On the convergence of the secant method under the gamma condition

Ioannis Argyros

Open Mathematics (2007)

  • Volume: 5, Issue: 2, page 205-214
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Ioannis Argyros. "On the convergence of the secant method under the gamma condition." Open Mathematics 5.2 (2007): 205-214. <http://eudml.org/doc/269636>.

@article{IoannisArgyros2007,
abstract = {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.},
author = {Ioannis Argyros},
journal = {Open Mathematics},
keywords = {Banach space; Secant method; Newton’s method; Gamma condition; majorizing sequence; semilocal convergence; radius of convergence; Newton-Kantorovich theorem; secant method; gamma condition; Newton's method; numerical example},
language = {eng},
number = {2},
pages = {205-214},
title = {On the convergence of the secant method under the gamma condition},
url = {http://eudml.org/doc/269636},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Ioannis Argyros
TI - On the convergence of the secant method under the gamma condition
JO - Open Mathematics
PY - 2007
VL - 5
IS - 2
SP - 205
EP - 214
AB - 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.
LA - eng
KW - Banach space; Secant method; Newton’s method; Gamma condition; majorizing sequence; semilocal convergence; radius of convergence; Newton-Kantorovich theorem; secant method; gamma condition; Newton's method; numerical example
UR - http://eudml.org/doc/269636
ER -

References

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  1. [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 Zbl1016.65035
  2. [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 Zbl1057.65029
  3. [3] I.K. Argyros: “New sufficient convergence conditions for the Secant method”, Che-choslovak Math. J., Vol. 55(130), (2005), pp. 175–187. Zbl1081.65043
  4. [4] I.K. Argyros: Approximate Solution of Operator Equations with Applications, World Scientific Publ. Comp., Hackensack, New Jersey, 2005, U.S.A. 
  5. [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 Zbl1024.65043
  6. [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 Zbl1019.65036
  7. [7] L.V. Kantorovich and G.P. Akilov: Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982. 
  8. [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. [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. [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 Zbl0884.30004
  11. [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 

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