Biconjugate gradient method

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Contents 1 Biconjugate gradient method 1.1 System of equation 1.2 Algorithm 1.3 Reference

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

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Allocate temperary vectors r,z,p,q, rtilde,ztilde,qtilde

Allocate temerary reals rho_1, rho_2 , alpha, beta

r := b - Ax

rtilde = r

for i := 1 step 1 until max_itr do

solve (Mz = r )

solve (M^Tztilde = rtilde )

rho_1 = zrtilde

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 := Ap

qtilde := A^Tptilde

alpha = rho_1 / (ptildeq)

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

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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"

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