No 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.

In this manuscript, we present a new class of highly efficient two-parameter optimal iterative methods for solving nonlinear systems that generalizes Ostrowski's method, King's Family, Chun's method and KLAM Family in multidimensional context. This class is an extension to systems of the Ermakov's Hyperfamily. The fourth order of convergence of the members of the class is demonstrated, thus obtaining optimal schemes for solving nonlinear systems. The high efficiency of the elements of the class is studied, compared with other known methods of the same order or even higher and some numerical proofs are presented and we analyze it robustness.