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

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