On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub

TL;DR

This work analyzes the dynamics of the damped Traub family $T_{p,\delta}$ for polynomial root finding, unifying Newton's method ($\delta=0$) and Traub's method ($\delta=1$). It derives local dynamical properties near roots, infinity, and critical points, showing simple roots yield superattracting fixed points for all $\delta$, while multiple roots become attracting under a $\delta$-dependent condition, and it provides a detailed critical-point census across parameter regimes. In the quadratic case, $T_{p,\delta}$ is conjugate to a Blaschke product $G_{\delta}$, leading to a 1D $\delta$-plane with a central hyperbolic component containing $\delta=1$ and unbounded, simply connected basins for $\delta\in[0,1]$, while Newton's method lies on the boundary. For $p_{n,\beta}(z)=z^n-\beta$ the authors extend these results, showing unbounded, simply connected basins for roots of unity and, via symmetry and conjugacy, similar properties for roots of $\beta$; the cubic case is analyzed in depth, revealing explicit critical-orbit behavior that supports a hyperbolic dynamical picture. Numerical evidence supports a conjecture that the immediate basins of roots under $T_{p,1}$ are simply connected and unbounded, and suggests the damped family as a useful avenue to construct universal initial-condition sets for all roots, paralleling the HowToNewton framework while exposing richer local convergence (cubic vs quadratic) properties. Overall, the paper contributes a rigorous dynamical-systems perspective on a broad class of root-finding algorithms and highlights the potential to design robust, globally convergent initial-condition schemes from the observed basin topology.

Abstract

In this paper we study the dynamics of damped Traub's methods $T_δ$ when applied to polynomials. The family of damped Traub's methods consists of root finding algorithms which contain both Newton's ($δ=0$) and Traub's method ($δ=1$). Our goal is to obtain several topological properties of the basins of attraction of the roots of a polynomial $p$ under $T_1$, which are used to determine a (universal) set of initial conditions for which convergence to all roots of $p$ can be guaranteed. We also numerically explore the global properties of the dynamical plane for $T_δ$ to better understand the connection between Newton's method and Traub's method.

Figures (6)

• Figure 1: In the left picture we illustrate the dynamical plane of Traub's method applied to the cubic polynomial $p(z)=(z^2+0.25)(z-0.439)$. In this case, $T_p$ has an attracting fixed point located at $\zeta \approx 0.155$. The basins of attraction of the three fixed points associated with the zeros of $p$ are shown in red. The basin of attraction of $\zeta$ is shown in black. In the right picture we illustrate the Newton's and Traub's maps restricted to $\mathbb R$. We can observe that $N_p|_{\mathbb{R}}$ has one attracting fixed point while $T_p|_{\mathbb{R}}$ has two attracting fixed points.
• Figure 2: Parameter plane of $G_{\delta}$ near $\delta=1$. The hyperbolic component $\mathcal{K}$ is the central red region and it is the hyperbolic component containing Traub's method ($\delta=1$). We also indicate the position of Newton's method ($\delta=0$) on the boundary of $\mathcal{K}$.
• Figure 3: Dynamical planes of Traub's method applied to the polynomial $p_n=z^n-1$.
• Figure 4: Graphic of $T_{p_3,\delta}(x)$ for $\delta=0.1$. We also draw the line $y=x$.
• Figure 5: Parameter plane of damped Traub's method applied to $p_3$.

Theorems & Definitions (24)

• Theorem 1.1
• Conjecture
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof