Convergence of Newton’s method under the gamma condition

  • Ioannis K. Argyros Cameron University.
Palabras clave: Banach space, Newton’s method, local/semilocal convergence, Newton—Kantorovich theorem, Frechet derivative, majorizing sequence, radius of convergence, gamma condition, analytic operator, espacio de Banach, método de Newton, convergencia local/semilocal.


We provide a semilocal as well as a local convergence analysis of Newton’s method using the gamma condition [1], [10], [11]. Using more precise majorizing sequences than before [4], [8]—[11] and under at least as weak hypotheses, we provide in the semilocal case: finer error bounds on the distances involved and an at least as precise information on the location of the solution; in the local case: a larger radius of convergence.

Biografía del autor/a

Ioannis K. Argyros, Cameron University.
Department of Mathematical Sciences.


[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).
Cómo citar
Argyros, I. (2017). Convergence of Newton’s method under the gamma condition. Proyecciones. Journal of Mathematics, 25(3), 293-306.