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Preconditioning for accurate solutions of ill‐conditioned linear systems
Author(s) -
Ye Qiang
Publication year - 2020
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2315
Subject(s) - preconditioner , linear system , mathematics , inverse , condition number , stability (learning theory) , algorithm , simple (philosophy) , computer science , mathematical analysis , eigenvalues and eigenvectors , machine learning , physics , geometry , quantum mechanics , philosophy , epistemology
Summary 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.