Frontiers of Mathematics in China

, Volume 13, Issue 6, pp 1397–1426 | Cite as

Structured backward error for palindromic polynomial eigenvalue problems, II: Approximate eigentriplets

  • Changli Liu
  • Ren-Cang LiEmail author
Research Article


A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEPs)
$$P(\lambda ) \equiv (\sum\limits_{\ell = 0}^d {{A_\ell }{\lambda ^\ell }} )x = 0,{A_{d - \ell }} = \varepsilon A_\ell ^*,\ell = 0,1...,\left\lfloor {\frac{d}{2}} \right\rfloor,$$
for an approximate eigentriplet is performed, where * is one of the two actions: transpose and conjugate transpose, and ε ∈ {±1} The analysis is concerned with estimating the smallest perturbation to P(λ); while preserving the respective palindromic structure, such that the given approximate eigentriplet is an exact eigentriplet of the perturbed PPEP. Previously, R. Li, W. Lin, and C. Wang [Numer. Math., 2010, 116(1): 95[122] had only considered the case of an approximate eigenpair for PPEP but commented that attempt for an approximate eigentriplet was unsuccessful. Indeed, the latter case is much more complicated. We provide computable upper bounds for the structured backward errors. Our main results in this paper are several informative and very sharp upper bounds that are capable of revealing distinctive features of PPEP from general polynomial eigenvalue problems (PEPs). In particular, they reveal the critical cases in which there is no structured backward perturbation such that the given approximate eigentriplet becomes an exact one of any perturbed PPEP, unless further additional conditions are imposed. These critical cases turn out to the same as those from the earlier studies on an approximate eigenpair.


Palindromic polynomial eigenvalue problem (PPEP) eigentriplet structured backward error error bound 


65F15 65G99 


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Changli Liu was supported in part by the International Visiting Program for Excellent Young Scholars of Sichuan University and the National Natural Science Foundation of China (Grant No. 11501388). Ren-Cang Li was supported in part by the Natural Science Foundation (Grants DMS-1317330, DMS-1719620, and CCF-1527104) and the Natural Science Foundation of China (Grant No. 11428104).


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© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSichuan UniversityChengduChina
  2. 2.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

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