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

作     者：Changli LIU Ren-Cang LI 

作者机构：School of MathematicsSichuan UniversityChengdu 610065China Department of MathematicsUniversity of Texas at ArlingtonArlingtonTX 76019USA 

出 版 物：《Frontiers of Mathematics in China》 (中国高等学校学术文摘·数学（英文）)

年 卷 期：2018年第13卷第6期

页      面：1397-1426页

核心收录：

学科分类：07[理学] 

基　　金：supported in part by the International Visiting Program for Excellent Young Scholars of Sichuan University 国家自然科学基金 supported in part by the Natural Science Foundation 国家自然科学基金 

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

摘      要：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.