Biconjugate gradient stabilized method

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(Difference between revisions)

Revision as of 08:18, 14 September 2005

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 precondioning matrix constructued 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 $\bullet$ x

rtilde = r

for i := 1 step 1 until max_itr do

rho_1 = rtilde $\bullet$ 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 $\bullet$ phat = p )

v = A $\bullet$ phat

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

s = r - alpha * v

solve (M $\bullet$ shat = s )

t = A * shat;

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

x = x + alpha * phat + omega * shat

r = s - omega * t

rho_2 = rho_1

end (i-loop)

deallocate all temp memory

return TRUE

@@ Line 12: / Line 12: @@
 === Algorithm ===
+----
-  Allocate temperary vectors p, phat, s, shat, t, v, rtilde  <br>
+:  Allocate temperary vectors p, phat, s, shat, t, v, rtilde  <br>
-  Allocate temerary reals rho_1, rho_2 , alpha, beta, omega <br>
+:  Allocate temerary reals rho_1, rho_2 , alpha, beta, omega <br>
-  r := b - A<math>\bullet</math>x <br>
+: <br>
-  rtilde = r <br>
+:  r := b - A<math>\bullet</math>x <br>
-  for i := 1 step 1 until max_itr do
+:  rtilde = r <br>
-      rho_1 = rtilde<math>\bullet</math>r <br>
+: <br>
-      if i = 1 then p := r else <br>
+:  for i := 1 step 1 until max_itr do
-         beta = (rho_1/rho_2) * (alpha/omega)<br>
+::      rho_1 = rtilde<math>\bullet</math>r <br>
-         p = r + beta * (p - omega * v) <br>
+::      if i = 1 then p := r else <br>
-      end if <br>
+:::         beta = (rho_1/rho_2) * (alpha/omega)<br>
-      solve (M<math>\bullet</math>phat  = p ) <br>
+:::         p = r + beta * (p - omega * v) <br>
-      v = A<math>\bullet</math>phat <br>
+::      end if <br>
-      alpha = rho_1 / (rtilde<math>\bullet</math>v) <br>
+::      solve (M<math>\bullet</math>phat  = p ) <br>
-      s = r - alpha * v <br>
+::      v = A<math>\bullet</math>phat <br>
-      solve (M<math>\bullet</math>shat = s ) <br>
+::      alpha = rho_1 / (rtilde<math>\bullet</math>v) <br>
-      t = A * shat;
+::      s = r - alpha * v <br>
-      omega = (t<math>\bullet</math>s) / (t<math>\bullet</math>t) <br>
+::      solve (M<math>\bullet</math>shat = s ) <br>
-      x = x + alpha * phat + omega * shat <br>
+::      t = A * shat;
-      r = s - omega * t <br>
+::      omega = (t<math>\bullet</math>s) / (t<math>\bullet</math>t) <br>
-      rho_2 = rho_1 <br>
+::      x = x + alpha * phat + omega * shat <br>
-  end (i-loop)
+::      r = s - omega * t <br>
+::      rho_2 = rho_1 <br>
-  deallocate all temp memory <br>
+:  end (i-loop)
-  return TRUE <br>
+:  <br>
+:  deallocate all temp memory <br>
+:  return TRUE <br>
+----

Biconjugate gradient stabilized method

From CFD-Wiki

Revision as of 08:18, 14 September 2005