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

Abstract

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.

Design, Convergence and Stability Analysis of a New Parametric Class of Fourth-Order Optimal Iterative Schemes for Solving Nonlinear Equations

“…This is because in practice, it has been observed that iterative schemes that are stable for such functions tend to perform better compared to other methods that exhibit some pathology when applied to more complicated functions. To this end, complex discrete dynamics are employed to analyze stability in functions like quadratic polynomials (see, for example, the work of Amat et al in [1,2], Behl et al in [3], Chicharro et al in [9], Cordero et al in [15][16][17]), Khirallah et al [22], Moccari et al in [25], among others.…”