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Structured backward error analysis of linearized structured polynomial eigenvalue problems

Authors: Froilán M. Dopico, Javier Pérez and Paul Van Dooren
Journal: Math. Comp. 88 (2019), 1189-1228
MSC (2010): Primary 65F15, 15A18, 15A21, 15A22, 15A54, 93B18
Published electronically: June 14, 2018
MathSciNet review: 3904143
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Abstract: We start by introducing a new class of structured matrix polynomials, namely, the class of $ \mathbf {M}_A$-structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the classes of (skew-)symmetric, (anti-)palindromic, and alternating matrix polynomials. Then, we introduce the families of $ \mathbf {M}_A$-structured strong block minimal bases pencils and of $ \mathbf {M}_A$-structured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Pérez and Van Dooren, and show that any $ \mathbf {M}_A$-structured odd-degree matrix polynomial can be strongly linearized via an $ \mathbf {M}_A$-structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the $ \mathbf {M}_A$-structured framework allows us to perform a global and structured backward stability analysis of complete structured polynomial eigenproblems, regular or singular, solved by applying to a $ \mathbf {M}_A$-structured block Kronecker pencil a structurally backward stable algorithm that computes its complete eigenstructure, like the palindromic-QR algorithm or the structured versions of the staircase algorithm. This analysis allows us to identify those $ \mathbf {M}_A$-structured block Kronecker pencils that yield a computed complete eigenstructure which is the exact one of a slightly perturbed structured matrix polynomial. These pencils include (modulo permutations) the well-known block-tridiagonal and block-anti-tridiagonal structure-preserving linearizations. Our analysis incorporates structure to the recent (unstructured) backward error analysis performed for block Kronecker linearizations by Dopico, Lawrence, Pérez and Van Dooren, and share with it its key features, namely, it is a rigorous analysis valid for finite perturbations, i.e., it is not a first order analysis, it provides precise bounds, and it is valid simultaneously for a large class of structure-preserving strong linearizations.

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Additional Information

Froilán M. Dopico
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain

Javier Pérez
Affiliation: Department of Mathematical Sciences, University of Montana, Montana

Paul Van Dooren
Affiliation: Department of Mathematical Engineering, Université catholique de Louvain, Avenue Georges Lemaître 4, B-1348 Louvain-la-Neuve, Belgium

Keywords: Structured backward error analysis, complete polynomial eigenproblems, structured matrix polynomials, structure-preserving linearizations, M\"obius transformations, matrix pertubation theory, dual minimal bases
Received by editor(s): January 12, 2017
Received by editor(s) in revised form: July 25, 2017, January 10, 2018, and January 23, 2018
Published electronically: June 14, 2018
Additional Notes: The first author was supported by Ministerio de Economía, Industria y Competitividad of Spain and Fondo Europeo de Desarrollo Regional (FEDER) of EU through grants MTM-2015-68805-REDT, MTM-2015-65798-P (MINECO/FEDER, UE)
The second author was supported by KU Leuven Research Council grant OT/14/074.
The second and third authors were supported by the Belgian network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office
Article copyright: © Copyright 2018 American Mathematical Society