Keywords: nonlinear equation, iterative method, derivative-free, complex dynamics, parameters plane, fixed point, critical point, immersed basin of attraction.

The dynamic behaviour of the Newton's iterative method has been widely studied. However, the dynamics of derivative-free methods such as Steffensen's one are not as well known. In this paper, Steffensen's [1] and M4 [2] methods are analysed and compared, when they are applied to quadratic and cubic polynomials.

Several basics of complex dynamics are recalled: fixed and critical points, superattracting and repelling points, basins of attraction for attracting fixed points, Julia and Fatou sets and immediate basins of attraction. Furthermore, the immersed basins of attraction are defined.

In [3] the scaling theorem for Newton's method is introduced. For Steffensen's and M4 methods it is not possible to find a similar theorem, as a result of the derivative-free nature of both methods. So that, particular cases of quadratic and cubic polynomials are analysed.

Despite no dynamic behaviour can be generalised for the failure or the scaling theorem, the parameter space is a useful tool in order to check similar conducts on each case. When Steffensen's method is applied to quadratic polynomials, the parameter space is obtained. In this case, there are two critical points different to the fixed points.

The dynamic planes improve the understanding of the dynamic behaviour of the iterative methods. The basins of attraction of each superattracting fixed point and the number of iterations needed to reach them are shown for all the analysed instances. They can be discriminated by different colours and intensities, respectively.

The only superattracting finite fixed points are, in Steffensen's and M4 methods, the roots of the polynomial. Moreover, the infinity is a superattracting fixed point in Steffensen's methods. The M4 method has full convergence in the complex plane. As a result of the basin of the infinity, Steffensen's methods have more problems than the M4 ones to ensure the stability.

The fixed point operators satisfy a symmetry property, as shown in the dynamic planes. In real cases of the parameter, it is only necessary for the analysis of a complex semiplane to find the complete plane dynamics. In complex cases of the parameter, the dynamic plane of an operator has a symmetry with the conjugate operator.

1

J.M. Ortega, W.G. Rheinboldt, "Iterative solutions of nonlinear equations in several variables", New York, Academic Press, 1970.

2

A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, "A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations", Journal of Computational and Applied mathematics, 2012. doi:10.1016/j.cam.2012.03.030

3

J. Curry, L. Garnet, D. Sullivan, "On the iteration of a rational function: Computer experiments with Newton's method", Comm. Math. Phys., 91, 267-277, 1983. doi:10.1007/BF01211162

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