A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n   evaluations could achieve optimal convergence order 2n-12n-1. The new family of eighth-order methods agrees with the conjecture of Kung–Traub for the case n=4n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.