, Volume 53, Issue 2, pp 201–215 | Cite as

Solving nonlinear equations by a derivative-free form of the King’s family with memory

  • Somayeh SharifiEmail author
  • Stefan Siegmund
  • Mehdi Salimi


In this paper, we present an iterative three-point method with memory based on the family of King’s methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newton’s interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.


Multi-point method Nonlinear equations Method with memory R-order of convergence Kung-Traub’s conjecture 

Mathematics Subject Classification



  1. 1.
    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)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dzunic, J., Petkovic, M.S., Petkovic, L.D.: Three-point methods with and without memory for solving nonlinear equatins. Appl. Math. Comput. 218, 4917–4927 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hazrat, R.: Mathematica: a problem-centered approach. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)CrossRefzbMATHGoogle Scholar
  6. 6.
    King, R.F.: Family of four order methods for nonlinear equations. SIAM. J. Numer. Anal. 10, 876–879 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 634–651 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S.: A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algor. 68, 261–288 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Neta, B.: On a family of multipoint methods for nonlinear equations. Int. J. Computer Math. 9, 353–361 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ostrowski, A.M.: Solution of equations and systems of equations. Academic Press, New York (1966)zbMATHGoogle Scholar
  11. 11.
    Petkovic, M.S., Dzunic, J., Neta, B.: Interpolatory multipoint methods with memory for solving nonlinear equations. Appl. Math. Comput. 218, 2533–2541 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Petkovic, M.S., Neta, B., Petkovic, L.D., Dzunic, J.: Multipoint methods for solving nonlinear equations. Elsevier, Waltham (2013)zbMATHGoogle Scholar
  13. 13.
    Salimi, M., Lotfi, T., Sharifi, S., Siegmund, S.: Optimal newton-secant like methods without memory for solving nonlinear equations with its dynamics. Under Revision, (2015)Google Scholar
  14. 14.
    Sharifi, S. Ferrara, M. Salimi, M. Siegmund, S.: New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations. Under Revision, (2015)Google Scholar
  15. 15.
    Sharifi, S. Salimi, M. Siegmund, S. Lotfi, T.: A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations. Under Revision, (2015)Google Scholar
  16. 16.
    Sharma, J.R., Guha, R.K., Gupa, P.: Some efficient derivative free methods with memory for solving nonlinear equations. Appl. Math. Comput. 219, 699–707 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Traub, J.F.: Iterative methods for the solution of equations. Prentice Hall, New York (1964)zbMATHGoogle Scholar
  18. 18.
    Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  • Somayeh Sharifi
    • 1
    Email author
  • Stefan Siegmund
    • 2
  • Mehdi Salimi
    • 2
  1. 1.Young Researchers and Elite Club, Hamedan BranchIslamic Azad UniversityHamedanIran
  2. 2.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany

Personalised recommendations