Dynamics of a Family of Rational Operators of Arbitrary Degree

TL;DR

The paper analyzes the dynamics of a rational operator family arising from Behl's optimal fourth-order root-finding methods. It introduces a reparametrization to remove parameter redundancies and unboundedness, yielding the generalized family $O_{a,n,k}$ that encompasses other numerical-method-derived maps. Through fixed-point, critical-point, and parameter-plane analyses, it demonstrates how antenna-like parameter regions become bounded under reparametrization and proves the absence of Herman rings for the generalized family. The work provides concrete guidelines (e.g., $n>k+1$, $k\\ge 1$) to ensure numerically stable dynamics suitable for root-finding, with implications for the stability of related iterative methods.

Abstract

In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and unboundedness of problematic parameters. After reparametrization, we observe that these rational maps belong to a more general family $O_{a,n,k}$ of degree $n+k$ operators, which includes several other families of maps obtained from other numerical methods. We study the dynamics of $O_{a,n,k}$ and discuss for which parameters $n$ and $k$ these operators would be suitable from the numerical point of view.

Figures (6)

• Figure 1: Parameter plane of the operator $O_b$ (left) and a zoom in (right).
• Figure 2: Regions of stability of the strange fixed points for the operator $O_b$.
• Figure 3: Parameter plane of the operator $O_a$.
• Figure 4: Relation between the parameters $a$ and $b$ for $b$ real.
• Figure 5: Parameter planes of $O_{a,n,k}(z)$ for several values of $n$ and $k$.

Theorems & Definitions (24)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Lemma 6
• Lemma 7
• Proposition 8
• Proposition 9
• Proposition 10