Dynamics of Newton-like root finding methods

TL;DR

This paper investigates the consistent form of Newton-like root finding operators for quadratic polynomials, explores their symmetries for degree d polynomials, and analyzes their dynamical behavior to understand their properties.

Contribution

It provides a theoretical justification for the form of Newton-like operators, introduces a method to derive new algorithms, and studies their dynamical properties.

Findings

Operators have a universal form for quadratic polynomials

Symmetries explain the structure of Newton-like methods

Dynamical analysis reveals stability and convergence properties

Abstract

When exploring the literature, it can be observed that the operator obtained when applying \textit{Newton-like} root finding algorithms to the quadratic polynomials $z^2-c$ has the same form regardless of which algorithm has been used. In this paper we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree $d$ polynomials $p(z)=z^d-c$. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algoritms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods.