Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Livia J Quiñonez T, Carlos E Cadenas R

TL;DR

The paper addresses the complex dynamics of four root-finding method families formed by a convex weight between Newton's method and a Newton–Halley type step, specifically on quadratic polynomials with two roots. It constructs the associated rational iteration map, analyzes fixed and critical points as functions of the parameter $A$, and studies the stability of strange fixed points through conjugacy and scaling results, complemented by parameter-space and dynamical-plane visualizations. Key contributions include explicit expressions for fixed and critical points in terms of $A$, identification of stability regions around $A=\tfrac{42}{55}$ and $A=\tfrac{29}{35}$, and the demonstration of period-two orbits via dynamical planes and computations of six such orbits. The findings offer a structured framework for selecting well-behaved methods within this family for solving nonlinear equations on quadratics and reveal rich, conjugacy-consistent dynamical behavior, including universal Julia-set features under affine conjugacy.

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.

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

TL;DR

The paper addresses the complex dynamics of four root-finding method families formed by a convex weight between Newton's method and a Newton–Halley type step, specifically on quadratic polynomials with two roots. It constructs the associated rational iteration map, analyzes fixed and critical points as functions of the parameter , and studies the stability of strange fixed points through conjugacy and scaling results, complemented by parameter-space and dynamical-plane visualizations. Key contributions include explicit expressions for fixed and critical points in terms of , identification of stability regions around and , and the demonstration of period-two orbits via dynamical planes and computations of six such orbits. The findings offer a structured framework for selecting well-behaved methods within this family for solving nonlinear equations on quadratics and reveal rich, conjugacy-consistent dynamical behavior, including universal Julia-set features under affine conjugacy.

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.
Paper Structure (13 sections, 4 theorems, 33 equations, 10 figures, 1 table)

This paper contains 13 sections, 4 theorems, 33 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic preliminaries
  3. Dynamical behavior of the rational function associated with the family in study
  4. Conjugacy classes
  5. Newton's Method $(A=1)$
  6. Halley's Method $\left(A=0\right)$
  7. An interesting case $\left(A=\frac{2}{3}\right)$
  8. Super Halley method $\left(A=\frac{1}{2}\right)$
  9. Study of the fixed points
  10. Study of the critical points
  11. Study of parameter space
  12. Dynamical Planes
  13. Result and discussion

Key Result

Theorem 2.1

(The Scaling Theorem). Let $f(z)$ be an analytical function on the Riemann sphere, and let $T(z)=\alpha z+\beta$, $\alpha\neq0$, be an affine map. If $g(z)=f\circ T(z)$, then $T\circ R_g\circ T^{-1}=R_f(z)$. That is, $R_f$ is analytically conjugate to $R_g$ by $T$.

Figures (10)

  • Figure 1: Dynamical Behavior of strange fixed points for $-5 < A < 5$
  • Figure 2: Stability regions of the strange fixed point. Left: $z=1$. Right: $z=z_{2,3}$
  • Figure 3: Dynamical Behavior of critical points for $-1 < A < 5$
  • Figure 4: Parameter plane associated to the critical point $zc1$ and zoom
  • Figure 5: Various areas of the $zc1$-associated parameter space.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 2.1
  • Theorem 2.1
  • ...and 8 more