Dynamics of four families of methods with the same weight function to solve nonlinear equations

TL;DR

This paper analyzes the dynamics of four families of iterative methods combining Newton and Newton-Halley approaches for solving polynomials with two roots, focusing on stability, fixed points, and parameter spaces.

Contribution

It provides analytical expressions for fixed and critical points and explores the stability and behavior of these methods through dynamic planes and parameter analysis.

Findings

Identification of stable and unstable fixed points.

Parameter spaces for methods with good convergence behavior.

Observation of periodic orbits with period two.

Abstract

We study the dynamics of four families of methods obtained with a weight function from a convex combination of Newton's method and a Newton-Halley type method on polynomials with two roots. We find the analytical expressions for the fixed and critical points. We study the stable and unstable behavior of the strange fixed points. Also, parameters spaces for identify methods with good behavior are presented. Then, several dynamic planes are presented to confirm the results obtained. Finally, some periodic orbits with period two for a selected method are presented.