# On the superlinear convergence of the successive approximations method

## Abstract

The Ostrowski theorem is a classical result which ensures the attraction of all the successive approximations xk+1 = G(xk) 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 q-convergence orders in terms of some spectral elements of G′(x*); we obtain that the set of trajectories with high q-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.

## Authors

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

fixed point; successive approximations; nonlinear system of equations in Rn; inexact Newton method; perturbed Newton method; linear systems of equation in Rn; residual; local convergence; q-convergence orders.

## Cite this paper as:

E. Cătinaş, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl., 113 (2002) no. 3, pp. 473-485
doi: 10.1023/A:1015304720071

Scanned paper.

0022-3239

1573-2878

0022-3239

1573-2878