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Biconjugate gradient stabilized method

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m (Algorithm: * -> \cdot)
 
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A = coefficient matrix <br>
A = coefficient matrix <br>
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M = the precondioning matrix constructued by matrix A <br>
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M = the preconditioning matrix constructed by matrix A <br>
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=== 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>\bullet</math>phat <br>
+
::      v = A<math>\cdot</math>phat <br>
-
::      alpha = rho_1 / (rtilde<math>\bullet</math>v) <br>
+
::      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 * shat;
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::      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 ===
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#'''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>

Latest 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\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

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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