An optimal class of eighth-order iterative methods based on Kung and Traub's method with its dynamics

TL;DR

This paper introduces a new eighth-order iterative method for solving nonlinear equations, achieving optimal efficiency with four function evaluations per iteration, supported by convergence analysis and numerical comparisons.

Contribution

It presents a novel three-point, memoryless iterative method based on Kung and Traub's approach, achieving optimal eighth-order convergence with high computational efficiency.

Findings

Method achieves eighth-order convergence with four function evaluations.

Numerical tests demonstrate superior accuracy and convergence basins.

Method supports Kung-Traub conjecture on optimality.

Abstract

In this paper, we present a three-point without memory iterative method based on Kung and Traub's method for solving non-linear equations in one variable. The proposed method has eighth-order convergence and costs only four function evaluations each iteration which supports the Kung-Traub conjecture on the optimal order of convergence. Consequently, this method possesses very high computational efficiency. We present the construction, the convergence analysis, and the numerical implementation of the method. Furthermore, comparisons with some other existing optimal eighth-order methods concerning accuracy and basins of attraction for several test problems will be given.