A dynamical comparison between iterative methods with memory: Are the derivatives good for the memory?

The role of the derivatives at the iterative expression of methods with memory for solving nonlinear equations is analyzed in this manuscript. To get this aim, a known class of methods without memory is transformed into different families involving or not derivatives with an only accelerating parameter, then they are defined as discrete dynamical systems and the stability of the fixed points of their rational operators on quadratic polynomials are studied by means of real multidimensional dynamical tools, showing in all cases similar results. Finally, a different approach holding the derivatives, and by using different accelerating parameters, in the iterative methods involved present the most stable results, showing that the role of the appropriated accelerating factors is the most relevant fact in the design of this kind of iterative methods.