On an Aitken-Steffensen-Newton type method

We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We prove a local convergence result showing the -convergence order 7 of the iterations. Under certain supplementary conditions, we obtain monotone convergence of the iterations, providing an alternative to the usual ball attraction theorems.
Numerical examples show that this method may, in some cases, have larger (possibly sided) convergence domains than other methods with similar convergence orders.