On rational functions without Froissart doublets

TL;DR

This paper investigates methods to detect and prevent Froissart doublets in rational functions used in numerical modeling, introducing three parameters that improve upon previous techniques for ensuring function stability.

Contribution

It presents three novel parameters for monitoring Froissart doublets in rational functions, enhancing previous methods with sharper criteria.

Findings

The Euclidean condition number of a Sylvester matrix helps detect Froissart doublets.

Parameters for coprimeness of polynomials are effective in avoiding doublets.

Bounds on the spherical derivative improve stability analysis of rational functions.

Abstract

In this paper we consider the problem of working with rational functions in a numeric environment. A particular problem when modeling with such functions is the existence of Froissart doublets, where a zero is close to a pole. We discuss three different parameters which allow one to monitor the absence of Froissart doublets for a given general rational function. These include the euclidean condition number of an underlying Sylvester-type matrix, a parameter for determing coprimeness of two numerical polynomials and bounds on the spherical derivative. We show that our parameters sharpen those found in a previous paper by two of the autours.