PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2008 | 6 | 2 | 262-271
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Twórcy
autor
Bibliografia
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [12] Kantorovich L.V., Akilov G.P., Functional analysis in normed spaces, Pergamon Press, Oxford, 1982
  • [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
  • [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
  • [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
  • [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
  • [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
  • [18] Rheinboldt W.C., An adaptive continuation process for solving systems of nonlinear equations, Banach Center Publ., 1978, 3, 129–142
  • [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
  • [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
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-008-0015-y
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.