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

Mon Nov 20 08:52:54 EST 1995