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Three Step Kurchatov Method for Nondifferentiable Operators

  • Himanshu Kumar
  • P. K. ParidaEmail author
Original Paper
  • 110 Downloads

Abstract

In this paper we find the order of convergence and semilocal convergence of three step Kurchatov-type method. We also analyze the efficiency index and computational efficiency of this method. The semilocal convergence analysis of method has been established by using recurrence relations under the assumption of first order divided difference operators satisfy Lipschitz condition. The convergence theorem and domain of parameters of the method has also been included. The applicability of the proposed convergence analysis is illustrated by solving some numerical examples.

Keywords

Nonlinear equations Order of convergence Kurchatov’s method Efficiency index Computational efficiency Recurrence relations Semilocal convergence 

Mathematics Subject Classfication

47H99 65H10 

References

  1. 1.
    Ezquerro, J.A., Hernández-Verón, M.A.: Increasing the applicability of Steffensen’s method. J. Math. Anal. Appl. 418, 1062–1073 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Parhi, S.K., Gupta, D.K.: Semi-local convergence of Stirling’s method under Hölder continuous first derivative in Banach spaces. Int. J. Comput. Math. 87(12), 2752–2759 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Parhi, S.K., Gupta, D.K.: Relaxing convergence conditions for Stirling’s method. Math. Methods Appl. Sci. 33, 224–232 (2010)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Parhi, S.K., Gupta, D.K.: Convergence of Stirling’s method under weak differentiability condition. Math. Methods Appl. Sci. 34, 168–175 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: The Halley method. Computing 44, 169–184 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45, 355–367 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Prashanth, M., Gupta, D.K.: Recurrence relations for super-Halley’s method with Hölder continous second derivative in Banach spaces. Kodai Math. J. 36(1), 119–136 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Hernndez, M.A.: Reduced recurrence relations for the Chebyshev method. J. Optim. Theory Appl. 98, 385–397 (1998)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Parida, P.K., Gupta, D.K.: Semilocal convergence of a family of third-order Chebyshev-type methods under a mild differentiability condition. Int. J. Comput. Math. 87(15), 3405–3419 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Parida, P.K., Gupta, D.K.: Semilocal convergence of a family of third-order methods in Banach spaces under Hölder continuous second derivative. Nonlinear Anal. 69(11), 4163–4173 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Prashanth, M., Gupta, D.K., Singh, S.: Semilocal convergence for the super-Halley’s method. Numer. Anal. Appl. 7(1), 70–84 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Potra, F.-A., Pták, V.: Nondiscrete Induction and Iterative Processes. Pitman, Boston (1984)zbMATHGoogle Scholar
  13. 13.
    Amat, S., Busquier, S., Ezquerro, J.A., Hernández-Verón, M.A.: A Steffensen type method of two step in Banach spaces with applications. J. Comp. Appl. Math. 291, 317–331 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ezquerro, J.A., Hernández-Verón, M.A., Velasco, A.I.: An analysis of the semilocal convergence for secant-like methods. Appl. Math. Comput. 266, 883–892 (2015)MathSciNetGoogle Scholar
  15. 15.
    Hernández-Verón, M.A., Rubio, M.J.: Semilocal convergence of the secant method under mild convergence conditions of differentiability. Comput. Math. Appl. 44, 277–285 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Shakhno, S.M.: On a Kurchatov’s method of linear interpolation for solving nonlinear equations. Proc. Appl. Math. Mech. 4, 650–651 (2004)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ezquerro, J.A., Grau, A., Grau-Sánchez, M., Hernández-Verón, M.A.: On the efficiency of two variants of Kurchatov’s method for solving nonlinear systems. Numer. Algor. 64, 685–698 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Amat, S., Busquier, S.: Convergence and numerical analysis of a family of two-step Steffensen’s methods. Int. J. Comput. Math. Appl. 49, 13–22 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Dennis, J.E.: Toward a unified convergence theory for Newtonlike methods. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 425–472. Academic Press, NewYork (1971)CrossRefGoogle Scholar
  20. 20.
    Potra, F.A.: Sharp error bounds for a class of Newton-like methods. Libertas Math. 5, 71–84 (1985)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Argyros, I.K., Ren, Hongmin: Efficient Steffensen-type algorithms for solving nonlinear equations. Int. J. Comput. Math. 90(3), 691–704 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Argyros, I.K., Ezquerro, J.A., Gutiérrez, J.M., Hernández-Verón, M.A., Hilout, S.: On the semilocal convergence of efficient Chebyshev–Secant-type methods. J. Comput. Appl. Math. 235, 3195–3206 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ezquerro, J.A., Hernández-Verón, M.A.: An optimization of Chebyshev’s method. J. Complex. 25, 343–361 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Ren, Hongmin: New sufficient convergence conditions of the secant method for nondifferentiable operators. Appl. Math. Comput. 182, 1255–1259 (2006)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Hernández-Verón, M.A., Rubio, M.J.: The Secant method for nondifferentiable operators. Appl. Math. Lett. 15(4), 395–399 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Thim, J.: Continuous nowhere differentiable functions. Master thesis (2003). http://epubl.ltu.se/1402-1617/2003/320/LTU-EX-03320-SE
  27. 27.
    Ezquerro, J.A., Grau-Sánchez, M., Hernández-Verón, M.A., Noguera, M.: A Tarub type result for one-point iterative methods with memory. Anal. Appl. 12(3), 323–340 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer India Pvt. Ltd. 2017

Authors and Affiliations

  1. 1.Centre for Applied MathematicsCentral University of JharkhandRanchiIndia

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