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Recent Advances in Iterative Methods pp 69–94Cite as

Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 60))

Abstract

Recently, Freund and Nachtigal proposed a novel conjugate gradient-type method, the quasi-minimal residual algorithm (QMR), for the iterative solution of general non-Hermitian systems of linear equations. The QMR method is based on the nonsymmetric Lanczos process, and thus, like the latter, QMR requires matrix-vector multiplications with both the coefficient matrix of the linear system and its transpose. However, in certain applications, the transpose is not readily available, and generally, it is desirable to trade in multiplications with the transpose for matrix-vector products with the original matrix.

This paper gives a survey of transpose-free algorithms that are based on the quasi-minimal residual approach. First, it is shown that, in principle, the transpose in the standard QMR method can always be eliminated by choosing special starting vectors. Examples are given for which this approach is practical. Then, two transpose-free QMR methods, the TFQMR algorithm and the QMR squared algorithm, for general non-Hermitian systems axe described. Some theory for ideal transpose-free QMR and TFQMR is presented. Results of numerical experiments are reported. Finally, some open problems are mentioned.

This research was performed while the author was in residence at the Research Institute for Advanced Computer Science (RIACS), NASA Ames Research Center, Moffett Field, California 94035, it was supported by Cooperative Agreement NCC 2-387 between the National Aeronautics and Space Administration and the Universities Space Research Association.

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Author information

Authors and Affiliations

  1. AT&T Bell Laboratories, Room 2C-420, 600 Mountain Avenue, Murray Hill, New Jersey, 07974-0636, USA

    Roland W. Freund

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  1. Roland W. Freund
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Editors and Affiliations

  1. Department of Computer Science, Stanford University, 94035, Stanford, CA, USA

    Gene Golub

  2. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, 55455, Minneapolis, MN, USA

    Mitchell Luskin

  3. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, NY, USA

    Anne Greenbaum

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© 1994 Springer-Verlag New York, Inc

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Freund, R.W. (1994). Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems. In: Golub, G., Luskin, M., Greenbaum, A. (eds) Recent Advances in Iterative Methods. The IMA Volumes in Mathematics and its Applications, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9353-5_6

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  • DOI: https://doi.org/10.1007/978-1-4613-9353-5_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9355-9

  • Online ISBN: 978-1-4613-9353-5

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