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Relaxed Newton's Method as a Family of Root-finding Methods: Dynamics and Convergence

Soumen Pal

Abstract

Relaxed Newton's method is a one-parameter family of root-finding methods that generalizes the classical Newton's method. When viewed as a rational map on the Riemann sphere, this family exhibits rich and subtle global dynamics that depend both on the underlying polynomial and on the relaxation parameter. In this paper, we investigate the complex dynamical behavior of relaxed Newton maps associated with complex polynomials. We first characterize rational maps that arise as relaxed Newton maps in terms of the multipliers of their fixed points. Our main results identify several explicit classes of polynomials for which relaxed Newton's method is \textit{convergent} for all parameters $h$, in the sense that the Fatou set consists precisely of the basins of attraction of the roots. We further show that this favorable behavior does not hold in general: for any fixed relaxation parameter $h\in \{z:|z-1|<1\}$, there exists a generic cubic polynomial for which the relaxed Newton map fails to be convergent. Additional results include a complete characterization of when the Julia set is a straight line, an analysis of symmetry groups arising from rotational invariance, and sufficient conditions ensuring that all immediate basins of attraction are unbounded.

Relaxed Newton's Method as a Family of Root-finding Methods: Dynamics and Convergence

Abstract

Relaxed Newton's method is a one-parameter family of root-finding methods that generalizes the classical Newton's method. When viewed as a rational map on the Riemann sphere, this family exhibits rich and subtle global dynamics that depend both on the underlying polynomial and on the relaxation parameter. In this paper, we investigate the complex dynamical behavior of relaxed Newton maps associated with complex polynomials. We first characterize rational maps that arise as relaxed Newton maps in terms of the multipliers of their fixed points. Our main results identify several explicit classes of polynomials for which relaxed Newton's method is \textit{convergent} for all parameters , in the sense that the Fatou set consists precisely of the basins of attraction of the roots. We further show that this favorable behavior does not hold in general: for any fixed relaxation parameter , there exists a generic cubic polynomial for which the relaxed Newton map fails to be convergent. Additional results include a complete characterization of when the Julia set is a straight line, an analysis of symmetry groups arising from rotational invariance, and sufficient conditions ensuring that all immediate basins of attraction are unbounded.
Paper Structure (16 sections, 15 theorems, 38 equations, 3 figures)

This paper contains 16 sections, 15 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction and Main Results
  2. Background and Preliminary Discussions
  3. Relaxed Newton's Method: Formula, Properties and Characterization
  4. Fixed Points and Properties
  5. Characterization
  6. $h$-Convergent Classes: Proofs of Theorems \ref{['Conv_cls']} & \ref{['Cubic_nconv']}
  7. Polynomials having exactly two roots:
  8. $p$ is uncritical:
  9. $p(z)=(a z+b)^{m}\left[(a z+b)^{n}+c\right]$ where $a, b, c \in \mathbb{C}$, $a, c \neq 0$ and $m, n \in \mathbb{N}$ with $n\geq 2$:
  10. Proof of Theorem \ref{['Cubic_nconv']}
  11. Some Dynamical Remarks
  12. Declarations
  13. Funding
  14. Conflict of Interests
  15. Data Availability Statement
  16. ...and 1 more sections

Key Result

Theorem A

For each of the following classes of polynomials, relaxed Newton's method is $h$-convergent:

Figures (3)

  • Figure 1: Relaxed Newton's method applied to polynomials with exactly two roots: The Julia set is the boundary of any two differently colored regions.
  • Figure 2: Relaxed Newton's method applied to $z^3-1$ and $z(z^3-1)$: The Fatou set $\mathcal{F}(N_{h,p})$ is the union of basins corresponding to the roots of the respective polynomials
  • Figure 3: Non-convergent relaxed Newton's method: Red color represents the region where successive iterations of points do not converge to any of the roots of the polynomial

Theorems & Definitions (24)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1: Scaling property
  • proof
  • ...and 14 more