Convergence of Newton’s method under the gamma condition
DOI:
https://doi.org/10.4067/S0716-09172006000300006Keywords:
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.Abstract
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.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).
[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).
Published
2017-05-08
How to Cite
[1]
I. K. Argyros, “Convergence of Newton’s method under the gamma condition”, Proyecciones (Antofagasta, On line), vol. 25, no. 3, pp. 293-306, May 2017.
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