Dynamical Analysis to Explain the Numerical Anomalies in the Family of Ermakov-Kalitlin Type Methods

Cordero, Alicia; Torregrosa, Juan R.; Vindel, P.

doi:10.3846/mma.2019.021

Cited by 2 publications

(2 citation statements)

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“…We denote scheme (10) as the Budzco x-family or PM. Cordero et al in [13] carried out an in-depth study on the dynamics of the Budzco family and determined the convergence properties that allow this method to have a stable dynamical behavior for parameter values close to 0. They called the accelerating factor of the second step,…”

Section: Introductionmentioning

confidence: 99%

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Fractal Complexity of a New Biparametric Family of Fourth Optimal Order Based on the Ermakov–Kalitkin Scheme

et al. 2023

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In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods.

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Section: Introductionmentioning

confidence: 99%

“…This uniparametric family of methods managed to improve Newton's, both in order of convergence [12] and in providing much larger basins of attraction. They showed that, for small values of the parameter, the basins of attraction that did not correspond to the roots of the polynomial were indeed small [13].…”

Section: Introductionmentioning

confidence: 99%