Stable polefinding and rational least-squares fitting via eigenvalues

Abstract

A common way of finding the poles of a meromorphic function f in a domain, where an explicit expression of f is unknown but f can be evaluated at any given z, is to interpolate f by a rational function such that at prescribed sample points , and then find the roots of q. This is a two-step process and the type of the rational interpolant needs to be specified by the user. Many other algorithms for polefinding and rational interpolation (or least-squares fitting) have been proposed, but their numerical stability has remained largely unexplored. In this work we describe an algorithm with the following three features: (1) it automatically finds an appropriate type for a rational approximant, thereby allowing the user to input just the function f, (2) it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values in a one-step fashion, and (3) it computes rational approximants in a numerically stable manner, in that with small at the sample points, making it the first rational interpolation (or approximation) algorithm with guaranteed numerical stability. Our algorithm executes an implicit change of polynomial basis by the QR factorization, and allows for oversampling combined with least-squares fitting. Through experiments we illustrate the resulting accuracy and stability, which can significantly outperform existing algorithms.

The AAA Algorithm for Rational Approximation

“…¿From an algorithmic point of view, problems with continuous sample sets suggest variants of AAA based on a discrete sample set Z that is enlarged as the computation proceeds. Such an approach is used in [43] and is also an option in the Chebfun aaa code, but we make no claims that success is guaranteed.…”

“…Rational approximation is an old subject, and many algorithms have been proposed over the years. Probably nobody has a complete view of this terrain, as the developments have occurred in diverse fields including often highly theoretical approximation theory [3,4,16,30,47,48,50,54,55,56,57,60,61], physics [6,29,31,32,33], systems and control [1,2,9,20,22,24,36,37,38,42,49,52], extrapolation of sequences and series [17,18,19], and numerical linear algebra [10,39,43]. The languages and emphases differ widely among the fields, with the systems and control literature, for example, giving particular attention to vector and matrix approximation problems and to approximation on the imaginary axis.…”

Generalized Maxwell projections for multi-mode network Photonics

“…(40). For this task, we used the stable pole extraction algorithm recently developed and detailed in 55 , which is mathematically exact for rational models and can automatically detect the order of the rational polynomial in the DOS from its singular matrix.…”

Efficient Reduction of Compressed Unitary Plus Low Rank Matrices to Hessenberg Form

“…Matrices of the form A = D + U V H , where D is a unitary n \times n block diagonal matrix and U, V \in \BbbC n\times k , k < n, arise commonly in the numerical treatment of structured (generalized) eigenvalue problems [1,2]. In particular any unitary plus low rank matrix can be reduced in this form by a similarity (unitary) transformation and additionally matrices of this form can be directly generated by linearization techniques based on interpolation schemes applied for the solution of nonlinear eigenvalue problems [6,7,8,10,19]. The class of unitary block upper Hessenberg matrices perturbed in the first block row or in the last block column includes block companion linearizations of matrix polynomials.…”