Biconjugate gradient stabilized method
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A = coefficient matrix <br> | A = coefficient matrix <br> | ||
| - | M = the | + | M = the preconditioning matrix constructed by matrix A <br> |
| - | + | ||
=== Algorithm === | === Algorithm === | ||
| Line 26: | Line 25: | ||
:: end if <br> | :: end if <br> | ||
:: solve (M<math>\cdot</math>phat = p ) <br> | :: solve (M<math>\cdot</math>phat = p ) <br> | ||
| - | :: v = A<math>\ | + | :: v = A<math>\cdot</math>phat <br> |
| - | :: alpha = rho_1 / (rtilde<math>\ | + | :: alpha = rho_1 / (rtilde<math>\cdot</math>v) <br> |
:: s = r - alpha * v <br> | :: s = r - alpha * v <br> | ||
:: solve (M<math>\cdot</math>shat = s ) <br> | :: solve (M<math>\cdot</math>shat = s ) <br> | ||
| - | :: t = A | + | :: t = A<math>\cdot</math>shat; |
:: omega = (t<math>\cdot</math>s) / (t<math>\cdot</math>t) <br> | :: omega = (t<math>\cdot</math>s) / (t<math>\cdot</math>t) <br> | ||
:: x = x + alpha * phat + omega * shat <br> | :: x = x + alpha * phat + omega * shat <br> | ||
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=== Reference === | === 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" | + | #'''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" [http://www.netlib.org/linalg/html_templates/node41.html | http://www.netlib.org/linalg/html_templates/] |
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<i> Return to [[Numerical methods | Numerical Methods]] </i> | <i> Return to [[Numerical methods | Numerical Methods]] </i> | ||
Revision as of 18:07, 28 July 2006
Contents |
Biconjugate gradient stabilized method
Biconjugate gradient stabilized 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 preconditioning matrix constructed by matrix A
Algorithm
- Allocate temperary vectors p, phat, s, shat, t, v, rtilde
- Allocate temerary reals rho_1, rho_2 , alpha, beta, omega
-
- r := b - A
x
- rtilde = r
-
- for i := 1 step 1 until max_itr do
- rho_1 = rtilder
- if i = 1 then p := r else
- beta = (rho_1/rho_2) * (alpha/omega)
- p = r + beta * (p - omega * v)
- beta = (rho_1/rho_2) * (alpha/omega)
- end if
- solve (Mphat = p )
- v = Aphat
- alpha = rho_1 / (rtildev)
- s = r - alpha * v
- solve (Mshat = s )
- t = Ashat;
- omega = (ts) / (tt)
- x = x + alpha * phat + omega * shat
- r = s - omega * t
- rho_2 = rho_1
- rho_1 = rtilde
- 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" | http://www.netlib.org/linalg/html_templates/
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