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.