An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

TL;DR

This paper introduces a new three-point iterative method that achieves eighth-order convergence for solving nonlinear equations, offering high efficiency and improved basin of attraction compared to existing methods.

Contribution

It presents a novel eighth-order iterative method without memory that aligns with Kung and Traub's conjecture and enhances computational efficiency.

Findings

Achieves eighth-order convergence with four function evaluations per iteration.

Demonstrates superior accuracy and basin of attraction in numerical tests.

Supports theoretical convergence analysis and practical implementation.

Abstract

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction.