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

TL;DR

This paper explores the complex dynamics of relaxed Newton's methods as rational maps, identifying conditions for convergence and divergence, and analyzing their global behavior on the Riemann sphere.

Contribution

It characterizes when relaxed Newton maps are globally convergent and provides explicit classes of polynomials with guaranteed convergence for all parameters.

Findings

Certain polynomial classes ensure convergence for all relaxation parameters.

Generic cubic polynomials can exhibit divergence for specific relaxation parameters.

Complete characterization of when the Julia set is a straight line.

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.