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Biconjugate gradient method

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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\cdotx
rtilde := r

for i := 1 step 1 until max_itr do
solve (M\cdotz = r )
solve (MT\cdotztilde = rtilde )
rho_1 := z\cdotrtilde
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\cdotp
qtilde := AT\cdotptilde
alpha := rho_1 / (ptilde\cdotq)
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

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

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