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 === | ||
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:: 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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: return TRUE <br> | : return TRUE <br> | ||
---- | ---- | ||
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| + | |||
| + | == '''[[Sample code for BiCGSTAB - Fortran 90]]''' == | ||
=== 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/] |
---- | ---- | ||
<i> Return to [[Numerical methods | Numerical Methods]] </i> | <i> Return to [[Numerical methods | Numerical Methods]] </i> | ||
Latest revision as of 17:01, 31 October 2016
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
Sample code for BiCGSTAB - Fortran 90
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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