Dynamics and Fractal Dimension of Steffensen-Type Methods

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.

Keywords

• nonlinear equation
• derivative-free
• dynamical plane
• fractal dimension
• Padé-like approximant