A characterization of the dynamics of Schr\"oder's method for polynomials with two roots

TL;DR

This paper analyzes the dynamical behavior of Schr"oder's iterative method for solving complex polynomials with two roots, characterizing their basins of attraction and comparing them to Newton's method.

Contribution

It provides a topological characterization of the basins of attraction for Schr"oder's method applied to polynomials with two roots, including visual comparisons with Newton's method.

Findings

Basins are half-planes or circles depending on root multiplicities.

Topological conjugations used to characterize basins.

Graphical comparison of Schr"oder's and Newton's methods.

Abstract

The purpose of this work is to give a first approach to the dynamical behavior of Schr\"oder's method, a well known iterative process for solving nonlinear equations. In this context we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schr\"oder's method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We finish our study with a graphical gallery that allow us to compare the basins of attraction of Newton's and Schr\"ooder's method applied to some given polynomials. Key: Schr\"oder's method; basin of attraction; nonlinear equation.