Some efficient seventh-order derivative-free families in root-finding (Q2842011)

The author proposes some 7th-order root-finding algorithms for solving nonlinear equations with simple roots. Moreover, the algorithms require only a few function evaluations per iteration and hence they can be used to solve nonlinear equations without computing the derivatives. Detail derivations of the methods and rigorious convergence analysis are presented. The techniques used are based on some existing theories by \textit{H. T. Kung} and \textit{J. F. Traub} [J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)]. Finally, some numerical examples are given to illustrate that the proposed methods are able to locate the simple roots.