Computing parameter planes of iterative root-finding methods with several free critical points

TL;DR

The paper addresses the challenge of constructing parameter planes for families of iterative root-finding methods when multiple free critical points are present. It introduces an escaping-algorithm based approach that plots a single parameter plane while tracking all free critical points simultaneously, incorporating the symmetry $z\to 1/z$ and the palindromic structure of Newton-like operators to compute critical points efficiently. The authors demonstrate the method on several Newton-like families with two and three free critical points, deriving explicit expressions for critical points via palindromic polynomials and showing how simultaneous plotting resolves inconsistencies inherent in separate-plane analyses. The results provide a robust framework for understanding bifurcations in high-order root-finding methods and offer a practical tool for selecting methods with favorable dynamical behavior in complex settings.

Abstract

In this paper we present an algorithm to obtain the parameter planes of families of root-finding methods with several free critical points. The parameter planes show the joint behaviour of all critical points. This algorithm avoids the inconsistencies arising from the relationship between the different critical points as well as the indeterminacy caused by the square roots involved in their computation. We analyse the suitability of this algorithm by drawing the parameter planes of different Newton-like methods with two and three critical points. We also present some results of the expressions of the Newton-like operators and their derivatives in terms of palindromic polynomials, and we show how to obtain the expression of the critical points of a Newton-like method with real coefficients.

Figures (12)

• Figure 1: Upper figures show the parameter planes of the operator \ref{['op2']} using the critical points $c_1(a)$ (left) and $c_3(a)$ (right) separately. Lower figure shows the parameter plane obtained when using both critical points simultaneously.
• Figure 2: Dynamical planes of the operator \ref{['op2']} for different values of the parameter $a$.
• Figure 3: Stability regions for $z=1$, $z_1$ and $z_2$ under the operator \ref{['op3']}.
• Figure 4: Parameter planes of the operator \ref{['op3']} obtained using the critical points $c_1(a)$ (left), the critical point $c_3(a)$ (centre), and both critical points simultaneously (right).
• Figure 5: Parameter plane of the operator \ref{['op3']} obtained using the critical points $c_1$ and $c_3$. We use colour pink if only one free critical orbit converges to the roots, blue if the orbits od $c_1$ and $c_3$ converge to two different non-symmetric attracting cycles, and black if both free critical orbits converge to symmetric attracting cycles.

Theorems & Definitions (14)

• Definition 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Lemma 2.7
• proof
• Lemma 2.8