A modular framework for the backward error analysis of GMRES
Résumé
The Generalized Minimal Residual methods (GMRES) for the solution of general square linear systems is a class of Krylov-based iterative solvers for which there exist backward error analyses that guarantee the computed solution in inexact arithmetic to reach certain attainable accuracies. Unfortunately, these existing backward error analyses cover a relatively small subset of the possible GMRES variants and cannot be used straightforwardly in general to derive new backward error analyses for variants that do not yet have one. We propose a backward error analysis framework for GMRES that substantially simplifies the process of determining error bounds of most existing and future variants of GMRES. This framework describes modular bounds for the attainable normwise backward and forward errors of the computed solution that can be specialized for a given GMRES variant under minimal assumptions. To assess the relevance of our framework we first show that it is compatible with the previous rounding error analyses of GMRES in the sense that it delivers (almost) the same error bounds under (almost) the same conditions. Second, we explain how to use this framework to determine new error bounds for GMRES algorithms that do not have yet a backward error analysis, such as simpler GMRES, CGS2-GMRES, mixed precision GMRES, and more.
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