QMR: A Quasi-minimal Residual Method for Non-Hermitian Linear Systems

Authors

Roland W. Freund, Noel M. Nachtigal

Year

1991

Journal Paper

Algorithm

QMR

Source

Numerische Mathematik

Identifiers

DOI 10.1007/BF01385726

Abstract¶

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.

Summary¶

Introduces the Quasi-Minimal Residual (QMR) method, a BCG-like Krylov solver for non-Hermitian linear systems designed to avoid the instability and breakdown behavior of classical biconjugate gradients. The paper combines a look-ahead nonsymmetric Lanczos implementation with error analysis and numerical evidence, making QMR a standard reference point for robust nonsymmetric iterative solvers.