Preconditioning for accurate solutions of ill-conditioned linear systems

Qiang Ye

doi:10.1002/nla.2315

Preconditioning for accurate solutions of ill-conditioned linear systems

Qiang Ye

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

This article develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner M for an ill-conditioned linear system Ax=b, we show that, if the inverse of the preconditioner M⁻¹ can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse-equivalent accuracy is introduced to describe the high accuracy that is achieved and an error analysis will be presented. Numerical examples are presented to illustrate the error analysis and the performance of the methods.

Original language	English
Article number	e2315
Journal	Numerical Linear Algebra with Applications
Volume	27
Issue number	4
DOIs	https://doi.org/10.1002/nla.2315
State	Published - Aug 1 2020

Bibliographical note

Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.

Funding

The author would like to thank Prof. Jinchao Xu for some interesting discussions on multilevel preconditioners that have inspired this work. The author would also like to thank Kasey Bray for many helpful comments on a draft of this article and also like to thank anonymous referees for many detailed and insightful comments that have significantly improved the paper. This research was supported in part by NSF under Grants DMS‐1318633, DMS‐1620082, and DMS‐1821144. This work does not have any conflicts of interest. information NSF Division of Mathematical Sciences, 1318633; 1620082; 1821144The author would like to thank Prof. Jinchao Xu for some interesting discussions on multilevel preconditioners that have inspired this work. The author would also like to thank Kasey Bray for many helpful comments on a draft of this article and also like to thank anonymous referees for many detailed and insightful comments that have significantly improved the paper. This research was supported in part by NSF under Grants DMS-1318633, DMS-1620082, and DMS-1821144. This work does not have any conflicts of interest.

Funders	Funder number
National Science Foundation (NSF)	DMS‐1620082, 1821144, DMS‐1821144, DMS‐1318633
Division of Mathematical Sciences	1318633, 1620082

Keywords

accuracy
error analysis
ill-conditioned linear systems
preconditioning

ASJC Scopus subject areas

Algebra and Number Theory
Applied Mathematics

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10.1002/nla.2315

Cite this

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Preconditioning for accurate solutions of ill-conditioned linear systems

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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