{"title":"A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems","authors":"Dario Bini, L. Robol","doi":"10.1145/2631948.2631952","DOIUrl":null,"url":null,"abstract":"Many applications of the real world are modelled by matrix polynomials [EQUATION] where <i>A</i><sub><i>i</i></sub> are <i>m</i> x <i>m</i> matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of <i>P</i>(<i>x</i>), that is, the solutions of the polynomial equation det <i>P</i>(<i>x</i>) = 0. This task is generally accomplished by reducing <i>P</i>(<i>x</i>) to a linear pencil of the kind <i>xL</i> -- <i>K</i> for suitable matrices <i>K, L</i> of size <i>mn</i>, and to solving the eigenvalue problem (λ<i>L</i> -- <i>K</i>)<i>v</i> = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"37 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symbolic-Numeric Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2631948.2631952","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Many applications of the real world are modelled by matrix polynomials [EQUATION] where Ai are m x m matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of P(x), that is, the solutions of the polynomial equation det P(x) = 0. This task is generally accomplished by reducing P(x) to a linear pencil of the kind xL -- K for suitable matrices K, L of size mn, and to solving the eigenvalue problem (λL -- K)v = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.