## Abstract

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 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.

Latex-pdf version of the paper.

## 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

## Google Scholar citations

[1] Ortega, J. M., and Rheinboldt, W. C., *Iterative Solution of Nonlinear Equations in Several Variables*, Academic Press, New York, NY, 1970. 484 JOTA: VOL. 113, NO. 3, JUNE 2002

[2] Potra, F. A., *On Q-Order and R-Order of Convergence*, Journal of Optimization Theory and Applications, Vol. 63, pp. 415–431, 1989.

[3] Rheinboldt, W. C., *Methods for Solûing Systems of Nonlinear Equations*, SIAM, Philadelphia, Pennsylvania, 1998.

[4] Potra, F. A., *Q-Superlinear Convergence of the Iterates in Primal–Dual InteriorPoint Methods*, Mathematical Programming (to appear).

[5] Ostrowski, A. M., *Solution of Equations and Systems of Equations*, Academic Press, New York, NY, 1966.

[6] Householder, A. S., *The Theory of Matrices in Numerical Analysis*, Dover, New York, NY, 1974.

[7] Argyros, I., and Szidarovszky, F., *The Theory and Applications of Iteration Methods*, CRC Press, Boca Raton, Florida, 1993.

[8] Argyros, I., *On the Convergence of the Modified Contractions*, Journal of Computational and Applied Mathematics, Vol. 55, pp. 183–189, 1994.

[9] Brown, P. N., *A Local Convergence Theory for Combined Inexact-Newton Finite-Difference Projection Methods*, SIAM Journal on Numerical Analysis, Vol. 24, pp. 407–434, 1987.

[10] Brown, P. N., and Saad, Y., *Convergence Theory of Nonlinear Newton–Krylov Algorithms*, SIAM Journal on Optimization, Vol. 4, pp. 297–330, 1994.

[11] Catinas, E., *Newton and Newton–Krylov Methods for Solving Nonlinear Systems in n , PhD Thesis*, Babes–Bolyai University of Cluj–Napoca, Cluj– Napoca, Romania, 1999.

[12] Catinas, E., *On the High Convergence Orders of the Newton–GMBACK Methods*, Revue d’Analyse Numerique et de Theorie de l’Approximation, Vol. 28, pp. 125–132, 1999.

[13] Catinas, E., *A Note on the Quadratic Convergence of the Inexact Newton Methods*, Revue d’Analyse Numerique et de Theorie de l’Approximation, Vol. 29, pp. 129–134, 2000.

[14] Catinas, E., *Inexact Perturbed Newton Methods and Applications to a Class of Krylov Solvers*, Journal of Optimization Theory and Applications, Vol. 108, pp. 543–570, 2001.

[15] Catinas, E., *The Inexact, Inexact Perturbed, and Quasi-Newton Methods are Equivalent Models*, Mathematics of Computation (to appear).

[16] Dembo, R. S., Eisenstat, S. C., and Steihaug, T*., Inexact Newton Methods*, SIAM Journal on Numerical Analysis, Vol. 19, pp. 400–408, 1982.

[17] DENNIS, J. E., JR., and MORE´ , J. J., *A Characterization of Superlinear Convergence and Its Application to Quasi-Newton Methods*, Mathematics of Computation, Vol. 28, pp. 549–560, 1974.

[18] Dennis, J. E., JR., and More , J. J., *Quasi-Newton Methods*, Motivation and Theory, SIAM Review, Vol. 19, pp. 46–89, 1977.

[19] Dennis, J. E., JR., and Schnabel, R. B., *Numerical Methods for Unconstrained Optimization and Nonlinear Equations*, Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.

[20] Deuflhard, P., and Potra, F. A., *Asymptotic Mesh Independence of Newton– Galerkin Methods via a Refined Mysovskii Theorem*, SIAM Journal on Numerical Analysis, Vol. 29, pp. 1395–1412, 1992.