Block cross-interactive residual smoothing for Lanczos-type solvers for linear systems with multiple right-hand sides

TL;DR

This paper introduces a block cross-interactive residual smoothing (Bl-CIRS) method to improve convergence and accuracy of Lanczos-type solvers for large sparse linear systems with multiple right-hand sides, addressing residual oscillations and gaps.

Contribution

It extends the global CIRS scheme to a block version, providing theoretical analysis, numerical validation, and emphasizing the importance of orthonormalization in reducing residual gaps.

Findings

Bl-CIRS effectively reduces residual gaps in block Lanczos solvers.

Theoretical analysis confirms convergence improvements with Bl-CIRS.

Numerical experiments demonstrate enhanced accuracy and stability.

Abstract

Lanczos-type solvers for large sparse linear systems often exhibit large oscillations in the residual norms. In finite precision arithmetic, large oscillations increase the residual gap (the difference between the recursively updated residual and the explicitly computed residual) and a loss of attainable accuracy of the approximations. This issue is addressed using cross-interactive residual smoothing (CIRS). This approach improves convergence behavior and reduces the residual gap. Similar to how the standard Lanczos-type solvers have been extended to global and block versions for solving systems with multiple right-hand sides, CIRS can also be extended to these versions. While we have developed a global CIRS scheme (Gl-CIRS) in our previous study [K. Aihara, A. Imakura, and K. Morikuni, SIAM J. Matrix Anal. Appl., 43 (2022), pp.1308--1330], in this study, we propose a block version (Bl-CIRS). Subsequently, we demonstrate the effectiveness of Bl-CIRS from various perspectives, such as theoretical insights into the convergence behaviors of the residual and approximation norms, numerical experiments on model problems, and a detailed rounding error analysis for the residual gap. For Bl-CIRS, orthonormalizing the columns of direction matrices is crucial in effectively reducing the residual gap. This analysis also complements our previous study and evaluates the residual gap of the block Lanczos-type solvers.