Computing multiple zeros by using a parameter in Newton-Secant method

TL;DR

This paper introduces a modified third-order Newton-Secant method for efficiently finding multiple roots of nonlinear equations, requiring fewer evaluations per iteration and demonstrating superior convergence and dynamics compared to existing methods.

Contribution

The paper presents a new third-order convergence method for multiple roots, with an efficiency index of approximately 1.44225, and provides analysis and numerical comparisons.

Findings

The modified method achieves third-order convergence for multiple roots.

Numerical experiments show improved efficiency over existing methods.

Dynamics analysis demonstrates the robustness of the proposed method.

Abstract

In this paper, we modify the Newton-Secant method with third order of convergence for finding multiple roots of nonlinear equations. Per iteration this method requires two evaluations of the function and one evaluation of its first derivative. This method has the efficiency index equal to $3^{\frac{1}{3}}\approx 1.44225$. We describe the analysis of the proposed method along with numerical experiments including comparison with existing methods. Moreover, the dynamics of the proposed method are shown with some comparisons to the other existing methods.