Affine invariant conditions for the inexact perturbed Newton method


The high q-convergence orders of the inexact Newton iterates were characterized by Ypma in terms of some affine invariant conditions. Using these results, we obtain affine invariant characterizations for the q-convergence orders of the inexact perturbed Newton iterates.


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


inexact perturbed Newton methods; affine invariant conditions; q-convergence orders.

Cite this paper as:

E. Cătinaş, Affine invariant conditions for the inexact perturbed Newton method, Rev. Anal. Numér. Théor. Approx., 31 (2002) no. 1, pp. 17-20.


Scanned paper.

Latex-pdf version of the paper.

About this paper

Print ISSN


Online ISSN




Online ISSN


Google Scholar citations


Paper in html form


[1] I. Argyros and F. Szidarovsky, The Theory and Applications of Iteration Methods, C.R.C. Press Inc., Boca Raton, FL, 1993.

[2] E. Catinas, On the high convergence orders of the Newton-GMBACK methods, Rev.Anal. Numer. Th ́eor. Approx.,28(1999) no. 2, 125-132,

[3] E. Catinas, Newton and Newton-Krylov Methods for Solving Nonlinear Systems inRn, Ph.D. thesis, “Babes–Bolyai” University of Cluj-Napoca, Romania, 1999.

[4] E. Catinas, A note on the quadratic convergence of the inexact Newton methods, Rev.Anal. Numer. Theor. Approx.,29(2000) no. 2, 129-133,

[5] E. Catinas, Inexact perturbed Newton methods and applications to a class of Krylovsolvers, J. Optim. Theory Appl.,108(2001) no. 3, 543-570,

[6] E. Catinas, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl.,113(2002) no. 3, 473-485,

[7] E. Catinas, The inexact, inexact perturbed and quasi-Newton methods are equivalentmodels, Math. Comp.74(2005) no. 249, 291-301,

[8] Dembo, R. S., Eisenstat, S. C. and Steihaug, T., Inexact Newton methods, SIAMJ. Numer. Anal.,19(1982), pp. 400-408.

[9] Deuflhard, P.and Heindl, G., Affine invariant convergence theorems for Newton’smethod and extensions to related methods, SIAM J. Numer. Anal.,16(1979), pp. 1-10.

[10] Deuflhard, P.and Potra, F. A., Asymptotic mesh independence of Newton-Galerkin methods via a refined Mysovskii theorem, SIAM J. Numer. Anal.,29(1992), pp. 1395-1412.

[11] Ortega, J. M.and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

[12] Walker, H. F., An approach to continuation using Krylov subspace methods, Computational Science in the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J. L.Lions, J. Periaux and M. F. Wheeler, eds., John Wiley and Sons, Ltd., 1997, pp. 72-82.

[13] Ypma, T. J., Local convergence of inexact Newton methods, SIAM J. Numer. Anal.,21(1984), pp. 583-590.Received by the editors: October 3, 2001.

Related Posts