On the superlinear convergence of the successive approximations method

The Ostrowski theorem is a classical result which ensures the attraction of all the successive approximations x(k+1) = G(x(k)) near a fixed point x*. Different conditions [ultimately on the magnitude of G'(x*)] provide lower bounds for the convergence order of the process as a whole. In this paper, we consider only one such sequence and we characterize its high convergence orders in terms of some spectral elements of G'(x*); we obtain that the set of trajectories with high convergence orders is restricted to some affine subspaces, regardless of the nonlinearity of G. We analyze also the stability of the successive approximations under perturbation assumptions.