On two families of high order Newton type methods

We study a general class of high order Newton type methods. The schemes consist of the application of several steps of Newton type methods with frozen derivatives. We are interested to improve the order of convergence in each sub-step. In particular, we should finish the computation after some stop criteria and before the full computation of the current approximation. We prove that only two sequences of parameters can be derived verifying these properties. One corresponds to a very well known family and the other is a little (but not natural) modification. Finally, we study some dynamical aspects of these families in order to find differences. Surprisingly, the less natural family seems to have a simpler dynamic.