We find asymptotic error estimates for the method of simple iteration and for the modified and generalized Newton methods. The results, in contrast with the classical ones, provide explicit error estimates for these iterative processes in terms of their parameters, and this plays a decisive role not only for the proof of the convergence of combined methods, but also for determining the order of convergence. Moreover, in practice this allows us to theoretically evaluate the number of iterations sufficient for constructing the combined method of maximal order, and therefore to find the optimal number of iterations.

CEMAT - Center for Computational and Stochastic Mathematics