{"title":"On Asymptotics of Eigenvalues of Seven-Diagonal Toeplitz Matrices","authors":"I. V. Voronin","doi":"10.1134/s0965542524700404","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size <span>\\(n\\)</span> as <span>\\(n \\to \\infty \\)</span> in the case when their symbol <span>\\(a(t)\\)</span> has the form <span>\\(a(t) = (t - 2{{a}_{0}} + {{t}^{{ - 1}}}{{)}^{3}}\\)</span>. This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case <span>\\({{a}_{0}} = 1\\)</span>. The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":"23 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mathematics and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0965542524700404","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size \(n\) as \(n \to \infty \) in the case when their symbol \(a(t)\) has the form \(a(t) = (t - 2{{a}_{0}} + {{t}^{{ - 1}}}{{)}^{3}}\). This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case \({{a}_{0}} = 1\). The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues.
期刊介绍：
Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.
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