Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides

TL;DR

This paper introduces a novel cross-interactive residual smoothing technique for global and block Lanczos-type solvers, significantly reducing residual gaps and oscillations in solving large sparse linear systems with multiple right-hand sides.

Contribution

It extends residual smoothing to multiple right-hand sides, improving accuracy and stability with minimal additional computational cost.

Findings

Reduces residual gap effectively in numerical experiments.

Maintains low additional cost compared to original methods.

Enhances stability of Krylov subspace methods for large systems.

Abstract

Global and block Krylov subspace methods are efficient iterative solvers for large sparse linear systems with multiple right-hand sides. However, global or block Lanczos-type solvers often exhibit large oscillations in the residual norms and may have a large residual gap relating to the loss of attainable accuracy of the approximations. Conventional residual smoothing schemes suppress these oscillations but cannot improve the attainable accuracy, whereas a recent residual smoothing scheme enables the improvement of the attainable accuracy for single right-hand side Lanczos-type solvers. The underlying concept of this scheme is that the primary and smoothed sequences of the approximations and residuals influence one another, thereby avoiding the severe propagation of rounding errors. In the present study, we extend this cross-interactive residual smoothing to the case of solving linear systems with multiple right-hand sides. The resulting smoothed methods can reduce the residual gap with a low additional cost compared to their original counterparts. We demonstrate the effectiveness of the proposed approach through rounding error analysis and numerical experiments.