A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

Abstract

Kung and Traub conjectured that a multipoint iterative scheme without memory based on $m$ evaluations of functions has an optimal convergence order $p=2^{m-1}$ . In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be $p=2^{3-1}=4$ .