Matrix function in conjugate gradient module

Question

I am solving simply linear problem A*x=b by using conjugate gradient method. I want to find x unknown.

Note that conjGrad calls the function Av that returns the product Av The code is given below:

Inputs:

• A - sparse matrix. 2D array;
• b - right hand-side vector. 1D array;
• x - initial guess. Here, it is just 1D array with zero values.

Code:

import numpy as np
import math

A = np.array([[ 0.56244579,  0.        ,  0.        ,  0.        ,  0.52936075,
        0.59553084,  0.        ,  0.        ,  0.        ,  1.1248915 ,
        0.        ,  0.        ,  0.        ,  0.46319065,  0.43672262,
        0.        ],
      [ 0.5       ,  1.        ,  1.        ,  0.5       ,  0.        ,
        0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
        0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
        0.        ],
      [ 0.        ,  0.        ,  0.        ,  0.58009067,  0.        ,
        0.        ,  0.75411788,  0.40606347,  0.        ,  0.        ,
        0.23203627,  0.        ,  0.        ,  0.        ,  0.        ,
        0.        ]])

x = np.array([ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
    0.,  0.,  0.])

b = np.array([ 3.99464617,  1.81663614,  1.86413003])

def Av(v):
return np.dot(A,v)

def conjGrad(Av, x, b, tol=1.0e-9):
     n = len(b)
     r = b - Av(x)
     s = r.copy()
     for i in range(n):
           u = Av(s)
           alpha = np.dot(s,r)/np.dot(s,u)
           x = x + aplha*s
           r = b - Av(x)
           if(math.sqrt(np.dot(r,r))) < tol:
                 break
           else:
                 beta = - np.dot(r,u)/np.dot(s,u)
                 s = r + beta * s
     return x,i

if __name__ == '__main__':
    x, iter_number = conjGrad(Av, x, b) 


 Traceback (most recent call last):
   File "C:\Python27\Conjugate_Gradient.py", line 59, in <module>
     x, iter_number = conjGrad(Av, x, b)
   File "C:\Python27\Conjugate_Gradient.py", line 47, in conjGrad
     u = Av(s)
   File "C:\Python27\Conjugate_Gradient.py", line 40, in Av
     return np.dot(A,v)
 ValueError: matrices are not aligned

Is there any simple solution to avoid this message? Any answers will be appreciated

Answer 1

You have implemented the CG method wrong. The error message shows you one of the lines where there is a problem.

In particular, your matrix is not square.

Answer 2

The conjugate gradients method solves for Ax=b when A is SPD.

If A is not SPD, like in your case, then you can still use conjugate gradients to find the least squares solution for your problem:

A^t A x = A^tb

The matrix A^t A is SPD and well suited for your method.