Eighth-order Derivative-Free Family of Iterative Methods for Nonlinear Equations

In this note, we present an eighth-order derivative-free family of itera- tive methods for nonlinear equations. The proposed family shows optimal eight-order of convergence in the sense of the Kung and Traub conjecture (1) and is based on the Steensen derivative approximation used in the Newton-method. As a nal step, having in mind computational purposes, a derivative-free polynomial base interpo- lation is used in order to get optimal order of convergence with only four functional evaluations. Numerical esperiments and few issues are discussed at the end of this note.