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

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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.

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Author notes

  1. Roland W. Freund

    Present address: RIACS, NASA Ames Research Center, Mail Stop Ellis Street, 94035, Moffett Field, CA, USA

Authors and Affiliations

  1. Institut für Angewandte Mathematik und Statistik, Universität Würzburg, W-8700, Würzburg, Federal Republic of Germany

    Roland W. Freund

  2. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Noël M. Nachtigal

Authors
  1. Roland W. Freund
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  2. Noël M. Nachtigal
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This work was supported in part by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).

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Freund, R.W., Nachtigal, N.M. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991). https://doi.org/10.1007/BF01385726

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  • DOI: https://doi.org/10.1007/BF01385726

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