Biconjugate gradient stabilized method

From CFD-Wiki

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 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 - Ax 

 rtilde = r 

 for i := 1 step 1 until max_itr do
     rho_1 = rtilder 

     if i = 1 then p := r else 

        beta = (rho_1/rho_2) * (alpha/omega)

        p = r + beta * (p - omega * v) 

     end if 

     solve (Mphat  = p ) 

     v = Aphat 

     alpha = rho_1 / (rtildev) 

     s = r - alpha * v 

     solve (Mshat = s ) 

     t = A * shat;
     omega = (ts) / (tt) 

     x = x + alpha * phat + omega * shat 

     r = s - omega * t 

     rho_2 = rho_1 

 end (i-loop)
 
 deallocate all temp memory 

 return TRUE