Optimal Steffensen-type methods with eighth order of convergence

This paper proposes two classes of three-step without memory iterations based on the well known second-order method of Steffensen. Per computing step, the methods from the developed classes reach the order of convergence eight using only four evaluations, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung–Traub for providing multi-point iterations without memory. As things develop, numerical examples are employed to support the underlying theory developed for the contributed classes of optimal Steffensen-type eighth-order methods.