(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) Newer edit → Line 16: Line 16: :  Allocate temerary reals rho_1, rho_2 , alpha, beta
:  Allocate temerary reals rho_1, rho_2 , alpha, beta
:
:
- :  r := b - A$\bullet$x
+ :  r := b - A$\cdot$x
:  rtilde = r
:  rtilde = r
:
:
:  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 )
- ::      solve (MT$\bullet$ztilde = rtilde )
+ ::      solve (MT$\cdot$ztilde = rtilde )
- ::      rho_1 = z$\bullet$rtilde
+ ::      rho_1 = z$\cdot$rtilde
::      if i = 1 then ::      if i = 1 then :::        p := z
:::        p := z
Line 31: Line 31: :::        ptilde = ztilde + beta * ptilde
:::        ptilde = ztilde + beta * ptilde
::      end if
::      end if
- ::      q := A$\bullet$p
+ ::      q := A$\cdot$p
- ::      qtilde := AT$\bullet$ptilde
+ ::      qtilde := AT$\cdot$ptilde
- ::      alpha = rho_1 / (ptilde$\bullet$q)
+ ::      alpha = rho_1 / (ptilde$\cdot$q)
::      x = x + alpha * p
::      x = x + alpha * p
::      r = r - alpha * q
::      r = r - alpha * q

## Contents

Biconjugate 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 r,z,p,q, rtilde,ztilde,qtilde
Allocate temerary reals rho_1, rho_2 , alpha, beta

r := b - A$\cdot$x
rtilde = r

for i := 1 step 1 until max_itr do
solve (M$\cdot$z = r )
solve (MT$\cdot$ztilde = rtilde )
rho_1 = z$\cdot$rtilde
if i = 1 then
p := z
ptilde := ztilde
else
beta = (rho_1/rho_2)
p = z + beta * p
ptilde = ztilde + beta * ptilde
end if
q := A$\cdot$p
qtilde := AT$\cdot$ptilde
alpha = rho_1 / (ptilde$\cdot$q)
x = x + alpha * p
r = r - alpha * q
rtilde = rtilde - alpha * qtilde
rho_2 = rho_1
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"