Structured backward error analysis of linearized structured polynomial eigenvalue problems

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Show simple item record Martínez Dopico, Froilán César Perez, Javier Van Dooren, Paul 2021-03-26T09:34:21Z 2021-03-26T09:34:21Z 2019
dc.identifier.bibliographicCitation Dopico, F. M., Pérez, J. & Van Dooren, P. (2018). Structured backward error analysis of linearized structured polynomial eigenvalue problems. Mathematics of Computation, 88(317), pp. 1189–1228.
dc.identifier.issn 0025-5718
dc.description.abstract We start by introducing a new class of structured matrix polynomials, namely, the class of 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 M-A-structured strong block minimal bases pencils and of M-A-structured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Perez and Van Dooren, and show that any M-A-structured odd-degree matrix polynomial can be strongly linearized via an M-A-structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the 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 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 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, Perez 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-pre
dc.format.extent 40
dc.language.iso eng
dc.publisher American Mathematical Society
dc.rights © 2018 American Mathematical Society
dc.rights Atribución-NoComercial-SinDerivadas 3.0 España
dc.subject.other Structured backward error analysis
dc.subject.other Complete polynomial eigenproblems
dc.subject.other Structured matrix polynomials
dc.subject.other Structure-preserving linearizations
dc.subject.other M\"Obius transformations
dc.subject.other Matrix pertubation theory
dc.subject.other Dual minimal bases
dc.title Structured backward error analysis of linearized structured polynomial eigenvalue problems
dc.type article
dc.subject.eciencia Matemáticas
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM2015-65798-P
dc.relation.projectID Gobierno de España. MTM2015-68805-REDT
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 1189
dc.identifier.publicationissue 317
dc.identifier.publicationlastpage 1228
dc.identifier.publicationtitle Mathematics of Computation
dc.identifier.publicationvolume 88
dc.identifier.uxxi AR/0000023047
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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