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Previous: Generalized Minimal Residual (GMRES)
Up: Nonstationary Iterative Methods
Next: Quasi-Minimal Residual (QMR)
Previous Page: Implementation
Next Page: Convergence
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
.
