An optimal sixteenth order family of methods for solving nonlinear equation and their basins of attraction

Abstract

We propose a new family of iterative methods for finding the simple roots of nonlinear equation. The proposed method is four-point method with convergence order 16, which consists of four steps: the Newton step, an optional fourth order iteration scheme, an optional eighth order iteration scheme and the step constructed using the divided difference. By reason of the new iteration scheme requiring four function evaluations and one first derivative evaluation per iteration, the method satisfies the optimality criterion in the sense of Kung-Traub's conjecture and achieves a high efficiency index $16^{1/5} \approx 1.7411$. Computational results support theoretical analysis and confirm the efficiency.
The basins of attraction of the new presented algorithms are also compared to the existing methods with encouraging results.