Conjugate gradient method of Golub and van Loan
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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
- solve (Mz = r )
- end (i-loop)
-
- deallocate all temp memory
- return TRUE
Reference
- Richard Barret, Michael Berry, Tony F. Chan, James Demmel, June M. Donato, Jack Dongarra, Victor Eijihout, Roldan Pozo, Charles Romine, Henk Van der Vorst, "Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods"
- Ferziger, J.H. and Peric, M. 2002. Computational Methods for Fluid Dynamics, 3rd rev. ed., Springer-Verlag, Berlin.
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