{"title":"Method for verifying solutions of sparse linear systems with general coefficients","authors":"Takeshi Terao , Katsuhisa Ozaki","doi":"10.1016/j.amc.2024.129204","DOIUrl":null,"url":null,"abstract":"<div><div>This paper proposes a verification method for sparse linear systems <span><math><mi>A</mi><mi>x</mi><mo>=</mo><mi>b</mi></math></span> with general and nonsingular coefficient matrices. A verification method produces the error bound for a given approximate solution. Practical methods use one of two approaches. One approach is to verify the computed solution of the normal equation <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>A</mi><mi>x</mi><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>b</mi></math></span> by exploiting symmetric and positive definiteness; however, the condition number of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>A</mi></math></span> is the square of that for <em>A</em>. The other approach applies an approximate inverse of <em>A</em>; however, the approximate inverse of <em>A</em> may be dense even if <em>A</em> is sparse. Additionally, several other methods have been proposed; however, they are considered impractical due to various issues. Here, this paper provides a computing method for verified error bounds using the previous verification method and the latest equilibration. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, we will show the efficiency of an iterative refinement method to obtain accurate solutions.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"490 ","pages":"Article 129204"},"PeriodicalIF":3.5000,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0096300324006659","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper proposes a verification method for sparse linear systems with general and nonsingular coefficient matrices. A verification method produces the error bound for a given approximate solution. Practical methods use one of two approaches. One approach is to verify the computed solution of the normal equation by exploiting symmetric and positive definiteness; however, the condition number of is the square of that for A. The other approach applies an approximate inverse of A; however, the approximate inverse of A may be dense even if A is sparse. Additionally, several other methods have been proposed; however, they are considered impractical due to various issues. Here, this paper provides a computing method for verified error bounds using the previous verification method and the latest equilibration. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, we will show the efficiency of an iterative refinement method to obtain accurate solutions.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.