Numerische Mathematik

, Volume 60, Issue 1, pp 315–339 | Cite as

QMR: a quasi-minimal residual method for non-Hermitian linear systems

  • Roland W. Freund
  • Noël M. Nachtigal
Article

Summary

The biconjugate gradient (BCG) method is the “natural” generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.

Mathematics Subject Classification (1991)

65F10 

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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Roland W. Freund
    • 1
    • 2
  • Noël M. Nachtigal
    • 3
  1. 1.RIACSNASA Ames Research CenterMoffett FieldUSA
  2. 2.Institut für Angewandte Mathematik und StatistikUniversität WürzburgWürzburgFederal Republic of Germany
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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