A new family of fourth-order optimal iterative schemes and remark on Kung and Traub's conjecture (Q2034931)

Summary: 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 \(p=2^{3-1} =4\).