Optimal eighth-order Steffensen type methods for solving nonlinear equations

Abstract

Based on Petković-Ilić-Džunić method [3], we derive a family of three-step eighth-order Steffensen type iterative methods for solving nonlinear equations. The new methods without memory use two suitable parametric functions at the second and third steps and are free from any derivatives. Per iteration the new methods require four functional evaluations, which implies that the efficiency index of the new methods is 1.682. The advantage of the new methods is that the computing speed of the new methods is faster than that of other eighth-order Steffensen type methods without memory. Numerical examples are made to show the performance of our methods and support the developed theory.