Efficient two-step with memory methods and their dynamics

In this work,a fourth-order without-memory method is proposed that has a self-accelerator parameter.This method doesn’t need to compute a derivative function forsolving nonlinear equations.We have approximated the self-accelerator parameter andhave increased the convergence order to %50 without increase function evaluation.Theefficiency index of the with-memory method sixth-order is equal to 1.81712. Which ishigher than one-, two-, three-, and four-step optimal methods.The attraction basin ofthe proposed methods is compared by the famous Newton’s method and Kung-Traub’s method.