Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics

TL;DR

This paper introduces two new high-order Newton-Secant like iterative methods for solving nonlinear equations, achieving optimal convergence rates with efficient function evaluations, supported by numerical experiments and complex plane analysis.

Contribution

The paper develops two novel optimal iterative methods with convergence orders four and eight, supporting the Kung and Traub conjecture and enhancing computational efficiency.

Findings

Methods achieve convergence orders four and eight.

Numerical experiments demonstrate superior performance.

Basins of attraction are analyzed in the complex plane.

Abstract

We construct two optimal Newton-Secant like iterative methods for solving non-linear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.