Development of Optimal Iterative Methods with Their Applications and Basins of Attraction

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