Dynamics of a Family of Rational Operators of Arbitrary Degree

TL;DR

This paper investigates the dynamics of a family of rational operators derived from root-finding algorithms, reparametrizes them for better analysis, and explores their suitability for numerical methods.

Contribution

It introduces a reparametrization of rational operators from root-finding algorithms and studies their dynamics within a broader family of operators.

Findings

Reparametrization prevents redundancies and unbounded parameters.

The operators belong to a general family $O_{a,n,k}$ of degree $n+k$.

Certain parameter choices make the operators more suitable for numerical methods.

Abstract

In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and unboundedness of problematic parameters. After reparametrization, we observe that these rational maps belong to a more general family $O_{a,n,k}$ of degree $n+k$ operators, which includes several other families of maps obtained from other numerical methods. We study the dynamics of $O_{a,n,k}$ and discuss for which parameters $n$ and $k$ these operators would be suitable from the numerical point of view.