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

Authors: Roland W. Freund,

Publish Date: 1994
Volume: , Issue:, Pages: 69-94
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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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