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Cross-Interactive Residual Smoothing for Global and Block Lanczos-Type Solvers for Linear Systems with Multiple Right-Hand Sides

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.

Keywords

  1. multiple right-hand sides
  2. global Lanczos-type solver
  3. block Lanczos-type solver
  4. residual smoothing
  5. residual gap

MSC codes

  1. 65F10
  2. 65F45

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Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1308 - 1330
ISSN (online): 1095-7162

History

Submitted: 27 July 2021
Accepted: 27 May 2022
Published online: 8 August 2022

Keywords

  1. multiple right-hand sides
  2. global Lanczos-type solver
  3. block Lanczos-type solver
  4. residual smoothing
  5. residual gap

MSC codes

  1. 65F10
  2. 65F45

Authors

Affiliations

Funding Information

Japan Society for the Promotion of Science https://doi.org/10.13039/501100001691 : JP16K17639, JP17K12690, JP18H03250, JP18K18064, JP19KK0255, JP20K14356, JP21H03451, JP21K11925

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