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2007 | 5 | 2 | 205-214
Tytuł artykułu

On the convergence of the secant method under the gamma condition

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.
Wydawca
Czasopismo
Rocznik
Tom
5
Numer
2
Strony
205-214
Opis fizyczny
Daty
wydano
2007-06-01
online
2007-06-01
Twórcy
Bibliografia
  • [1] S. Amat, S. Busquier and V. Candela: “A class of quasi-Newton generalized Steffensen methods on Banach spaces”, J. Comput. Appl. Math., Vol. 149, (2002), pp. 397–408. http://dx.doi.org/10.1016/S0377-0427(02)00484-3
  • [2] I.K. Argyros: “A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space”, J. Math. Anal. Applic., Vol. 298, (2004), pp. 374–397. http://dx.doi.org/10.1016/j.jmaa.2004.04.008
  • [3] I.K. Argyros: “New sufficient convergence conditions for the Secant method”, Che-choslovak Math. J., Vol. 55(130), (2005), pp. 175–187.
  • [4] I.K. Argyros: Approximate Solution of Operator Equations with Applications, World Scientific Publ. Comp., Hackensack, New Jersey, 2005, U.S.A.
  • [5] M.A. Hernandez and M.J. Rubio: “The Secant method and divided differences Hölder continuous”, Appl. Math. Comput., Vol. 15, (2001), pp. 139–149. http://dx.doi.org/10.1016/S0096-3003(00)00079-5
  • [6] M.A. Hernandez and M.J. Rubio: “A uniparametric family of iterative processes for solving nondifferentiable equations”, J. Math. Anal. Appl., Vol. 275, (2005), pp. 821–834. http://dx.doi.org/10.1016/S0022-247X(02)00432-8
  • [7] L.V. Kantorovich and G.P. Akilov: Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982.
  • [8] S. Smale: “Newton’s method estimate from data at one point”, In: R. Ewing et al. (Eds.): The Merging of Disciplines New Directions in Pure, Applied and Computational Mathematics, Springer-Verlag, New York, 1986.
  • [9] D. Wang and F. Zhao: “The theory of Smale’s point estimation and its applications”, J. Comput. Appl. Math., Vol. 60, (1995), pp. 253–269. http://dx.doi.org/10.1016/0377-0427(94)00095-I
  • [10] X.H. Wang: “Convergence of the iteration of Halley family in weak conditions”, Chinese Sci. Bull., Vol. 42, (1997), pp. 552–555. http://dx.doi.org/10.1007/BF03182614
  • [11] J.C. Yakoubsohn: “Finding zeros of analytic functions: α theory for the secant type methods”, J. Complexity, Vol. 15, (1999), pp. 239–281. http://dx.doi.org/10.1006/jcom.1999.0501
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-007-0007-3
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