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

Roland W. Freund | Zendy; No�l M. Nachtigal | Zendy

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

Author(s) -

Roland W. Freund,

No�l M. Nachtigal

Publication year - 1991

Publication title -

numerische mathematik

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.214

H-Index - 90

eISSN - 0945-3245

pISSN - 0029-599X

DOI - 10.1007/bf01385726

Subject(s) - biconjugate gradient method , hermitian matrix , mathematics , lanczos resampling , conjugate gradient method , residual , iterated function , generalization , biconjugate gradient stabilized method , lanczos algorithm , numerical analysis , linear system , positive definite matrix , algorithm , mathematical analysis , conjugate residual method , eigenvalues and eigenvectors , pure mathematics , computer science , gradient descent , physics , quantum mechanics , machine learning , artificial neural network

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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