The main effects of rounding errors in Krylov solvers for symmetric linear systems

Abstract

The 3-term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of linear systems, by solving the reduced system in one way or another. This leads to well-known methods: MINRES (GMRES), CG, CR, and SYMMLQ. We will discuss in what way and to what extent the various approaches are sensitive to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the dierent methods (except CR), and we will not consider the errors in the Lanczos process itself. These errors may lead to large perturbations with espect to the exact process, but convergence takes still place. Our attention is focussed to what happens in the solution phase. We will show that the way of solution may lead, under circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our ndings are supported and illustrated by numerical examples.