Abstract

Given a nonlinear mapping \(G:D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}^n\) differentiable at a fixed point \(x^\ast\), the Ostrowski theorem offers the sharp sufficient condition \[\rho(G′(x^\ast))<1\] for \(x^\ast\) to be an attraction point, where \(\rho\) denotes the spectral radius. However, no estimate for the size of an attraction ball is known.

We show in this note that such an estimate may be readily obtained in terms of \(\|G^\prime(x^\ast)\|<1\) (with \(\|\cdot \|\) an arbitrary given norm) and of the Hölder (in particular Lipschitz) continuity constant of \(G’\).

An elementary example shows that this estimate may be sharp. Our assumptions do not necessarily require \(G\)  to be of contractive type on the whole estimated ball.

Authors

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

Keywords

fixed point; attraction points; attraction ball; local convergence; convergence order.

Cite this paper as:

E. Cătinaş, Estimating the radius of an attraction ball, Appl. Math. Lett., 22 (2009) no. 5, pp. 712-714.

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Publisher Name

Elsevier

Print ISSN

0893-9659

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References

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

[2] A. M. Ostrowski, Solution of Equations and Systems of Equations, 2nd ed., Academic Press, New York (1966).

[3] W. C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, 2nd ed., SIAM, Philadelphia (1998).

[4] E. Zeidler, Nonlinear Functional Analysis and Its Applications, I. Fixed Point Theorems, Springer-Verlag, New York (1986).

[5] E. Catinas, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl. 113 473-485 (2002).

[6] F. A. Potra, On Q-order and R-order of convergence, J. Optim. Theory Appl. 63 415-431 (1989).

[7] F. A. Potra, Q-superlinear convergence of the iterates in primal-dual interior-point methods, Math. Progr. 91 1 ser. A 99-115 (2001).

[8] T. J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 583-590 (1984).

[9] E. Catinas, Inexact perturbed Newton methods and applications to a class of Krylov solvers, J. Optim. Theory Appl. 108 543-570 (2001).

[10] E. Catinas, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp. 74 249 291-301 (2005).

[11] J. E. Dennis, Jr., and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics, Englewood Cliffs (1983).

[12] W. C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Polish Academy of Science, Banach Ctr. Publ. 3 129-142 (1977).

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

[14] I. K. Argyros, On the convergence and application of Newton’s method under weak Holder continuity assumptions, Int. J. Comp. Math. 80 6 767-780 (2003).

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