Advertisement

Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 405–421 | Cite as

Local convergence analysis for Chebyshev’s method

  • Chandni Kumari
  • P. K. ParidaEmail author
Original Research
  • 79 Downloads

Abstract

In this work, we are working to present a local convergence analysis for Chebyshev’s method by using majorizing sequence. The given method is a third order iterative process, used in order to approximate a zero of an nonlinear operator equation in a Banach space. Here we are using a new type of majorant conditions to prove the convergence. We will also try to establish relations between this majorant conditions with results of based on Kantorovich-type and Smale-type assumptions.

Keywords

Chebyshev’s method Newton’s method Banach space Convex majorant Ball convergence Radius of convergence 

Mathematics Subject Classification

65G99 65J15 47H17 47J05 

References

  1. 1.
    Argyros, I.K.: Convergence and Application of Newton-Type Iterations. Springer, New York (2008)zbMATHGoogle Scholar
  2. 2.
    Argyros, I.K., George, S.: Local convergence for deformed Chebyshev-type method in Banach space under weak conditions. Cogent Math. 2, 1036958 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Argyros, I.K., Hongmin, R.: Ball convergence theorems for Halleys method in Banach space. J. Appl. Math. Comput. 38, 453–465 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Argyros, I.K., Khattri, S.K.: An improved semilocal convergence analysis for the Chebyshev method. J. Appl. Math. Comput. 42, 509–528 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: The Chebyshev method. Computing 45, 113–130 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diloné, M.A., García-Olivo, M., Gutiérrez, J.M.: A note on the semilocal convergence of Chebyshev’s method. Bull. Aust. Math. Soc 88, 98–105 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ezquerro, J.A.: A modification of the Chebyshev method. IMA J. Numer. Anal. 17, 511–525 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hilliart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms, Part 1. Springer, Berlin (1993)zbMATHGoogle Scholar
  9. 9.
    Ivanov, S.: On the convergence of Chebyshevs method for multiple polynomial zeros. Results. Math. 69, 93–103 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis in Normed Spaces. Pergamon Press, Oxford (1982)zbMATHGoogle Scholar
  11. 11.
    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kogan, T., Sapir, L., Sapir, A., Sapir, A.: To the question of efficiency of iterative methods. Appl. Math. Lett. 66, 40–46 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kyncheva, V.K., Yotov, V.V., Ivanov, S.I.: Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros. Appl. Numer. Math. 112, 146–154 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ling, Y., Xu, X.: On the semilocal convergence behavior for Halley’s method. Comput. Optim. Appl. 58, 597–618 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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, 3405–3419 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rall, L.B.: Computational Solution of Nonlinear Operator Equations. E. Robert Krieger, New York (1969)zbMATHGoogle Scholar
  17. 17.
    Wang, X.: Convergence on Newtons method and inverse function theorem in Banach space. Math. Comput. 68, 169–186 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, X., Kou, J.: Convergence for a family of modified Chebyshev methods under weak condition. Numer. Algorithm 66, 33–48 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Centre for Applied MathematicsCentral University of JharkhandRanchiIndia

Personalised recommendations