Boundedness and simple connectivity of the basins of attraction for some numerical methods

TL;DR

This paper investigates the geometric properties of basins of attraction for Halley's and Traub's root-finding methods applied to certain polynomials, revealing conditions for boundedness and connectivity of these basins.

Contribution

It provides new insights into the boundedness and simple connectivity of basins of attraction for specific numerical methods on polynomial roots.

Findings

Existence of polynomials with bounded immediate basins under Halley's method

Analysis of the connectivity of basins for different root-finding algorithms

Conditions under which basins of attraction are bounded or unbounded

Abstract

In this paper we study the dynamics of Halley's and Traub's root-finding algorithms applied to a symmetric family of polynomials of degree $d+1\geq 3$. We discuss the (un)boundedness and simple connectivity of the immediate basins of attraction of the fixed points associated to the roots of the polynomials. In particular, we show the existence of polynomials for which the immediate basin of attraction of a root is bounded under Halley's method