Biconjugate gradient stabilized method

Biconjugate gradient stabilized method could be summarized as follows

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

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

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/