Abstract
A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEPs)
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.
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References
Byers R, Mackey D S, Mehrmann V, Xu H. Symplectic, BVD, and palindromic approaches to discrete-time control problems. Technical Report, Preprint 14–2008, Institute of Mathematics, Technische Universität Berlin, 2008
Chu E K-W, Hwang T M, Lin W W, Wu C T. Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms. J Comput Appl Math, 2008, 219: 237–252
Guo C, Lin W W. Solving a structured quadratic eigenvalue problem by a structurepreserving doubling algorithm. SIAM J Matrix Anal Appl, 2010, 31(5): 2784–2801
Higham N J, Tisseur F, van Dooren P. Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems. Linear Algebra Appl, 2002, 351–352: 455–474
Hilliges A. Numerische Lösung von quadratischen Eigenwertproblemen mit Anwendungen in der Schiendynamik. Master’s Thesis, Technical University Berlin, Germany, July 2004
Hilliges A, Mehl C, Mehrmann V. On the solution of palindramic eigenvalue problems. In: Proceedings of 4th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Jyväskylä, Finland, 2004
Huang T M, Li T, Lin W W, Wu C T. Numerical studies on structure-preserving algorithms for surface acoustic wave simulations. J Comput Appl Math, 2013, 244(1): 140–154
Huang T M, Lin W W, Qian J. Numerically stable, structure-preserving algorithms for palindromic quadratic eigenvalue problems arising from vibration of fast trains. SIAM J Matrix Anal Appl, 2009, 30(4): 1566–1592
Huang T M, Lin W W, Su W S. Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree. Numer Math, 2011, 118(4): 713–735
Ipsen I C F. Accurate eigenvalues for fast trains. SIAM News, 2004, 37(9)
Li R C, Lin W W, Wang C S. Structured backward error for palindromic polynomial eigenvalue problems. Numer Math, 2010, 116(1): 95–122
Liu C, Li R C. Structured backward error for palindromic polynomial eigenvalue problems, ii: Approximate eigentriplets. Technical Report 2016–12, Dept Math, Univ of Texas at Arlington, December 2016. https://doi.org/www.uta.edu/math/preprint/
Liu X G, Wang Z X. A note on the backward errors for Hermite eigenvalue problems. Appl Math Comput, 2005, 165(2): 405–417
Lu L, Wang T, Kuo Y C, Li R C, Lin W W. A fast algorithm for fast train palindromic quadratic eigenvalue problems. SIAM J Sci Comput, 2016, 38(6): 3410–3429
Lu L, Yuan F, Li R C. A new look at the doubling algorithm for a structured palindromic quadratic eigenvalue problem. Numer Linear Algebra Appl, 2015, 22: 393–409
Mackey D S, Mackey N, Mehl C, Mehrmann V. Structured polynomial eigenvalue problems: Good vibrations from good linearizations. SIAM J Matrix Anal Appl, 2006, 28(4): 1029–1051
Schröder C. URV decomposition based structured methods for palindromic and even eigenvalue problems. Technical Report, Preprint 375, TU Berlin, Matheon, Germany, 2007
Schröder C. A QR-like algorithm for the palindromic eigenvalue problem. Technical Report, Preprint 388, TU Berlin, Matheon, Germany, 2007
Tisseur F. Backward error and condition of polynomial eigenvalue problems. Linear Algebra Appl, 2000, 309(1–3): 339–361
Xu H. On equivalence of pencils from discrete-time and continuous-time control. Linear Algebra Appl, 2006, 414: 97–124
Zaglmayr S, Schöberl J, Langer U. Eigenvalue problems in surface acoustic wave filter simulations. In: Bucchianico A, Mattheij R M M, Peletier M A, eds. Progress in Industrial Mathematics at ECMI 2004. Math Ind, Vol 8. Berlin: Springer, 2006, 74–98
Acknowledgements
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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Liu, C., Li, RC. Structured backward error for palindromic polynomial eigenvalue problems, II: Approximate eigentriplets. Front. Math. China 13, 1397–1426 (2018). https://doi.org/10.1007/s11464-018-0738-4
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DOI: https://doi.org/10.1007/s11464-018-0738-4