# On the convergence of the secant method under the gamma condition

Open Mathematics (2007)

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

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topIoannis 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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- [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 Zbl0884.30004
- [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

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