Preconditioning for accurate solutions of ill‐conditioned linear systems

DOI10.1002/NLA.2315zbMATH Open1463.65037arXiv1705.04340OpenAlexW3037029117MaRDI QIDQ5122243

Qiang Ye

Publication date: 22 September 2020

Published in: Numerical Linear Algebra with Applications (Search for Journal in Brave)

Abstract: This paper 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 higher accuracy that is achieved and an error analysis will be presented. As an application, we use the preconditioning approach to accurately compute a few smallest eigenvalues of certain ill-conditioned matrices. Numerical examples are presented to illustrate the error analysis and the performance of the methods.

Full work available at URL: https://arxiv.org/abs/1705.04340

zbMATH Keywords

preconditioningaccuracyerror analysisill-conditioned linear systems

Mathematics Subject Classification ID

Iterative numerical methods for linear systems (65F10) Preconditioners for iterative methods (65F08)

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