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

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About this paper

Publisher Name

Kluwer Academic Publishers-Plenum Publishers

Print ISSN

0022-3239

Online ISSN

1573-2878

MR

0022-3239

Online ISSN

1573-2878

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