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Biconjugate gradient 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 r,z,p,q, rtilde,ztilde,qtilde
Allocate temerary reals rho_1, rho_2 , alpha, beta

r := b - A\bulletx
rtilde = r

for i := 1 step 1 until max_itr do
solve (M\bulletz = r )
solve (MT\bulletztilde = rtilde )
rho_1 = z\bulletrtilde
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 := A\bulletp
qtilde := AT\bulletptilde
alpha = rho_1 / (ptilde\bulletq)
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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