Generation of orthogonal rational functions by procedures for structured matrices

by   Niel Van Buggenhout, et al.

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.



There are no comments yet.


page 1

page 2

page 3

page 4


Backward Error of Matrix Rational Function

We consider a minimal realization of a rational matrix functions. We per...

On constructing orthogonal generalized doubly stochastic matrices

A real quadratic matrix is generalized doubly stochastic (g.d.s.) if all...

The Short-term Rational Lanczos Method and Applications

Rational Krylov subspaces have become a reference tool in dimension redu...

Fast approximation of orthogonal matrices and application to PCA

We study the problem of approximating orthogonal matrices so that their ...

A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix

In this paper we explore orthogonal systems in L_2(R) which give rise to...

Spectral analysis of a mixed method for linear elasticity

The purpose of this paper is to analyze a mixed method for linear elasti...

Twice is enough for dangerous eigenvalues

We analyze the stability of a class of eigensolvers that target interior...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.