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.

Resumen

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.

Citas

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

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[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).
Publicado
2017-05-08
Cómo citar
Argyros, I. (2017). Convergence of Newton’s method under the gamma condition. Proyecciones. Journal of Mathematics, 25(3), 293-306. https://doi.org/10.4067/S0716-09172006000300006
Sección
Artículos