Biconjugate gradient 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 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