## Publication: Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

 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 Martínez Dopico, Froilán César dc.contributor.author Pomés Portal, Kenet Jorge dc.contributor.funder Ministerio de Economía y Competitividad (España) es dc.date.accessioned 2021-04-05T08:39:06Z dc.date.available 2021-04-05T08:39:06Z dc.date.issued 2016-12 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. en dc.format.extent 28 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. en dc.identifier.doi https://doi.org/10.1007/s11075-016-0133-8 dc.identifier.issn 1017-1398 dc.identifier.publicationfirstpage 1131 dc.identifier.publicationissue 4 dc.identifier.publicationlastpage 1158 dc.identifier.publicationtitle Numerical Algorithms en dc.identifier.publicationvolume 73 dc.identifier.uri https://hdl.handle.net/10016/32252 dc.identifier.uxxi AR/0000018593 dc.language.iso eng en dc.publisher Springer Nature en dc.relation.projectID Gobierno de España. MTM2015-68805-REDT es dc.relation.projectID Gobierno de España. MTM2012-32542 es dc.relation.projectID Gobierno de España. MTM2015-65798-P es dc.rights © 2016, Springer Science Business Media New York en dc.rights.accessRights open access en dc.subject.eciencia Matemáticas es dc.subject.other Condition numbers en dc.subject.other Linear systems en dc.subject.other Low-rank structured matrices en dc.subject.other Quasiseparable matrices en dc.subject.other Quasiseparable representation en dc.subject.other Givens-vector representation en dc.subject.other Inversion algorithms en dc.title Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices en dc.type research article * dc.type.hasVersion AM * dspace.entity.type Publication
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