A new family of optimal eighth order methods with dynamics for nonlinear equations

We propose a simple yet efficient family of three-point iterative methods for solving nonlinear equations. Each method of the family requires four evaluations, namely three functions and one derivative, per full iteration and possesses eighth order of convergence. Thus, the family is optimal in the sense of Kung–Traub conjecture and has the efficiency index 1.682 which is better than that of Newton’s and many other higher order methods. Various numerical examples are considered to check the performance and to verify the theoretical results. Computational results including the elapsed CPU-time, confirm the efficient and robust character of proposed technique. Moreover, the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature.