Backward error and conditioning of Fiedler linearizations

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Show simple item record Terán Vergara, Fernando de 2021-04-27T09:46:40Z 2021-04-27T09:46:40Z 2020-05-01
dc.identifier.bibliographicCitation Mathematics of Computation, (2020), 89(323), pp.: 1259-1300.
dc.identifier.issn 0025-5718
dc.description.abstract The standard way to solve polynomial eigenvalue problems is through linearizations. The family of Fiedler linearizations, which includes the classical Frobenius companion forms, presents many interesting properties from both the theoretical and the applied point of view. These properties make the Fiedler pencils a very attractive family of linearizations to be used in the solution of polynomial eigenvalue problems. However, their numerical features for general matrix polynomials had not yet been fully investigated. In this paper, we analyze the backward error of eigenpairs and the condition number of eigenvalues of Fiedler linearizations in the solution of polynomial eigenvalue problems. We get bounds for: (a) the ratio between the backward error of an eigenpair of the matrix polynomial and the backward error of the corresponding (computed) eigenpair of the linearization, and (b) the ratio between the condition number of an eigenvalue in the linearization and the condition number of the same eigenvalue in the matrix polynomial. A key quantity in these bounds is ρ, the ratio between the maximum norm of the coefficients of the polynomial and the minimum norm of the leading and trailing coefficient. If the matrix polynomial is well scaled (i. e., all its coefficients have a similar norm, which implies ρ ≈ 1), then solving the Polynomial Eigenvalue Problem with any Fiedler linearization will give a good performance from the point of view of backward error and conditioning. In the more general case of badly scaled matrix polynomials, dividing the coefficients of the polynomial by the maximum norm of its coefficients allows us to get better bounds. In particular, after this scaling, the ratio between the eigenvalue condition number in any two Fiedler linearizations is bounded by a quantity that depends only on the size and the degree of the polynomial. We also analyze the effect of parameter scaling in these linearizations, which improves significantly the backward error and conditioning in some cases where ρ is large. Several numerical experiments are provided to support our theoretical results.
dc.description.sponsorship This work was partially supported by the Ministerio de Ciencia e Innovación of Spain through grant MTM-2009-09281, and by the Ministerio de Economía y Competitividad of Spain through grants MTM-2012-32542, MTM2015-68805-REDT, and MTM2015-65798-P.
dc.format.extent 41
dc.language.iso eng
dc.publisher American Mathematical Society
dc.rights © Copyright 2019 American Mathematical Society
dc.rights Atribución-NoComercial-SinDerivadas 3.0 España
dc.subject.other Matrix polynomial
dc.subject.other Matrix pencil
dc.subject.other Eigenvalue
dc.subject.other Eigenvector
dc.subject.other Polynomial eigenvalue problem
dc.subject.other Companion linearization
dc.subject.other Fiedler Pencil
dc.subject.other Conditioning
dc.subject.other Backward error
dc.subject.other Scaling
dc.title Backward error and conditioning of Fiedler linearizations
dc.type article
dc.description.status Publicado
dc.subject.eciencia Matemáticas
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM-2009-09281
dc.relation.projectID Gobierno de España. MTM-2012-32542
dc.relation.projectID Gobierno de España. MTM2015-68805-REDT
dc.relation.projectID Gobierno de España. MTM2015-65798-P
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 1259
dc.identifier.publicationissue 323
dc.identifier.publicationlastpage 1300
dc.identifier.publicationtitle MATHEMATICS OF COMPUTATION
dc.identifier.publicationvolume 89
dc.identifier.uxxi AR/0000027178
dc.contributor.funder Ministerio de Economía y Competitividad (España)
dc.contributor.funder Ministerio de Ciencia e Innovación (España)
dc.affiliation.dpto UC3M. Departamento de Matemáticas
dc.affiliation.grupoinv UC3M. Grupo de Investigación: Matemática Aplicada a Control, Sistemas y Señales
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