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.
@article{bwmeta1.element.doi-10_2478_s11533-007-0007-3,
author = {Ioannis Argyros},
title = {On the convergence of the secant method under the gamma condition},
journal = {Open Mathematics},
volume = {5},
year = {2007},
pages = {205-214},
zbl = {1141.65034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0007-3}
}
Ioannis Argyros. On the convergence of the secant method under the gamma condition. Open Mathematics, Tome 5 (2007) pp. 205-214. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0007-3/
[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 | Zbl 1016.65035
[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 | Zbl 1057.65029
[3] I.K. Argyros: “New sufficient convergence conditions for the Secant method”, Che-choslovak Math. J., Vol. 55(130), (2005), pp. 175–187. | Zbl 1081.65043
[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 | Zbl 1024.65043
[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 | Zbl 1019.65036
[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 | Zbl 0884.30004
[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