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Biconjugate gradient stabilized method

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::      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 * shat;
+
::      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>
Line 39: Line 39:
:  return TRUE <br>
:  return TRUE <br>
----
----
 +
 +
 +
[[Sample code for BiCGSTAB - Fortran 90]]
=== Reference ===
=== Reference ===

Latest revision as of 09:56, 18 May 2015

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\cdotx
rtilde = r

for i := 1 step 1 until max_itr do
rho_1 = rtilde\cdotr
if i = 1 then p := r else
beta = (rho_1/rho_2) * (alpha/omega)
p = r + beta * (p - omega * v)
end if
solve (M\cdotphat = p )
v = A\cdotphat
alpha = rho_1 / (rtilde\cdotv)
s = r - alpha * v
solve (M\cdotshat = s )
t = A\cdotshat;
omega = (t\cdots) / (t\cdott)
x = x + alpha * phat + omega * shat
r = s - omega * t
rho_2 = rho_1
end (i-loop)

deallocate all temp memory
return TRUE


Sample code for BiCGSTAB - Fortran 90

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" | http://www.netlib.org/linalg/html_templates/



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