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On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub. (English) Zbl 1509.30017

Summary: In this paper we study the dynamics of damped Traub’s methods \(T_\delta\) when applied to polynomials. The family of damped Traub’s methods consists of root finding algorithms which contain both Newton’s (\( \delta =0\)) and Traub’s method (\( \delta =1\)). 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 \(T_\delta\) to better understand the connection between Newton’s method and Traub’s method.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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