Structured eigenvalue condition numbers for parameterized quasiseparable matrices

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Show simple item record Martínez Dopico, Froilán César Pomés Portal, Kenet Jorge 2015-12-15T12:31:12Z 2016-11-17T23:00:08Z 2015-09-16
dc.identifier.bibliographicCitation Numerische Mathematik (2015) 16-November
dc.identifier.issn 0029-599X
dc.description.abstract The development of fast algorithms for performing computations with n x n low-rank structured matrices has been a very active area of research during the last two decades, as a consequence of the numerous applications where these matrices arise. The key ideas behind these fast algorithms are that low-rank structured matrices can be described in terms of O(n) parameters and that these algorithms operate on the parameters instead on the matrix entries. Therefore, the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters de ning these matrices may su er, since this determines the maximum accuracy of a given fast computation. In other words, it is necessary to develop condition numbers with respect to parameters for di erent magnitudes and classes of low-rank structured matrices, but, as far as we know, this has not yet been accomplished in any case. In this paper, we derive structured relative eigenvalue condition numbers for the important class of low-rank structured matrices known as {1;1}-quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the Givens-vector representations of these matrices, and we provide fast algorithms for computing them. Comparisons among the new structured condition numbers and the unstructured one are also presented, as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge. In addition, the approach presented in this paper is general and may be extended to other problems and classes of low-rank structured matrices.
dc.description.sponsorship This research was partially supported by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542.
dc.format.extent 29
dc.format.mimetype application/pdf
dc.language.iso eng
dc.publisher Springer
dc.rights © Springer-Verlag Berlin Heidelberg 2015
dc.subject.other Condition numbers
dc.subject.other simple eigenvalues
dc.subject.other low-rank structured matrices
dc.subject.other quasiseparable matrices
dc.subject.other quasiseparable representation
dc.subject.other Givens-vector representation.
dc.title Structured eigenvalue condition numbers for parameterized quasiseparable matrices
dc.type article
dc.description.status Publicado
dc.subject.eciencia Matemáticas
dc.identifier.doi 10.1007/s00211-015-0779-5
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM2012-32542
dc.type.version acceptedVersion
dc.identifier.publicationissue 16-November
dc.identifier.publicationtitle Numerische Mathematik
dc.identifier.uxxi AR/0000017460
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