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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 [208] or Faber and Manteuffel [92]). 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 similar relations, 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 .