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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.25 n.3 Antofagasta dic. 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).         [ Links ]

[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).         [ Links ]

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

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

[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).         [ Links ]

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

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

[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).         [ Links ]

[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).         [ Links ]

[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).         [ Links ]

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

 

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