Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ENLARGING THE BALL OF CONVERGENCE OF SECANT-LIKE METHODS FOR NON-DIFFERENTIABLE OPERATORS

  • Argyros, Ioannis K. (Department of Mathematical Sciences Cameron University) ;
  • Ren, Hongmin (College of Information and Engineering Hangzhou Polytechnic)
  • Received : 2016.09.25
  • Accepted : 2017.11.24
  • Published : 2018.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we enlarge the ball of convergence of a uniparametric family of secant-like methods for solving non-differentiable operators equations in Banach spaces via using ${\omega}$-condition and centered-like ${\omega}$-condition meantime as well as some fine techniques such as the affine invariant form. Numerical examples are also provided.

Keywords

References

  1. I. K. Argyros, On the secant method, Publ. Math. Debrecen 43 (1993), no. 3-4, 223-238.
  2. I. K. Argyros, Computational theory of iterative methods, Series: Studies in Computational Mathematics 15, Editors, C. K. Chui and L. Wuytack, Elservier Publ. Co. New York, USA, 2007.
  3. I. K. Argyros and H. Ren, On an improved local convergence analysis for the secant method, Numer. Algorithms 52 (2009), no. 2, 257-271. https://doi.org/10.1007/s11075-009-9271-6
  4. P. Deuflhard, Newton methods for nonlinear problems, Springer Series in Computational Mathematics, 35, Springer-Verlag, Berlin, 2004.
  5. J. A. Ezquerro, M. Grau-Sanchez, M. A. Hernandez, and M. Noguera, Semilocal convergence of secant-like methods for differentiable and nondifferentiable operator equations, J. Math. Anal. Appl. 398 (2013), no. 1, 100-112. https://doi.org/10.1016/j.jmaa.2012.08.040
  6. M. A. Hernandez and M. J. Rubio, A uniparametric family of iterative processes for solving nondifferentiable equations, J. Math. Anal. Appl. 275 (2002), no. 2, 821-834. https://doi.org/10.1016/S0022-247X(02)00432-8
  7. M. A. Hernandez and M. J. Rubio, On the ball of convergence of secant-like methods for non-differentiable operators, Appl. Math. Comput. 273 (2016), 506-512.
  8. M. A. Hernandez, M. J. Rubio, and J. A. Ezquerro, Solving a special case of conservative problems by secant-like methods, Appl. Math. Comput. 169 (2005), no. 2, 926-942. https://doi.org/10.1016/j.amc.2004.09.070
  9. Z. Huang, The convergence ball of Newton's method and the uniqueness ball of equations under Holder-type continuous derivatives, Comput. Math. Appl. 47 (2004), no. 2-3, 247-251. https://doi.org/10.1016/S0898-1221(04)90021-1
  10. L. V. Kantorovich and G. P. Akilov, Functional Analysis, translated from the Russian by Howard L. Silcock, second edition, Pergamon Press, Oxford, 1982.
  11. F. A. Potra and V. Ptak, Nondiscrete induction and iterative processes, Research Notes in Mathematics, 103, Pitman (Advanced Publishing Program), Boston, MA, 1984.
  12. H. Ren and I. K. Argyros, Local convergence of effcient Secant-type methods for solving nonlinear equations, Appl. Math. Comput. 218 (2012), no. 14, 7655-7664. https://doi.org/10.1016/j.amc.2012.01.036
  13. H. Ren and Q. B. Wu, The convergence ball of the Secant method under Hoder continuous divided differences, J. Comput. Appl. Math. 194 (2006), 284-293. https://doi.org/10.1016/j.cam.2005.07.008
  14. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.
  15. X. H. Wang, Error estimates for some numerical root-finding methods, Acta Math. Sinica 22 (1979), no. 5, 638-642.
  16. X. H. Wang, On the domain of convergence of Newton's method, Kexue Tongbao (Chinese) 25 (1980), Special Issue, 36-37.