Citation: Chanu, W.H.; Sunil, P.; Thangkhenpau, G. Development of Optimal Iterative Methods with Their Applications and Basins of Attraction. Symmetry 2022, 14, 2020. https:// doi.org/10.3390/sym14102020 Academic Editors: Juan Luis García Guirao and Alexander Zaslavski Received: 25 August 2022 Accepted: 19 September 2022 Published: 26 September 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Development of Optimal Iterative Methods with Their Applications and Basins of Attraction Waikhom Henarita Chanu *, Sunil Panday and G. Thangkhenpau Department of Mathematics, National Institute of Technology Manipur, Langol, Imphal 795004, India * Correspondence: henaritawai@gmail.com or waikhomhenarita@nitmanipur.ac.in Abstract: In this paper, we construct variants of Bawazir’s iterative methods for solving nonlinear equations having simple roots. The proposed methods are two-step and three-step methods, with and without memory. The Newton method, weight function and divided differences are used to develop the optimal fourth- and eighth-order without-memory methods while the methods with memory are derivative-free and use two accelerating parameters to increase the order of convergence without any additional function evaluations. The methods without memory satisfy the Kung–Traub conjecture. The convergence properties of the proposed methods are thoroughly investigated using the main theorems that demonstrate the convergence order. We demonstrate the convergence speed of the introduced methods as compared with existing methods by applying the methods to various nonlinear functions and engineering problems. Numerical comparisons specify that the proposed methods are efficient and give tough competition to some well known existing methods. Keywords: simple roots; nonlinear equation; iterative methods; error 1. Introduction Finding the roots of nonlinear equations is one of the most challenging problems in ap- plied mathematics, engineering and scientific computing. Analytical methods are generally ineffective for finding the roots of a nonlinear equation. Consequently, iterative methods are employed to obtain the approximate roots of nonlinear equations. Many iterative methods for solving nonlinear equations have been developed and studied. Among these, Newton’s method is one of the most widely used [1], which is defined as follows: x n+1 = x n − f ( x n ) f ′ ( x n ) , n = 0, 1, 2, 3, ... (1) Other well-known iterative approaches for solving nonlinear equations include the Chebyshev [2], Halley [2] and Ostrowski [3], methods. Most of the authors try to improve the order of convergence. As the order of convergence rises, so does the quantity of functional evaluations. As a result, iterative methods’ efficiency index falls. The efficiency index [2,3] of an iterative method determines the method’s efficiency, which is defined by the formula below: E = ρ 1 λ (2) where ρ is the order of convergence and λ is the number of functional evaluations per step. Kung–Traub conjectured [2] that the order of convergence of an iterative method without memory is at most 2 λ−1 . The optimal method is one in which the order of convergence is 2 λ−1 . In 2022, S. Panday et al. created optimal methods [4]. In 2015, M. Kumar et al. developed a fifth-order derivative-free method [5]. N, Choubey et al. introduced the derivative-free eighth-order method [6] in 2015. Y. Tao et al. developed optimal methods [7]. B. Neta also developed a derivative-free method [8]. M. Kumar Singh et al. developed the eighth-order optimal method in 2021 [9]. In 2021, O. Said Solaiman et al. [10] developed Symmetry 2022, 14, 2020. https://doi.org/10.3390/sym14102020 https://www.mdpi.com/journal/symmetry