# On a quadratically convergent method using divided differences of order one under the gamma condition

Open Mathematics (2008)

- Volume: 6, Issue: 2, page 262-271
- ISSN: 2391-5455

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topIoannis Argyros, and Hongmin Ren. "On a quadratically convergent method using divided differences of order one under the gamma condition." Open Mathematics 6.2 (2008): 262-271. <http://eudml.org/doc/269199>.

@article{IoannisArgyros2008,

abstract = {We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.},

author = {Ioannis Argyros, Hongmin Ren},

journal = {Open Mathematics},

keywords = {Banach space; local/semilocal convergence; majorizing sequence; Fréchet-derivative; two-point iterative method; Steffensen’s method; gamma condition; local and semilocal convergence; Fréchet derivative; Steffensen method; Newton-Kantorovich condition; nonlinear operator equation},

language = {eng},

number = {2},

pages = {262-271},

title = {On a quadratically convergent method using divided differences of order one under the gamma condition},

url = {http://eudml.org/doc/269199},

volume = {6},

year = {2008},

}

TY - JOUR

AU - Ioannis Argyros

AU - Hongmin Ren

TI - On a quadratically convergent method using divided differences of order one under the gamma condition

JO - Open Mathematics

PY - 2008

VL - 6

IS - 2

SP - 262

EP - 271

AB - We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.

LA - eng

KW - Banach space; local/semilocal convergence; majorizing sequence; Fréchet-derivative; two-point iterative method; Steffensen’s method; gamma condition; local and semilocal convergence; Fréchet derivative; Steffensen method; Newton-Kantorovich condition; nonlinear operator equation

UR - http://eudml.org/doc/269199

ER -

## References

top- [1] Amat S., Busquier S., Candela V.F., A class of quasi-Newton generalized Steffensen’s methods on Banach spaces, J. Comput. Appl. Math., 2002, 149, 397–406 http://dx.doi.org/10.1016/S0377-0427(02)00484-3 Zbl1016.65035
- [2] Amat S., Busquier S., Gutiąäerrez J.M., On the local convergence of Secant-type methods, Int. J. Comput. Math., 2004, 81, 1153–1161 http://dx.doi.org/10.1080/00207160412331284123 Zbl1068.65068
- [3] Argyros I.K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 2004, 298, 374–397 http://dx.doi.org/10.1016/j.jmaa.2004.04.008 Zbl1057.65029
- [4] Argyros I.K., On a two-point Newton-like method of convergent order two, Int. J. Comput. Math., 2005, 82, 219–234 http://dx.doi.org/10.1080/00207160412331296661 Zbl1068.65070
- [5] Argyros I.K., A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations, J. Math. Anal. Appl., 2007, 332, 97–108 http://dx.doi.org/10.1016/j.jmaa.2006.09.075 Zbl1121.65061
- [6] Argyros I.K., On the convergence of the Secant method under the gamma condition, Cent. Eur. J. Math., 2007, 5, 205–214 http://dx.doi.org/10.2478/s11533-007-0007-3 Zbl1141.65034
- [7] Argyros I.K., Computation theory of iterative methods, In: Chui C.K., Wuytack L. (Eds.), Studies in Computational Mathematics 15, Elsevier, New York, 2007 Zbl1147.65313
- [8] Catinas E., On some iterative methods for solving nonlinear equations, Revue d’ Analyse Numerique et de Theorie de l’ Approximation, 1994, 23, 47–53 Zbl0818.65050
- [9] Chandrasekhar S., Radiative transfer, Dover Publications, New York, 1960
- [10] Hernandez M.A., Rubio M.J., Ezquerro J.A., Secant-like methods for solving nonlinear integral equations of the Hammerstein-type, Comput. Appl. Math., 2000, 115, 245–254 http://dx.doi.org/10.1016/S0377-0427(99)00116-8 Zbl0944.65146
- [11] Hernandez M.A., Rubio M.J., Semilocal convergence for the Secant method under mild convergence conditions of differentiability, Comput. Math. Appl., 2002, 44, 277–285 http://dx.doi.org/10.1016/S0898-1221(02)00147-5 Zbl1055.65069
- [12] Kantorovich L.V., Akilov G.P., Functional analysis in normed spaces, Pergamon Press, Oxford, 1982 Zbl0127.06102
- [13] Pavaloiu I., A convergence theorem concerning the method of Chord, Revue d’ Analyse Numerique et de Theorie de l’ Aapproximation, 1992, 21, 59–65 Zbl0773.65040
- [14] Potra F.A., An iterative algorithm of order 1.839 ... for solving nonlinear operator equations, Numer. Funct. Anal. Optim., 1984/85, 7, 75–106 http://dx.doi.org/10.1080/01630568508816182 Zbl0556.65049
- [15] Ren H.M., Wu Q.B., The convergence ball of the Secant method under Hölder continuous divided differences, J. Comput. Appl. Math., 2006, 194, 284–293 http://dx.doi.org/10.1016/j.cam.2005.07.008 Zbl1101.65057
- [16] Ren H.M., New sufficient convergence conditions of the Secant method for nondifferentiable operators, Appl. Math. Comput., 2006, 182, 1255–1259 http://dx.doi.org/10.1016/j.amc.2006.05.009 Zbl1111.65050
- [17] Ren H.M., Yang S.J., Wu Q.B., A new semilocal convergence theorem for the Secant method under Hölder continuous divided differences, Appl. Math. Comput., 2006, 182, 41–48 http://dx.doi.org/10.1016/j.amc.2006.03.034 Zbl1112.65050
- [18] Rheinboldt W.C., An adaptive continuation process for solving systems of nonlinear equations, Banach Center Publ., 1978, 3, 129–142 Zbl0378.65029
- [19] Wang X.H., Convergence of the iteration of Halley family in weak condition, Chinese Science Bulletin, 1997, 42, 552–555 http://dx.doi.org/10.1007/BF03182614 Zbl0884.30004
- [20] Wang D.R., Zhao F.G., The theory of Smale’s point estimation and its applications, J. Comput. Appl. Math., 1995, 60, 253–269 http://dx.doi.org/10.1016/0377-0427(94)00095-I

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