Numerical Algorithms

, Volume 72, Issue 4, pp 937–958 | Cite as

A stable class of improved second-derivative free Chebyshev-Halley type methods with optimal eighth order convergence

  • Alicia Cordero
  • Munish Kansal
  • Vinay Kanwar
  • Juan R. Torregrosa
Original Paper


In this paper, we present a uniparametric family of modified Chebyshev-Halley type methods with optimal eighth-order of convergence. In terms of computational cost, each member of the family requires only four functional evaluations per step, and hence is optimal in the sense of Kung-Traub conjecture. Moreover, in order to have additional information to choose some elements of the class, in particular some stable enough, we use complex dynamics tools to analyze their stability. Then, some ranges of values of the parameter are found to be avoided but we show that the region of stable members of this family is vast. It is found by way of illustration that these proposed methods are very useful in high precision computations.


Nonlinear equations Optimal iterative schemes Newton method Chebyshev-Halley scheme Efficiency index Complex dynamics Stability functions Dynamical planes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Petković, M.S., Neta, B., Petković, L.D., Dz̆unić, J. (eds.): Multipoint Methods for Solving Nonlinear Equations. Elsevier, New York (2013)Google Scholar
  2. 2.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, New Jersey (1964)MATHGoogle Scholar
  3. 3.
    Ostrowski, A.M.: Solutions of Equations and System of Equations. Academic Press, New York (1966)MATHGoogle Scholar
  4. 4.
    Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev-Halley type methods in Banach spaces. Bull. Austral. Math. Soc. 55, 113–130 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. ACM 21, 643–651 (1974)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    King, R.F.: A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)CrossRefMATHGoogle Scholar
  8. 8.
    Li, D., Liu, P., Kou, J.: An improvement of Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cordero, A., Lotfi, T., Mahdiani, K., Torregrosa, J.R.: A stable family with high order of convergence for solving nonlinear equations. Appl. Math. Comput. 254, 240–251 (2015)MathSciNetGoogle Scholar
  10. 10.
    Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intelligencer 24, 37–46 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A: Math. Sci. 10, 3–35 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cordero, A., García-Maimó, C., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev-Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)MathSciNetMATHGoogle Scholar
  15. 15.
    Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. Comput. Appl. Math. 233, 2688–2695 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equation. App. Math. Comput. 227, 567–592 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)MathSciNetMATHGoogle Scholar
  18. 18.
    Blanchard, P.: Complex Analytic Dynamics on the Riemann Sphere. Bull. of the AMS 11(1), 85–141 (1984)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Babajee, D.K.R., Cordero, A., Torregrosa, J.R.: Study of iterative methods through the Cayley Quadratic Test. Comput. Appl. Math. 291, 358–369 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. The Sci. World J. (2013). Article ID 780153Google Scholar
  22. 22.
    Chun, C., Lee, M.Y.: A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl. Math. Comput. 223, 506–519 (2013)MathSciNetMATHGoogle Scholar
  23. 23.
    Liu, L., Wang, X.: Eighth-order methods with high efficiency index for solving nonlinear equations. Appl. Math. Comput. 215, 3449–3454 (2010)MathSciNetMATHGoogle Scholar
  24. 24.
    Thukral, R., Petković, M.S.: A family of three-point methods of optimal order for solving nonlinear equations. J. Comput. Appl. Math. 233, 2278–2284 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Alicia Cordero
    • 1
  • Munish Kansal
    • 2
  • Vinay Kanwar
    • 2
  • Juan R. Torregrosa
    • 1
  1. 1.Multidisciplinary Institute of MathematicsUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia

Personalised recommendations