The stability of split-preconditioned FGMRES in four precisions

TL;DR

This paper analyzes the stability of split-preconditioned FGMRES in a mixed precision setting, providing error bounds and guidance on choosing precisions to ensure numerical stability.

Contribution

It offers the first comprehensive error analysis of split-preconditioned FGMRES across four different precisions, applicable to general preconditioners.

Findings

Error bounds for split-preconditioned FGMRES in mixed precision

Guidelines for selecting precisions to maintain stability

Insights into the impact of different precisions on convergence

Abstract

We consider the split-preconditioned FGMRES method in a mixed precision framework, in which four potentially different precisions can be used for computations with the coefficient matrix, application of the left preconditioner, application of the right preconditioner, and the working precision. Our analysis is applicable to general preconditioners. We obtain bounds on the backward and forward errors in split-preconditioned FGMRES. Our analysis further provides insight into how the various precisions should be chosen; under certain assumptions, a suitable selection guarantees a backward error on the order of the working precision.