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

Abstract

In this note, we present an eighth-order derivative-free family of iterative methods for nonlinear equations. The proposed family shows optimal eight-order of convergence in the sense of the Kung and Traub conjecture \cite{5} and is based on the Steffensen derivative approximation used in the Newton-method. As a final step, having in mind computational purposes, a derivative-free polynomial base interpolation 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.