An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub 1974 conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2n−1. Thus, the family agrees with Kung-Traub conjecture for the case n 4. Computational results demonstrate that the developedmethods are efficient and robust as compared with many well-known methods.