{"title":"Computing eigenvalues for products of two classes of sign regular matrices to high relative accuracy","authors":"Xiaoxiao Ma , Yingqing Xiao , Zhao Yang","doi":"10.1016/j.laa.2024.11.006","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we consider how to accurately solve the product eigenvalue problem for the class of sign regular (SR) matrices with signature <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and the class of totally nonnegative (TN) matrices, which tend to be extremely ill-conditioned. We present algorithms with <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> complexity to accurately compute the parameter matrices of products of TN matrices and SR matrices with signature <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. Based on the accurate parameter matrices, all eigenvalues of the product matrix are computed to high relative accuracy. Numerical experiments are provided to confirm the claimed high relative accuracy.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 80-106"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524004269","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this paper, we consider how to accurately solve the product eigenvalue problem for the class of sign regular (SR) matrices with signature and the class of totally nonnegative (TN) matrices, which tend to be extremely ill-conditioned. We present algorithms with complexity to accurately compute the parameter matrices of products of TN matrices and SR matrices with signature . Based on the accurate parameter matrices, all eigenvalues of the product matrix are computed to high relative accuracy. Numerical experiments are provided to confirm the claimed high relative accuracy.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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