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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 appropri-
ate initial approximation of the root. we have solved a few nonlinear problems
to justify the theoretical study. Finally, are described the dynamics of the with-
memory method for complex polynomials of degree two.