Conjugate gradient method of Golub and van Loan

From CFD-Wiki

Conjugate gradient method

Conjugate gradient method could be summarized as follows

System of equation

For the given system of equation
Ax = b ;
b = source vector
x = solution variable for which we seek the solution
A = coefficient matrix

M = the precondioning matrix constructued by matrix A

Algorithm

 Allocate temperary vectors p,z,q 

 Allocate temerary reals rho_0, rho_1 , alpha, beta 


 r := b - Ax 

 for i := 1 step 1 until max_itr do
     solve (Mz = r ) 

     beta := rho_0 / rho_1 

     p := z + betap 

     q := Ap 

     alpha = rho_0 / ( pq  ) 

     x := x + alphap 

     r := r - alphaq 

     rho_1 = rho_0 

 end (i-loop)
 
 deallocate all temp memory 

 return TRUE 

Reference

Ferziger, J.H. and Peric, M. 2002. "Computational Methods for Fluid Dynamics", 3rd rev. ed., Springer-Verlag, Berlin.