# Conjugate gradient method of Golub and van Loan

(Difference between revisions)
 Revision as of 06:25, 3 October 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 20:35, 15 December 2005 (view source)Tsaad (Talk | contribs) (fixed dot product notation---again!)Newer edit → Line 17: Line 17: :  Allocate temerary reals rho_0, rho_1 , alpha, beta
:  Allocate temerary reals rho_0, rho_1 , alpha, beta
:
:
- :  r := b - A$\bullet$x
+ :  r := b - A$\cdot$x
:
:
:  for i := 1 step 1 until max_itr do :  for i := 1 step 1 until max_itr do - ::    solve (M$\bullet$z = r )
+ ::    solve (M$\cdot$z = r )
::      beta := rho_0 / rho_1
::      beta := rho_0 / rho_1
- ::      p := z + beta$\bullet$p
+ ::      p := z + beta$\cdot$p
::      q := A$\bullet$p
::      q := A$\bullet$p
- ::      alpha = rho_0 / ( p$\bullet$q  )
+ ::      alpha = rho_0 / ( p$\cdot$q  )
- ::      x := x + alpha$\bullet$p
+ ::      x := x + alpha$\cdot$p
- ::      r := r - alpha$\bullet$q
+ ::      r := r - alpha$\cdot$q
::      rho_1 = rho_0
::      rho_1 = rho_0
:  end (i-loop) :  end (i-loop)

## Contents

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 - A$\cdot$x

for i := 1 step 1 until max_itr do
solve (M$\cdot$z = r )
beta := rho_0 / rho_1
p := z + beta$\cdot$p
q := A$\bullet$p
alpha = rho_0 / ( p$\cdot$q )
x := x + alpha$\cdot$p
r := r - alpha$\cdot$q
rho_1 = rho_0
end (i-loop)

deallocate all temp memory
return TRUE

## Reference

1. 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"
2. Ferziger, J.H. and Peric, M. 2002. Computational Methods for Fluid Dynamics, 3rd rev. ed., Springer-Verlag, Berlin.