Biconjugate gradient method
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(Difference between revisions)
| Line 16: | Line 16: | ||
: Allocate temerary reals rho_1, rho_2 , alpha, beta<br> | : Allocate temerary reals rho_1, rho_2 , alpha, beta<br> | ||
: <br> | : <br> | ||
| - | : r := b - A<math>\ | + | : r := b - A<math>\cdot</math>x <br> |
: rtilde = r <br> | : rtilde = r <br> | ||
: <br> | : <br> | ||
: for i := 1 step 1 until max_itr do | : for i := 1 step 1 until max_itr do | ||
| - | :: solve (M<math>\ | + | :: solve (M<math>\cdot</math>z = r ) <br> |
| - | :: solve (M<sup>T</sup><math>\ | + | :: solve (M<sup>T</sup><math>\cdot</math>ztilde = rtilde ) <br> |
| - | :: rho_1 = z<math>\ | + | :: rho_1 = z<math>\cdot</math>rtilde <br> |
:: if i = 1 then | :: if i = 1 then | ||
::: p := z <br> | ::: p := z <br> | ||
| Line 31: | Line 31: | ||
::: ptilde = ztilde + beta * ptilde <br> | ::: ptilde = ztilde + beta * ptilde <br> | ||
:: end if <br> | :: end if <br> | ||
| - | :: q := A<math>\ | + | :: q := A<math>\cdot</math>p <br> |
| - | :: qtilde := A<sup>T</sup><math>\ | + | :: qtilde := A<sup>T</sup><math>\cdot</math>ptilde <br> |
| - | :: alpha = rho_1 / (ptilde<math>\ | + | :: alpha = rho_1 / (ptilde<math>\cdot</math>q) <br> |
:: x = x + alpha * p <br> | :: x = x + alpha * p <br> | ||
:: r = r - alpha * q <br> | :: r = r - alpha * q <br> | ||
Revision as of 20:35, 15 December 2005
Contents |
Biconjugate gradient method
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
x
- rtilde = r
-
- for i := 1 step 1 until max_itr do
- solve (Mz = r )
- solve (MTztilde = rtilde )
- rho_1 = zrtilde
- if i = 1 then
- p := z
- ptilde := ztilde
- p := z
- else
- beta = (rho_1/rho_2)
- p = z + beta * p
- ptilde = ztilde + beta * ptilde
- beta = (rho_1/rho_2)
- end if
- q := Ap
- qtilde := ATptilde
- alpha = rho_1 / (ptildeq)
- x = x + alpha * p
- r = r - alpha * q
- rtilde = rtilde - alpha * qtilde
- rho_2 = rho_1
- solve (M
- 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"
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