Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A UNIFIED CONVERGENCE ANALYSIS FOR SECANT-TYPE METHODS

  • Received : 2013.11.16
  • Published : 2014.11.01
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Abstract

We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost our semilocal convergence criteria can be weaker; the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in earlier studies such as [1-3,7-14,16,20,21] at least for the cases of Newton's method and the secant method. Numerical examples are also presented to illustrate the theoretical results obtained in this study.

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References

  1. S. Amat, S. Busquier, and M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim. 25 (2004), no. 5-6, 397-405. https://doi.org/10.1081/NFA-200042628
  2. I. K. Argyros, Computational Theory of Iterative Methods, Studies in Computational Mathematics, 15. Elsevier B. V., Amsterdam, 2007.
  3. I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Method for Equations and Its Applications, CRC Press/Taylor and Francis, New York, 2012.
  4. I. K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387. https://doi.org/10.1016/j.jco.2011.12.003
  5. E. Catinas, The inexact, inexact perturbed, and quasi-Newton methods are equivalent models, Math. Comp. 74 (2005), no. 249, 291-301.
  6. S. Chandrasekhar, Radiative Transfer, Dover Publ., New York, 1960.
  7. J. E. Dennis, Toward a unified convergence theory for Newton-like methods, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) pp. 425-472 Academic Press, New York, 1971.
  8. J. A. Ezquerro, J. M. Gutierrez, M. A. Hernandez, N. Romero, and M. J. Rubio, The Newton method: from Newton to Kantorovich, (Spanish), Gac. R. Soc. Mat. Esp. 13 (2010), no. 1, 53-76.
  9. W. B. Gragg and R. A. Tapia, Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal. 11 (1974), 10-13. https://doi.org/10.1137/0711002
  10. M. A. Hernandez, M. J. Rubio, and J. A. Ezquerro, Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math. 115 (2000), no. 1-2, 245-254. https://doi.org/10.1016/S0377-0427(99)00116-8
  11. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
  12. A. A. Magrenan, Estudio de la dinamica del metodo de Newton amortiguado, PhD Thesis, Servicio de Publicaciones, Universidad de La Rioja, 2013.
  13. L. M. Ortega andW. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic press, New York, 1970.
  14. F. A. Potra and V. Ptak, Nondiscrete induction and iterative processes, Research Notes in Mathematics, 103. Pitman (Advanced Publishing Program), Boston, MA, 1984.
  15. P. D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton's method, J. Complexity 25 (2009), no. 1, 38-62. https://doi.org/10.1016/j.jco.2008.05.006
  16. P. D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity 26 (2010), no. 1, 3-42. https://doi.org/10.1016/j.jco.2009.05.001
  17. W. C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Mathematical models and numerical methods (Papers, Fifth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975), pp. 129-142, Banach Center Publ., 3, PWN, Warsaw, 1978.
  18. J. W. Schmidt, Untere Fehlerschranken fur Regula-Falsi Verhafren, Period. Math. Hungar. 9 (1978), no. 3, 241-247. https://doi.org/10.1007/BF02018090
  19. J. F. Traub, Iterative Methods for the Solutions of Equations, Prentice-Hall, New Jersey, 1964.
  20. T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987), no. 5, 545-557. https://doi.org/10.1007/BF01400355
  21. P. P. Zabrejko and D. F. Nguen, The majorant method in the theory of Newton- Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optim. 9 (1987), no. 5-6, 671-684. https://doi.org/10.1080/01630568708816254

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