Biconjugate gradient stabilized method

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(Difference between revisions)

Latest revision as of 17:01, 31 October 2016

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 $\cdot$ x

rtilde = r

for i := 1 step 1 until max_itr do

rho_1 = rtilde $\cdot$ r

if i = 1 then p := r else

beta = (rho_1/rho_2) * (alpha/omega)

p = r + beta * (p - omega * v)

end if

solve (M $\cdot$ phat = p )

v = A $\cdot$ phat

alpha = rho_1 / (rtilde $\cdot$ v)

s = r - alpha * v

solve (M $\cdot$ shat = s )

t = A $\cdot$ shat;

omega = (t $\cdot$ s) / (t $\cdot$ t)

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

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/

Return to Numerical Methods

@@ Line 1: / Line 1: @@
-== Biconjugate gradient method ==
+== Biconjugate gradient stabilized method ==
-Biconjugate gradient method could be summarized as follows <br>
+Biconjugate gradient stabilized method could be summarized as follows <br>
 === System of equation ===
@@ Line 8: / Line 8: @@
 A = coefficient matrix <br>
-M = the precondioning matrix constructued by matrix A <br>
+M = the preconditioning matrix constructed by matrix A <br>
 === Algorithm ===
+----
+:  Allocate temperary vectors p, phat, s, shat, t, v, rtilde  <br>
+:  Allocate temerary reals rho_1, rho_2 , alpha, beta, omega <br>
+: <br>
+:  r := b - A<math>\cdot</math>x <br>
+:  rtilde = r <br>
+: <br>
+:  for i := 1 step 1 until max_itr do
+::      rho_1 = rtilde<math>\cdot</math>r <br>
+::      if i = 1 then p := r else <br>
+:::         beta = (rho_1/rho_2) * (alpha/omega)<br>
+:::         p = r + beta * (p - omega * v) <br>
+::      end if <br>
+::      solve (M<math>\cdot</math>phat  = p ) <br>
+::      v = A<math>\cdot</math>phat <br>
+::      alpha = rho_1 / (rtilde<math>\cdot</math>v) <br>
+::      s = r - alpha * v <br>
+::      solve (M<math>\cdot</math>shat = s ) <br>
+::      t = A<math>\cdot</math>shat;
+::      omega = (t<math>\cdot</math>s) / (t<math>\cdot</math>t) <br>
+::      x = x + alpha * phat + omega * shat <br>
+::      r = s - omega * t <br>
+::      rho_2 = rho_1 <br>
+:  end (i-loop)
+:  <br>
+:  deallocate all temp memory <br>
+:  return TRUE <br>
+----
+== '''[[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/node41.html | http://www.netlib.org/linalg/html_templates/]
-  Allocate temperary vectors p, phat, s, shat, t, v, rtilde  <br>
+----
-  Allocate temerary reals rho_1, rho_2 , alpha, beta, omega <br>
+<i> Return to [[Numerical methods | Numerical Methods]] </i>
-  r := b - A<math>\bullet</math>x <br>
-  rtilde = r <br>
-  for i := 1 step 1 until max_itr do
-      rho_1 = rtilde<math>\bullet</math>r <br>
-      if i = 1 then p := r else <br>
-         beta = (rho_1/rho_2) * (alpha/omega)<br>
-         p = r + beta * (p - omega * v) <br>
-      end if <br>
-      solve (M<math>\bullet</math>phat  = p ) <br>
-      v = A<math>\bullet</math>phat <br>
-      alpha = rho_1 / (rtilde<math>\bullet</math>v) <br>
-      s = r - alpha * v <br>
-      solve (M<math>\bullet</math>shat = s ) <br>
-      t = A * shat;
-      omega = (t<math>\bullet</math>s) / (t<math>\bullet</math>t) <br>
-      x = x + alpha * phat + omega * shat <br>
-      r = s - omega * t <br>
-      rho_2 = rho_1 <br>
-  end (i-loop)
-  deallocate all temp memory <br>
-  return TRUE <br>

Biconjugate gradient stabilized method

From CFD-Wiki

Latest revision as of 17:01, 31 October 2016

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