Numerical Algorithms

, Volume 62, Issue 2, pp 253–260 | Cite as

On an Aitken–Newton type method

  • Ion Păvăloiu
  • Emil Cătinaş
Original Paper


We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is \(\sqrt[5]{6},\) which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.


Nonlinear equations Aitken method Newton method Monotone convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amat, S., Busquier, S.: A two step Steffenssen’s method under modified convergence conditions. J. Math. Anal. Appl. 324, 1084–1092 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beltyukov, B.A.: An analogue of the Aitken–Steffensen method with controlled step. URSS Comput. Math. Math. Phys. 27(3), 103–112 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chun, C.: A geometric construction of iterative formulas of order three. Appl. Math. Lett. 23, 512–516 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method for functions of several variables. Appl. Math. Comput. 183, 199–2008 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cordero A., Torregrosa R.J.: A class of Steffenssen type method with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hongmin, R., Qingbio, W., Welhong, B.: A class of two-step Steffensen type methods with fourth order convergence. Appl. Math. Comput. 209(2), 206–210 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Jain, P.: Steffensen type method for solving non-linear equations. Appl. Math. Comput. 194, 527–533 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)MATHGoogle Scholar
  10. 10.
    Ostrowski, M.A.: Solution of Equations in Euclidian and Banach Spaces. Academic Press, New York and London (1973)Google Scholar
  11. 11.
    Păvăloiu, I.: Approximation of the root of equations by Aitken–Steffensen-type monotonic sequences. Calcolo 32, 69–82 (1995)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Păvăloiu, I., Cătinaş, E.: On a Steffensen–Hermite method of order three. Appl. Math. Comput. 215(7), 2663–2672 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Păvăloiu, I., Cătinaş, E.: On a Steffensen type method. In: Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Proceedings, pp. 369–375 (2007)Google Scholar
  14. 14.
    Păvăloiu, I.: Optimal algorithms, concerning the solving of equations by interpolation. In: Popoviciu, E. (ed.) Research on Theory of Allure, Approximation, Convexity and Optimization, Editura Srima, Cluj-Napoca, pp. 222–248 (1999)Google Scholar
  15. 15.
    Păvăloiu, I.: Aitken–Steffensen-type method for nondifferentiable functions (I). Rev. Anal. Numér. Theor. Approx. 31(1), 109–114 (2002)MATHGoogle Scholar
  16. 16.
    Păvăloiu, I.: Aitken–Steffensen-type method for nonsmooth functions (II). Rev. Anal. Numér. Théor. Approx. 31(2), 195–198 (2002)MATHGoogle Scholar
  17. 17.
    Păvăloiu, I.: Aitken–Steffensen-type method for nonsmooth functions (III). Rev. Anal. Numér. Théor. Approx. 32(1), 73–78 (2003)MATHGoogle Scholar
  18. 18.
    Quan, Z., Peng, Z., Li, Z., Wenchao, M.: Variants of Steffensen secant method and applications. Appl. Math. Comput. 216(12), 3486–3496 (2010)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Sharma, R.J.: A composite thid order Newton–Steffensen method for solving nonlinear equations. Appl. Math. Comput. 169, 242–246 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Traub, Y.F.: Iterative Method for Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)Google Scholar
  21. 21.
    Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.“Tiberiu Popoviciu” Institute of Numerical AnalysisRomanian AcademyCluj-NapocaRomania

Personalised recommendations