A PhD thesis @ NALAG

TW 2019_02

Matthias Humet Cienfuegos Jovellan
Orthogonal polynomials and matrix computations, Applications of recurrence relations
April 30, 2019

Advisor(s): Marc Van Barel

Abstract

Univariate and multivariate polynomials play a fundamental role in pure and applied mathematics. In this text orthogonal polynomials and their corresponding recurrence relations are studied and connected to matrix computations and applied in polynomial interpolation and approximation and in polynomial system solving. For univariate orthogonal polynomials on the complex unit circle, efficient algorithms for the Geronimus transformation are designed and their accuracy is discussed based on several numerical experiments. The Uvarov transformation depends on a parameter m. The values of m for which the Uvarov transformation maintains the positive definiteness of the original bilinear functional are determined. Exploiting the correspondence between J-unitary Hessenberg matrices and quasi-definite functionals on the unit circle, a QR algorithm is developed for this type of matrices. For multivariate orthogonal polynomials with respect to a discrete inner product, a framework is presented that connects monomial orders, the structure of the corresponding recurrence matrices, the discrete inner product and the related inverse eigenvalue problem. Results are given concerning the algorithms to solve this inverse eigenvalue problem, the uniqueness of the solution and the multivariate vector monomial orders that can be used. As an application several algorithms are designed to compute point configurations for multivariate polynomial interpolation and least squares approximation that have a low or even almost minimal Lebesgue constant for a given geometry. The polynomial functions are represented using orthogonal bases with respect to a discrete inner product where the mass points are lying within the considered geometry. As another application, a generalized companion method to solve systems of polynomials is developed. The matrices of the generalized eigenvalue problem are sparse and structured and they contain the coefficients of recurrence relations of the multivariate basis polynomials and the coefficients of the system of polynomial equations.