53 research outputs found
A framework for structured linearizations of matrix polynomials in various bases
We present a framework for the construction of linearizations for scalar and
matrix polynomials based on dual bases which, in the case of orthogonal
polynomials, can be described by the associated recurrence relations. The
framework provides an extension of the classical linearization theory for
polynomials expressed in non-monomial bases and allows to represent polynomials
expressed in product families, that is as a linear combination of elements of
the form , where and
can either be polynomial bases or polynomial families
which satisfy some mild assumptions. We show that this general construction can
be used for many different purposes. Among them, we show how to linearize sums
of polynomials and rational functions expressed in different bases. As an
example, this allows to look for intersections of functions interpolated on
different nodes without converting them to the same basis. We then provide some
constructions for structured linearizations for -even and
-palindromic matrix polynomials. The extensions of these constructions
to -odd and -antipalindromic of odd degree is discussed and
follows immediately from the previous results
On the stability of computing polynomial roots via confederate linearizations
A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Based on the analysis we suggest how to choose between QR and QZ. Finally, we focus on the special case of the Chebyshev basis and finding real roots of a general function on an interval, and discuss how to compute accurate roots. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary
Linearizations of rational matrices
Mención Internacional en el título de doctorThis PhD thesis belongs to the area of Numerical Linear Algebra. Specifically, to
the numerical solution of the Rational Eigenvalue Problem (REP). This is a type
of eigenvalue problem associated with rational matrices, which are matrices whose
entries are rational functions. REPs appear directly from applications or as approx imations to arbitrary Nonlinear Eigenvalue Problems (NLEPs). Rational matrices
also appear in linear systems and control theory, among other applications. Nowa days, a competitive method for solving REPs is via linearization. This is due to the
fact that there exist backward stable and efficient algorithms to solve the linearized
problem, which allows to recover the information of the original rational problem.
In particular, linearizations transform the REP into a generalized eigenvalue pro blem in such a way that the pole and zero information of the corresponding rational
matrix is preserved. To recover the pole and zero information of rational matrices, it
is fundamental the notion of polynomial system matrix, introduced by Rosenbrock
in 1970, and the fact that rational matrices can always be seen as transfer functions
of polynomial system matrices.
This thesis addresses different topics regarding the problem of linearizing REPs.
On the one hand, one of the main objectives has been to develop a theory of li nearizations of rational matrices to study the properties of the linearizations that
have appeared so far in the literature in a general framework. For this purpose,
a definition of local linearization of rational matrix is introduced, by developing as
starting point the extension of Rosenbrock’s minimal polynomial system matrices to
a local scenario. This new theory of local linearizations captures and explains rigor ously the properties of all the different linearizations that have been used from the
1970’s for computing zeros, poles and eigenvalues of rational matrices. In particu lar, this theory has been applied to a number of pencils that have appeared in some
influential papers on solving numerically NLEPs through rational approximation.
On the other hand, the work has focused on the construction of linearizations
of rational matrices taking into account different aspects. In some cases, we focus
on preserving particular structures of the corresponding rational matrix in the li nearization. The structures considered are symmetric (Hermitian), skew-symmetric
(skew-Hermitian), among others. In other cases, we focus on the direct construc tion of the linearizations from the original representation of the rational matrix.
The representations considered are rational matrices expressed as the sum of their
polynomial and strictly proper parts, rational matrices written as general trans fer function matrices, and rational matrices expressed by their Laurent expansion
around the point at infinity. In addition, we describe the recovery rules of the
information of the original rational matrix from the information of the new lineari zations, including in some cases not just the zero and pole information but also the
information about the minimal indices. Finally, in this dissertation we tackle one of the most important open problems
related to linearizations of rational matrices. That is the analysis of the backward
stability for solving REPs by running a backward stable algorithm on a linearization.
On this subject, a global backward error analysis has been developed by considering
the linearizations in the family of “block Kronecker linearizations”. An analysis of
this type had not been developed before in the literature.Este trabajo ha sido desarrollado en el Departamento de
Matemáticas de la Universidad Carlos III de Madrid (UC3M)
bajo la dirección del profesor Froilán Martínez Dopico y codirección de la profesora Silvia Marcaida Bengoechea. Se contó
durante cuatro años con un contrato predoctoral FPI, referencia BES-2016-076744, asociado al proyecto ALGEBRA LINEAL NUMERICA ESTRUCTURADA PARA MATRICES CONSTANTES, POLINOMIALES Y RACIONALES,
referencia MTM2015-65798-P, del Ministerio de Economía
y Competitividad, y cuyo investigador principal fue Froilán
Martínez Dopico. Asociado a este contrato, se contó con
una ayuda para realizar parte de este trabajo durante dos es tancias internacionales de investigación. La primera estancia
de investigación se realizó del 30 de enero de 2019 hasta el
1 de marzo de 2019 en el Department of Mathematical En gineering, Université catholique de Louvain (Bélgica), bajo
la supervisión del profesor Paul Van Dooren. La segunda
estancia de investigación se realizó del 15 de septiembre de
2019 hasta el 19 de noviembre de 2019 en el Department
of Mathematical Sciences, University of Montana (EEUU),
bajo la supervisión del profesor Javier Pérez Alvaro. Dado que la entidad beneficiaria del contrato predoctoral es la
UC3M mientras que el otro codirector de tesis, la profesora
Silvia Marcaida Bengoechea, pertenece al Departamento de
Matemáticas de la Universidad del País Vasco (UPV/EHU),
el trabajo con la profesora Silvia Marcaida se reforzó mediante visitas a la UPV/EHU, financiadas por ayudas de
la RED temática de Excelencia ALAMA (Algebra Lineal, Análisis Matricial y Aplicaciones) asociadas al los proyectos
MTM2015-68805-REDT y MTM2017-90682-REDT.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Ion Zaballa Tejada.- Secretario: Fernando de Terán Vergara.- Vocal: Vanni Noferin
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