TransposeFree QuasiMinimal Residual Methods for

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 gradienttype method the quasiminimal residual algorithm QMR for the iterative solution of general nonHermitian systems of linear equations The QMR method is based on the nonsymmetric Lanczos process and thus like the latter QMR requires matrixvector 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 matrixvector products with the original matrixThis paper gives a survey of transposefree algorithms that are based on the quasiminimal 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 transposefree QMR methods the TFQMR algorithm and the QMR squared algorithm for general nonHermitian systems axe described Some theory for ideal transposefree QMR and TFQMR is presented Results of numerical experiments are reported Finally some open problems are mentionedThis 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 2387 between the National Aeronautics and Space Administration and the Universities Space Research Association

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