Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

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dc.contributor.author Martínez Dopico, Froilán César
dc.contributor.author Pomés Portal, Kenet Jorge
dc.date.accessioned 2021-04-05T08:39:06Z
dc.date.available 2021-04-05T08:39:06Z
dc.date.issued 2016-12
dc.identifier.bibliographicCitation Dopico, F. M. & Pomés, K. (2016). Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices. Numerical Algorithms, 73(4), pp. 1131–1158.
dc.identifier.issn 1017-1398
dc.identifier.uri http://hdl.handle.net/10016/32252
dc.description.abstract Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n) parameters, where nxn is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems, the solution of linear systems of equations is probably the most basic and relevant one. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. To develop and analyze such condition numbers is the main goal of this paper. To this purpose, a general expression is obtained for the condition number of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters. Since there are many different classes of low-rank structured matrices and many different types of parameters describing them, it is not possible to cover all of them in a single work. Therefore, the general expression of the condition number is particularized to the important case of {1,1}-quasiseparable matrices and to the quasiseparable and the Givens-vector representations, in order to obtain explicit expressions of the corresponding two condition numbers that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of these two condition numbers between themselves, and with respect to unstructured condition numbers, are provided, which show that there are situations in which the unstructured condition number is much larger than the structured ones, but that the opposite never happens. The approach presented in this manuscript can be generalized to other classes of low-rank structured matrices and parameterizations.
dc.format.extent 28
dc.language.iso eng
dc.publisher Springer Nature
dc.rights © 2016, Springer Science Business Media New York
dc.subject.other Condition numbers
dc.subject.other Linear systems
dc.subject.other Low-rank structured matrices
dc.subject.other Quasiseparable matrices
dc.subject.other Quasiseparable representation
dc.subject.other Givens-vector representation
dc.subject.other Inversion algorithms
dc.title Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices
dc.type research article
dc.subject.eciencia Matemáticas
dc.identifier.doi https://doi.org/10.1007/s11075-016-0133-8
dc.rights.accessRights open access
dc.relation.projectID Gobierno de España. MTM2015-68805-REDT
dc.relation.projectID Gobierno de España. MTM2012-32542
dc.relation.projectID Gobierno de España. MTM2015-65798-P
dc.identifier.publicationfirstpage 1131
dc.identifier.publicationissue 4
dc.identifier.publicationlastpage 1158
dc.identifier.publicationtitle Numerical Algorithms
dc.identifier.publicationvolume 73
dc.identifier.uxxi AR/0000018593
dc.contributor.funder Ministerio de Economía y Competitividad (España)
dc.affiliation.dpto UC3M. Departamento de Matemáticas
dc.affiliation.grupoinv UC3M. Grupo de Investigación: Matemática Aplicada a Control, Sistemas y Señales
dc.type.hasVersion AM
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