Biconjugate gradient method

From CFD-Wiki

(Difference between revisions)

Latest revision as of 15:04, 27 July 2006

Biconjugate gradient method

Biconjugate gradient 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 r,z,p,q, rtilde,ztilde,qtilde

Allocate temerary reals rho_1, rho_2 , alpha, beta

r := b - A $\cdot$ x

rtilde := r

for i := 1 step 1 until max_itr do

solve (M $\cdot$ z = r )

solve (M^T $\cdot$ ztilde = rtilde )

rho_1 := z $\cdot$ rtilde

if i = 1 then

p := z

ptilde := ztilde

else

beta := (rho_1/rho_2)

p := z + beta * p

ptilde := ztilde + beta * ptilde

end if

q := A $\cdot$ p

qtilde := A^T $\cdot$ ptilde

alpha := rho_1 / (ptilde $\cdot$ q)

x := x + alpha * p

r := r - alpha * q

rtilde := rtilde - alpha * qtilde

rho_2 := rho_1

end (i-loop)

deallocate all temp memory

return TRUE

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", Philadelphia, PA: SIAM, 1994. | http://www.netlib.org/linalg/html_templates/Templates.html

Return to Numerical Methods

@@ Line 16: / Line 16: @@
 : <br>
 :  r := b - A<math>\cdot</math>x <br>
-:  rtilde = r <br>
+:  rtilde := r <br>
 : <br>
 :  for i := 1 step 1 until max_itr do
 ::      solve (M<math>\cdot</math>z = r ) <br>
 ::      solve (M<sup>T</sup><math>\cdot</math>ztilde = rtilde ) <br>
-::      rho_1 = z<math>\cdot</math>rtilde <br>
+::      rho_1 := z<math>\cdot</math>rtilde <br>
 ::      if i = 1 then
 :::        p := z <br>
 :::        ptilde := ztilde <br>
 ::        else <br>
-:::         beta = (rho_1/rho_2) <br>
+:::         beta := (rho_1/rho_2) <br>
-:::         p = z + beta * p <br>
+:::         p := z + beta * p <br>
-:::         ptilde = ztilde + beta * ptilde <br>
+:::         ptilde := ztilde + beta * ptilde <br>
 ::      end if <br>
 ::      q := A<math>\cdot</math>p <br>
 ::      qtilde := A<sup>T</sup><math>\cdot</math>ptilde <br>
-::      alpha = rho_1 / (ptilde<math>\cdot</math>q) <br>
+::      alpha := rho_1 / (ptilde<math>\cdot</math>q) <br>
-::      x = x + alpha * p <br>
+::      x := x + alpha * p <br>
-::      r = r - alpha * q <br>
+::      r := r - alpha * q <br>
-::      rtilde = rtilde - alpha * qtilde <br>
+::      rtilde := rtilde - alpha * qtilde <br>
-::      rho_2 = rho_1 <br>
+::      rho_2 := rho_1 <br>
 :  end (i-loop)
 :  <br>
@@ Line 42: / Line 42: @@
 :  return TRUE <br>
 ----
 === 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", Philadelphia, PA: SIAM, 1994. [http://www.netlib.org/linalg/html_templates/Templates.html | http://www.netlib.org/linalg/html_templates/Templates.html]
+#'''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", Philadelphia, PA: SIAM, 1994. [http://www.netlib.org/linalg/html_templates/node32.html | http://www.netlib.org/linalg/html_templates/Templates.html]
 ----
 <i> Return to [[Numerical methods | Numerical Methods]] </i>

Biconjugate gradient method

From CFD-Wiki

Latest revision as of 15:04, 27 July 2006

Contents