Biconjugate gradient stabilized method

== Biconjugate gradient stabilized method ==
Biconjugate gradient stabilized method could be summarized as follows <br>

=== System of equation ===
For the given system of equation <br> Ax = b ;<br> 
b = source vector <br>
x = solution variable for which we seek the solution <br>
A = coefficient matrix <br>

M = the preconditioning matrix constructed by matrix A <br>

=== Algorithm ===
----
:  Allocate temperary vectors p, phat, s, shat, t, v, rtilde  <br>
:  Allocate temerary reals rho_1, rho_2 , alpha, beta, omega <br>
: <br>
:  r := b - A<math>\cdot</math>x <br>
:  rtilde = r <br>
: <br>
:  for i := 1 step 1 until max_itr do
::      rho_1 = rtilde<math>\cdot</math>r <br>
::      if i = 1 then p := r else <br>
:::         beta = (rho_1/rho_2) * (alpha/omega)<br>
:::         p = r + beta * (p - omega * v) <br>
::      end if <br>
::      solve (M<math>\cdot</math>phat  = p ) <br>
::      v = A<math>\cdot</math>phat <br>
::      alpha = rho_1 / (rtilde<math>\cdot</math>v) <br>
::      s = r - alpha * v <br>
::      solve (M<math>\cdot</math>shat = s ) <br>
::      t = A * shat;
::      omega = (t<math>\cdot</math>s) / (t<math>\cdot</math>t) <br>
::      x = x + alpha * phat + omega * shat <br>
::      r = s - omega * t <br>
::      rho_2 = rho_1 <br>
:  end (i-loop)
:  <br>  
:  deallocate all temp memory <br>
:  return TRUE <br>
----

=== Reference ===
#'''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"


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