structure a family of three-step with-memory methods for solving nonlinear equations and their dynamics

In this work, we will first propose an optimal three-step without-memory method for solving nonlinear equations. then, by introducing the self-accelerating parameters, the with-memory-methods have been built. they have a fifty-nine percentage improvement in the convergence order. the proposed methods have not the problems of calculating the function derivative. we use these steffensen-type methods to solve nonlinear equations with simple zeroes with the appropriate initial approximation of the root. we have solved a few nonlinear problems to justify the theoretical study and finally have described the dynamics of the with-memory method for complex polynomials of degree two.