versión impresa ISSN 0716-0917
Proyecciones (Antofagasta) v.25 n.3 Antofagasta dic. 2006
CONVERGENCE OF NEWTONS METHOD UNDER THE GAMMA CONDITION
IOANNIS K. ARGYROS
CAMERON UNIVERSITY, U.S.A.
We provide a semilocal as well as a local convergence analysis ofNewtons method using the gamma condition , , . Usingmore precise majorizing sequences than before ,  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, Newtons method, local/semilocalconvergence, NewtonKantorovich theorem, Fréchet derivative, majorizingsequence, radius of convergence, gamma condition, analyticoperator.
 Argyros, I. K., A convergence analysis for Newtons method based on Lipschitz center-Lipschitz and analytic operators, Pan American Math. J. 13, 3, pp. 1924, (2003). [ Links ]
 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. 374397, (2004). [ Links ]
 Argyros, I. K., Approximate Solution of Operator Equations with Applications, World Scientific Publ. Comp., Hackensack,, New Jersey, U.S.A., (2005) [ Links ]
 Dedieu, J. P. and Shub, M., Multihomogeneous Newton methods, Math. Comput. 69, 231, pp. 10711098, (1999). [ Links ]
 Ezquerro, J. A. and Hernandez, M.A., On a convex acceleration of Newtons method, J. Optim. Th. Appl. 100, 2, pp. 311326, (1999). [ Links ]
 Gutierrez, J. M., A new semilocal convergence theorem for Newtons method, J. Comput. Appl. Math. 79, pp. 131145, (1997). [ Links ]
 Kantorovich, L. V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, (1982). [ Links ]
 Smale, S., Newtons 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 ]
 Wang, D. and Zhao, F., The theory of Smales point estimation and its applications, J. Comput. Appl. Math. 60, pp. 253269, (1995). [ Links ]
 Wang, X. H. and Han, D.F., On dominating sequence method in the point estimate and Smale theorem, Sci. Sinica Ser. A, 33, pp 135144, (1990). [ Links ]
 Wang, X. H., Convergence of the iteration of Halley family in weak conditions, Chinese Science Bulletin, 42, pp. 552555, (1997). [ Links ]
Ioannis K. Argyros
Received : April 2006. Accepted : October 2006