Linearizations of rational matrices

Abstract

This 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.