Some novel optimal eighth order derivative-free root solvers and their basins of attraction

We present two families of derivative-free methods with eighth order convergence for solving nonlinear equations. Each method of the families requires four function evaluations per full iteration, that means, the families are optimal in the sense of the hypothesis of Kung–Traub (1974). Computational results and comparison (including CPU time) with existing methods confirm the efficient and robust character of new methods. Moreover, the presented basins of attraction also confirm equal or better performance of the methods as compared to other established methods in literature.