Modified Potra-Pt\'ak method to determine the multiple zeros of nonlinear equations

TL;DR

This paper introduces a third-order iterative method based on Potra-Pták for efficiently finding multiple roots of nonlinear equations, with analysis, numerical experiments, and comparison to existing methods.

Contribution

It presents a novel third-order method requiring fewer function evaluations for multiple roots, improving efficiency over previous approaches.

Findings

The method achieves third-order convergence.

Numerical experiments confirm the method's efficiency.

Attraction basins are analyzed and compared.

Abstract

In this paper, we present a third-order iterative method based on Potra-Pt{\'a}k method to compute the approximate multiple roots of nonlinear equations. The method requires two evaluations of the function and one evaluation of its first derivative per iteration and it has the efficiency index equal to $3^{\frac{1}{3}}\approx 1.44225$. We describe the analysis of the proposed methods along with numerical experiments including comparison with existing methods. Moreover, the attraction basins are shown and compared with other existing methods.