Convergence and dynamics of a family of Secant-Like Methods with memory to solve nonlinear equations

Abstract

The without-memory methods based on the weight functions are considered,in this work. Also, by entering a self-accelerating parameter and approximatingit using the Secant-like techniques and Newton interpolation polynomials, withmemorymethods with convergence order 2+√5, 12 (5+√13), 12 (5+√17), 3+√5and 6 are proposed to solve nonlinear equations. It should be noted that the useof these new techniques to approximate the self-accelerator parameter does notincrease computational costs. Several examples are given to illustrate the efficiencyand performance of these new methods.We have investigated the basinsof attraction of the given weight functions from the proposed method. Hence,the second to fourth-degree polynomial equations for selecting the most appropriateones have been used.