A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations

TL;DR

This paper introduces a new class of optimal four-point iterative methods with convergence order 16 for solving nonlinear equations, using four function evaluations and one derivative per iteration, outperforming existing methods.

Contribution

The paper proposes a novel class of optimal four-point methods with high convergence order, validated through convergence analysis and numerical comparisons.

Findings

Achieves convergence order 16 with four function and one derivative evaluations.

Demonstrates superior accuracy and larger basins of attraction compared to existing methods.

Provides comprehensive numerical experiments confirming theoretical results.

Abstract

We introduce a new class of optimal iterative methods without memory for approximating a simple root of a given nonlinear equation. The proposed class uses four function evaluations and one first derivative evaluation per iteration and it is therefore optimal in the sense of Kung and Traub's conjecture. We present the construction, convergence analysis and numerical implementations, as well as comparisons of accuracy and basins of attraction between our method and existing optimal methods for several test problems.