# Biconjugate gradient stabilized 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 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 - A$\bullet$x
rtilde = r
for i := 1 step 1 until max_itr do
rho_1 = rtilde$\bullet$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$\bullet$phat  = p )
v = A$\bullet$phat
alpha = rho_1 / (rtilde$\bullet$v)
s = r - alpha * v
solve (M$\bullet$shat = s )
t = A * shat;
omega = (t$\bullet$s) / (t$\bullet$t)
x = x + alpha * phat + omega * shat
r = s - omega * t
rho_2 = rho_1
end (i-loop)

deallocate all temp memory
return TRUE
```