The Conjugate Gradient method is not suitable for nonsymmetric systems because the residual vectors cannot be made orthogonal with short recurrences (for proof of this see Voevodin [213] or Faber and Manteuffel [96]). The GMRES method retains orthogonality of the residuals by using long recurrences, at the cost of a larger storage demand. The BiConjugate Gradient method takes another approach, replacing the orthogonal sequence of residuals by two mutually orthogonal sequences, at the price of no longer providing a minimization.
The update relations for residuals in the Conjugate Gradient method
are augmented in the BiConjugate Gradient method by relations
that are similar
but based on 
instead of 
. Thus we update two sequences of
residuals
and two sequences of search directions
The choices
ensure the bi-orthogonality relations
The pseudocode for the Preconditioned BiConjugate Gradient
Method with preconditioner 
is given in Figure 
.
Figure: The Preconditioned BiConjugate Gradient Method