Thresholds of the Inner Steps in Multi-Step Newton Method

We investigate the efficiency of multi-step Newton method (the classical Newton method in which the first derivative is re-evaluated periodically after m steps) for solving nonlinear equations, F(x) = 0, F : D ⊆ Rn → Rn. We highlight the following property of multi-step Newton method with respect to some other Newton-type method: for a given n, there exist thresholds of m, that is an interval (mi, ms), such that for m inside of this interval, the efficiency index of multi-step Newton method is better than that of other Newton-type method. We also search for optimal values of m.