Eighth-order methods with high efficiency index for solving nonlinear equations

In this paper, we construct two new families of eighth-order methods for solving simple roots of nonlinear equations by using weight function and interpolation methods. Per iteration in the present methods require three evaluations of the function and one evaluation of its first derivative, which implies that the efficiency indexes are 1.682. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2n−1. The new families of eighth-order methods agree with the conjecture of Kung-Traub for the case n = 4. Numerical comparisons are made with several other existing methods to show the performance of the presented methods, as shown in the illustration examples. Key–Words: Eighth-order convergence; Nonlinear equations; Weight function methods; Convergence order; Efficiency index