Backward error and conditioning of Fiedler linearizations

Terán Vergara, Fernando de

Publication:
Backward error and conditioning of Fiedler linearizations

dc.affiliation.dpto	UC3M. Departamento de Matemáticas	es
dc.affiliation.grupoinv	UC3M. Grupo de Investigación: Matemática Aplicada a Control, Sistemas y Señales	es
dc.contributor.author	Terán Vergara, Fernando de
dc.contributor.funder	Ministerio de Economía y Competitividad (España)	es
dc.contributor.funder	Ministerio de Ciencia e Innovación (España)	es
dc.date.accessioned	2021-04-27T09:46:40Z
dc.date.available	2021-04-27T09:46:40Z
dc.date.issued	2020-05-01
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.	en
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.	en
dc.description.status	Publicado	es
dc.format.extent	41
dc.identifier.bibliographicCitation	Mathematics of Computation, (2020), 89(323), pp.: 1259-1300.	en
dc.identifier.doi	https://doi.org/10.1090/mcom/3480
dc.identifier.issn	0025-5718
dc.identifier.publicationfirstpage	1259
dc.identifier.publicationissue	323
dc.identifier.publicationlastpage	1300
dc.identifier.publicationtitle	MATHEMATICS OF COMPUTATION	en
dc.identifier.publicationvolume	89
dc.identifier.uri	https://hdl.handle.net/10016/32491
dc.identifier.uxxi	AR/0000027178
dc.language.iso	eng	en
dc.publisher	American Mathematical Society	en
dc.relation.projectID	Gobierno de España. MTM-2009-09281	es
dc.relation.projectID	Gobierno de España. MTM-2012-32542	es
dc.relation.projectID	Gobierno de España. MTM2015-68805-REDT	es
dc.relation.projectID	Gobierno de España. MTM2015-65798-P	es
dc.rights	© Copyright 2019 American Mathematical Society	en
dc.rights	Atribución-NoComercial-SinDerivadas 3.0 España	*
dc.rights.accessRights	open access	en
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/es/	*
dc.subject.eciencia	Matemáticas	es
dc.subject.other	Matrix polynomial	en
dc.subject.other	Matrix pencil	en
dc.subject.other	Eigenvalue	en
dc.subject.other	Eigenvector	en
dc.subject.other	Polynomial eigenvalue problem	en
dc.subject.other	Companion linearization	en
dc.subject.other	Fiedler Pencil	en
dc.subject.other	Conditioning	en
dc.subject.other	Backward error	en
dc.subject.other	Scaling	en
dc.title	Backward error and conditioning of Fiedler linearizations	en
dc.type	research article	*
dc.type.hasVersion	AM	*
dspace.entity.type	Publication