err >= err2:
        print 'The system converges to the solution ( %f , %f ) in %d iterations\n'%(X[i,0],X[i,1],i-1) 
        break
    
       

#calcution of true error i.e. difference between final result and results from each iteration
trueerr1 = abs(X[:,0] - X[i,0]*ones([i+1,1])) ;
trueerr2 = abs(X[:,1] - X[i,1]*ones([i+1,1])) ;

#displaying final results
print 'Final Result is as : '
for i in range(0,5):
    print '%.10f'%X[i,:][1],'\t%.10f'%X[i,:][0],'\t%.10f'%trueerr1[0,i],'\t',trueerr2[0,i]

Using a matrix to display the results after each iteration, first row represents initial assumption
x1 = (5-x2)/3
x2 = (x1 - 5)/3
The system converges to the solution ( 1.999949 , -1.000017 ) in 4 iterations

Final Result is as : 
0.0000000000 	0.0000000000 	1.9999491947 	1.00001693509
-1.1111111111 	1.6666666667 	0.3332825281 	0.111094176023
-0.9876543210 	2.0370370370 	0.0370878423 	0.0123626141002
-1.0013717421 	1.9958847737 	0.0040644211 	0.00135480702467
-0.9998475842 	2.0004572474 	0.0005080526 	0.000169350878084

Example No. 8_04 Page No.261

In [4]:
from numpy import mat
#Gauss Seidel

A = mat([[ 1, -3],[3, 1 ]])
B = mat([[ 5],[5] ])
x1old = 0 #intial assumption
x2old = 0  #intial assumption

print 'x1 = 5 + 3*x2 '
print 'x2 = 5 - 3*x1 '
 
for i in range(1,4):
    x1 = 5 + 3*x2old 
    x1old = x1
    x2 = 5 - 3*x1old 
    x2old = x2
    
    print '\n Iteration : %d  x1 = %d and x2 = %d\n'%(i,x1,x2)
     
print 'It is clear that the process do not converge towards the solution, rather it diverges.'

x1 = 5 + 3*x2 
x2 = 5 - 3*x1 

 Iteration : 1  x1 = 5 and x2 = -10


 Iteration : 2  x1 = -25 and x2 = 80


 Iteration : 3  x1 = 245 and x2 = -730

It is clear that the process do not converge towards the solution, rather it diverges.