In the paper we present a new result for evaluating the convergence rate of iterative Newton-type methods with respect to the number of iteration steps. We obtain an explicit asymptotically correct estimate that provides a fruitful basis for many practical situations. As examples of such applications, we solve four important problems arising in numerical integration of ordinary differential equations and index 1 semi-explicit differential-algebraic systems.

CEMAT - Center for Computational and Stochastic Mathematics