Abstract
In this paper, the dynamical behavior of different optimal iterative schemes for solving nonlinear equations with increasing order, is studied. The tendency of the complexity of the Julia set is analyzed and referred to the fractal dimension. In fact, this fractal dimension can be shown to be a powerful tool to compare iterative schemes that estimate the solution of a nonlinear equation. Based on the box-counting algorithm, several iterative derivative-free methods of different convergence orders are compared.
Highlights
A large number of problems in science and engineering require the solution of a nonlinear equation f (z) = 0
The Newton’s method is a well-known iterative scheme to estimate the solution of nonlinear equations zn+1 = zn −
Yn − 2wn ) + c5 f [zn, yn ], and f [·, ·, ·, ·] and f [·, ·, ·, ·, ·] are the divided differences of order three and four, respectively. This procedure can be extended in order to obtain optimal derivative-free iterative schemes with convergence order 2k−1, k = 2, 3, 4, 5
Summary
The efficiency index, introduced by Ostrowski (see [1]), feeds back the comparison between methods in terms of efficiency It is defined as I = p1/d , where p is the order of convergence and d is the number of functional evaluations per step. We analyze the dynamical behavior of four derivative-free schemes, with orders of convergence 2, 4, 8 and 16, on different quadratic and cubic polynomials. From this analysis, some results can be conjectured. The Julia sets of the corresponding rational functions have less complexity and the basins of attraction obtained by the different schemes are wider, becoming more similar to that of Newton’s method, as the order of convergence increases. From a numerical point of view, these facts are checked by the fractal dimension of each procedure, that gets increasingly close to Newton’s one
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