Exercise 2.1.9

$\mathit{Ax}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]$, so we have $x=\left[\begin{array}{c}\hfill \frac{3}{4}\hfill \\ \hfill \frac{1}{2}\hfill \\ \hfill \frac{1}{4}\hfill \end{array}\right]$.

Using Conjugate Gradients, the computer program agrees with manual solution.

import numpy as np 
 
def conjugate_gradient(S, b, N): 
   x = np.zeros(b.shape) 
   r = b 
   d = r 
   for k in range(1, N): 
       alpha = np.matmul(r.transpose(), r)/np.matmul(d.transpose(), np.matmul(S, d)) 
       x = x + alpha * d 
       r_next = r - alpha * np.matmul(S, d) 
       u = np.matmul(r_next.transpose(), r_next) 
       d = np.matmul(r.transpose(), r) 
       beta = u/d 
       d = r_next + beta * d 
       r = r_next 
   return x 
 
A = np.array([[2,-1,0],[-1,2,-1], [0,-1,2]]) 
b = np.array([1,0,0]).reshape(-1, 1) 
x = conjugate_gradient(A, b, 1000) 
print(’Conjugate␣Gradient␣Solution:␣’, x)

Conjugate Gradient Solution:  [[0.75]
 [0.5 ]
 [0.25]]