Stable polefinding and rational least-squares fitting via eigenvalues.

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 p q such that r ( γ i ) = f ( γ i ) at prescribed sample points { γ i } i = 1 L , 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 f ( γ i ) in a one-step fashion, and (3) it computes rational approximants p ^ , q ^ in a numerically stable manner, in that ( p ^ + Δ p ) / ( q ^ + Δ q ) = f with small Δ p , Δ q 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.