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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.25 no.3 Antofagasta Dec. 2006

http://dx.doi.org/10.4067/S0716-09172006000300006 

 

Proyecciones
Vol. 25, No 3, pp. 293-306, December 2006.
Universidad Católica del Norte
Antofagasta - Chile

 

CONVERGENCE OF NEWTON’S METHOD UNDER THE GAMMA CONDITION

 

IOANNIS K. ARGYROS

CAMERON UNIVERSITY, U.S.A.


Abstract

We provide a semilocal as well as a local convergence analysis ofNewton’s method using the gamma condition [1], [10], [11]. Usingmore precise majorizing sequences than before [4], [8]—[11] and underat least as weak hypotheses, we provide in the semilocal case: finererror bounds on the distances involved and an at least as precise informationon the location of the solution; in the local case: a largerradius of convergence.

AMS (MOS) Subject Classification Codes : 65H10, 65G99,47H17, 49M15.

Key Words: Banach space, Newton’s method, local/semilocalconvergence, Newton—Kantorovich theorem, Fréchet derivative, majorizingsequence, radius of convergence, gamma condition, analyticoperator.


 

REFERENCES

[1] Argyros, I. K., A convergence analysis for Newton’s method based on Lipschitz center-Lipschitz and analytic operators, Pan American Math. J. 13, 3, pp. 19—24, (2003).

[2] Argyros, I. K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Applic. 298, pp. 374—397, (2004).

[3] Argyros, I. K., Approximate Solution of Operator Equations with Applications, World Scientific Publ. Comp., Hackensack,, New Jersey, U.S.A., (2005)

[4] Dedieu, J. P. and Shub, M., Multihomogeneous Newton methods, Math. Comput. 69, 231, pp. 1071—1098, (1999).

[5] Ezquerro, J. A. and Hernandez, M.A., On a convex acceleration of Newton’s method, J. Optim. Th. Appl. 100, 2, pp. 311—326, (1999).

[6] Gutierrez, J. M., A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79, pp. 131—145, (1997).

[7] Kantorovich, L. V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, (1982).

[8] Smale, S., Newton’s method estimate from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics (eds., Ewing, R. et al.), Springer-Verlag, New York, (1986).

[9] Wang, D. and Zhao, F., The theory of Smale’s point estimation and its applications, J. Comput. Appl. Math. 60, pp. 253—269, (1995).

[10] Wang, X. H. and Han, D.F., On dominating sequence method in the point estimate and Smale theorem, Sci. Sinica Ser. A, 33, pp 135—144, (1990).

[11] Wang, X. H., Convergence of the iteration of Halley family in weak conditions, Chinese Science Bulletin, 42, pp. 552—555, (1997).

 

Ioannis K. Argyros
Department of Mathematical Sciences
Cameron University
Lawton, OK 73505
U. S. A.
e-mail address : iargyros@cameron.edu

 

Received : April 2006. Accepted : October 2006

 

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