Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

[EN] We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.5156(1/4) approximate to 1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under Grant MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089, and Schlumberger Foundation, Faculty for the Future Program.Cordero Barbero, A.; Junjua, M.; Torregrosa Sánchez, JR.; Yasmin, N.; Zafar, F. (2018). Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices. Mathematical Problems in Engineering. 1-12. https://doi.org/10.1155/2018/809367311