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Structured backward error for palindromic polynomial eigenvalue problems

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Abstract

A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEP)

$$ \left(\sum_{\ell=0}^d A_{\ell} \lambda^{\ell} \right)x=0, \quad A_{d-\ell}=\varepsilon A_{\ell}^{\star} \quad{\rm for}\,\ell=0,1,\ldots,\lfloor d/2\rfloor, $$

where \({\star}\) is one of the two actions: transpose and conjugate transpose, and \({\varepsilon\in\{\pm 1\}}\). Each of them has its application background with the case \({\star}\) taking transpose and ε = 1 attracting a great deal of attention lately because of its application in the fast train modeling. Computable formulas and bounds for the structured backward errors are obtained. The analysis reveals distinctive features of PPEP from general polynomial eigenvalue problems (PEP) investigated by Tisseur (Linear Algebra Appl 309:339–361, 2000) and by Liu and Wang (Appl Math Comput 165:405–417, 2005).

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, TX, 76019, USA

    Ren-Cang Li

  2. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Wen-Wei Lin

  3. Department of Mathematics, National Cheng Kung University, Tainan, 701, Taiwan

    Chern-Shuh Wang

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  1. Ren-Cang Li
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  2. Wen-Wei Lin
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Correspondence to Ren-Cang Li.

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Li, RC., Lin, WW. & Wang, CS. Structured backward error for palindromic polynomial eigenvalue problems. Numer. Math. 116, 95–122 (2010). https://doi.org/10.1007/s00211-010-0297-4

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  • DOI: https://doi.org/10.1007/s00211-010-0297-4

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