Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Ramandeep Behl,Majed Aali Alsulami,Sonia Bhalla,Eulalia Martínez

doi:10.3390/math9111242

Ramandeep Behl, Majed Aali Alsulami + Show 2 more

Open Access

https://doi.org/10.3390/math9111242

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Abstract

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

Highlights

The most demanding task of science and engineering problems [1,2,3] is to find the solutions of nonlinear equations
Along with the simple roots of nonlinear equations, multiple roots of nonlinear equations play a significant role in many areas such as the Ideal Gas Law [6], which describes the nature of a real gas and the relation between molecular size and attraction forces
We focused on the study of multiple roots of function f : D ⊂ C → C

Summary

Introduction

The most demanding task of science and engineering problems [1,2,3] is to find the solutions of nonlinear equations. The major drawback of these schemes is the computation of the first-order derivative at each step, which consumes much time To reduce this complexity, researchers [17,18,19,20,21,22] have worked on derivative-free schemes of multiple roots of scalar equations with the concept of the divided difference introduced by. Behl et al [6] proposed an optimal derivative-free Chebyshev–Halley family for multiple roots of a nonlinear equation. This scheme utilized three functional evaluations, one weighted function H (τ ), and one parameter α to obtain the fourth-order convergence.

Construction of the Higher Order Scheme

Numerical Experimentation

Concluding Remarks