Dynamics of Newton-like root finding methods

TL;DR

The paper addresses why Newton-like root-finding operators applied to quadratics share a universal rational form. It develops a symmetry-based framework using $\lambda^d$-odd/$\lambda^d$-even maps to derive a general operator for $p(z)=z^d-c$, then specializes to $p(z)=z^2-c$ to obtain a canonical operator $O(z)=z^n \prod_{i=1}^{k}\frac{z-r_i}{1-r_i z}$ and analyzes its fixed points and stability. Key contributions include a general symmetry theorem for Newton-like schemes and a detailed dynamical analysis that yields explicit stability regions (e.g., a disk in parameter space for $z=1$) and fixed-point structure, with extensive demonstrations across Chebyshev-Halley, King, Jarratt, and Ostrowski-Chun families. The findings provide a unified understanding of the dynamical behavior of Newton-like methods on quadratics and offer a constructive path to generate new Newton-type algorithms, aided by parameter-planes that reveal robust regions free of undesirable attractors.

Abstract

When exploring the literature, it can be observed that the operator obtained when applying \textit{Newton-like} root finding algorithms to the quadratic polynomials $z^2-c$ has the same form regardless of which algorithm has been used. In this paper we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree $d$ polynomials $p(z)=z^d-c$. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algoritms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods.

Figures (7)

• Figure 1: Parameter plane of the Chebyshev-Halley family, $n=3$, $k=1$.
• Figure 2: Parameter planes of King's family (left) and a family coming from Jarrat method (right). In both cases $n=4$, $k=2$.
• Figure 3: Parameter plane of the subfamily S2, $n=5$, $k=3$
• Figure 4: The subfamily S5. The parameter plane (top) and dynamical planes for $a=2-9.3i$ (bottom left) and $a=0$ (bottom right).
• Figure 5: Parameter plane of $c-$family for $n=k=3$ (left) and the M4 family for $\alpha=\frac{-1+5\beta}{\beta}$$n=k=4$ (right).

Theorems & Definitions (22)

• Definition 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• Proposition 2.6
• proof
• Theorem 2.7
• proof