On a New Three-Step Class of Methods and Its Acceleration for Nonlinear Equations

A class of derivative-free methods without memory for approximating a simple zero of a nonlinear equation is presented. The proposed class uses four function evaluations per iteration with convergence order eight. Therefore, it is an optimal three-step scheme without memory based on Kung-Traub conjecture. Moreover, the proposed class has an accelerator parameter with the property that it can increase the convergence rate from eight to twelve without any new functional evaluations. Thus, we construct a with memory method that increases considerably efficiency index from 8(1/4) ≈ 1.681 to 12(1/4) ≈ 1.861. Illustrations are also included to support the underlying theory.