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.