Numerical Algorithms

, Volume 71, Issue 1, pp 1–23 | Cite as

A study on the local convergence and the dynamics of Chebyshev–Halley–type methods free from second derivative

Original Paper


We study the local convergence of Chebyshev-Halley-type methods of convergence order at least five to approximate a locally unique solution of a nonlinear equation. Earlier studies such as Behl (2013), Bruns and Bailey (Chem. Eng. Sci 32, 257–264, 1977), Candela and Marquina (Computing 44, 169–184, 1990), (Computing 45(4):355–367, 1990), Chicharro et al. (2013), Chun (Appl. Math. Comput, 190(2):1432–1437, 1990), Cordero et al. (Appl.Math. Lett. 26, 842–848, 2013), Cordero et al. (Appl. Math. Comput. 219, 8568–8583, 2013), Cordero and Torregrosa (Appl. Math. Comput. 190, 686–698, 2007), Ezquerro and Hernández (Appl. Math. Optim. 41(2):227–236, 2000), (BIT Numer. Math. 49, 325–342, 2009), (J. Math. Anal. Appl. 303, 591–601, 2005), Gutiérrez and Hernández (Comput. Math. Applic. 36(7):1–8, 1998), Ganesh and Joshi (IMA J. Numer. Anal. 11, 21–31, 1991), Hernández (Comput. Math. Applic. 41(3–4):433–455, 2001), Hernández and Salanova (Southwest J. Pure Appl. Math. 1, 29–40, 1999), Jarratt (Math. Comput. 20(95):434–437, 1996), Kou and Li (Appl. Math. Comput. 189, 1816–1821, 2007), Li (Appl. Math. Comput. 235, 221–225, 2014), Ren et al. (Numer. Algorithm. 52(4):585–603, 2009), Wang et al. (Numer. Algorithm. 57, 441–456, 2011), Kou et al. (Numer. Algorithm. 60, 369–390, 2012) show convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analyses of these methods are also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.


Chebyshev–Halley–type methods Local convergence Order of convergence Dynamics 

Mathematics Subject Classification (2010)

65D10 65D99 65G99 90C30 


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  1. 1.
    Amat, S., Busquier, S.: Plaza, Dynamics of the King and Jarratt iterations. Aequationes Math. 69(3), 212–223 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amat, S., Busquier, S.: Plaza, Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl. 366(1), 24–32 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Amat, M.A., Hernández, N., Romero, A.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206(1), 164–174 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Argyros, I.K.: Convergence and Application of Newton-type Iterations. Springer (2008)Google Scholar
  5. 5.
    Argyros, I.K, Hilout, S.: Numerical methods in Nonlinear Analysis. World Scientific Publ. Comp, New Jersey (2013)Google Scholar
  6. 6.
    Behl, R.: Development and analysis of some new iterative methods for numerical solutions of nonlinear equations (PhD Thesis). Punjab University (2013)Google Scholar
  7. 7.
    Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci 32, 257–264 (1977)CrossRefGoogle Scholar
  8. 8.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: The Halley method. Computing 169–184, 44 (1990)Google Scholar
  9. 9.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: The Chebyshev method. Computing 45(4), 355–367 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. The Scientific World Journal Volume Article ID 780153 (2013)Google Scholar
  11. 11.
    Chun, C.: Some improvements of Jarratt’s method with sixth-order convergence. Appl Math. Comput 190(2), 1432–1437 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)Google Scholar
  13. 13.
    Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev-Halley type methods. Appl. Math. Comput. 219, 8568–8583 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ezquerro, J. A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41(2), 227–236 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ezquerro, J. A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)CrossRefMATHGoogle Scholar
  17. 17.
    Ezquerro, J. A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Applic. 36(7), 1–8 (1998)CrossRefMATHGoogle Scholar
  19. 19.
    Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Applic. 41(3-4), 433–455 (2001)CrossRefMATHGoogle Scholar
  21. 21.
    Hernández, M.A., Salanova, M.A.: Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1, 29–40 (1999)MATHGoogle Scholar
  22. 22.
    Jarratt, P.: Some fourth order multipoint methods for solving equations. Math. Comput. 20(95), 434–437 (1966)CrossRefMATHGoogle Scholar
  23. 23.
    Kou, J., Li, Y.: An improvement of the Jarratt method. Appl. Math. Comput. 189, 1816–1821 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Li, D., Liu, P., Kou, J.: An improvement of the Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Magreñán, Á. A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Magreñán, Á.A.: A new tool to study real dynamics: The convergence plane. Appl. Math. Comput. 248, 215–224 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Parhi, S.K., Gupta, D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206(2), 873–887 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rall, L.B.: Computational solution of nonlinear operator equations. In: Robert E. Krieger (ed.). New York (1979)Google Scholar
  29. 29.
    Ren, H., Wu, Q., Bi, W.: New variants of Jarratt method with sixth-order convergence. Numer. Algorithm. 52(4), 585–603 (2009)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rheinboldt, W.C.: An adaptive continuation process for solving systems of nonlinear equations. Pol. Acad. Sci., Banach. Ctr. Publ. 3, 129–142 (1978)MathSciNetGoogle Scholar
  31. 31.
    Traub, J.F.: Iterative methods for the solution of equations. Prentice- Hall Series in Automatic Computation Englewood Cliffs, N. J. (1964)Google Scholar
  32. 32.
    Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithm. 57, 441–456 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kou, J., Wang, X.: Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity conditions. Numer. Algorithm. 60, 369–390 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics SciencesCameron UniversityLawtonUSA
  2. 2.Universidad Internacional de La RiojaEscuela de Ingeniería C/Gran Vía 41Logroño (La Rioja)Spain

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