An eighth-order two-step with-memory adaptive method base on Hansen-Patrick’s method and its dynamic

Abstract

A tri-parametric family of two-point iterative methods with 7.5 and 8-order convergence for solving nonlinear equations has been proposed. Each derivative-free method member of the family requires only three evaluations of the given function per iteration. It is optimal in the sense of the Kung and Traub conjecture.
The proposed family has an efficiency index of 1.96 and 2. Numerical comparisons have been made to reveal the high efficiency of the developed method. The dynamical study of iterative schemes reflects a good overview of their stability, convergence properties, and graphical aspects by drawing attraction basins in the complex plane. Also, we have examined the dynamic
behavior of new methods to select the best weight function has the largest attraction basins for different polynomials.