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

TL;DR

This paper introduces an eighth-order derivative-free iterative family for solving nonlinear equations, achieving optimal convergence with minimal function evaluations using polynomial interpolation and Steffensen's derivative approximation.

Contribution

It presents a novel eighth-order derivative-free method that is optimal and efficient, based on polynomial interpolation and Steffensen's derivative approximation.

Findings

Achieves eighth-order convergence with four function evaluations.

Demonstrates efficiency through numerical experiments.

Maintains optimality in convergence order as per Kung and Traub conjecture.

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.