Numerical Algorithms

, Volume 58, Issue 4, pp 513–527 | Cite as

Accurate fourteenth-order methods for solving nonlinear equations

  • Parviz Sargolzaei
  • Fazlollah SoleymaniEmail author
Original Paper


We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.


Nonlinear equations Three-step methods Four-step methods Efficiency index Order of convergence Simple root 

Mathematics Subject Classifications (2010)

65H05 65B99 


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  1. 1.
    Chun, C., Ham, Y.: Some sixth-order variants of Ostrowski root finding methods. Appl. Math. Comput. 193, 389–394 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cordero, A., Hueso, J.L., Martinez, M., Torregrosa, J.R.: Efficient three-step iterative methods with sixth order convergence for nonlinear equations. Numer. Algorithms 53, 485–495 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Frontini, M., Sormani, E.: Modified Newton’s method with third order convergence and multiple roots. J. Comput. Appl. Math. 156, 345–354 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Galantai, A., Hegedus, C.J.: A study of accelerated Newton methods for multiple polynomial roots. Numer. Algorithms 54, 219–243 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Geum, Y.H., Kim, Y.I.: A multi-parameter family of three-step eighth-order iterative methods locating a simple root. Appl. Math. Comput. 215, 3375–3382 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Grau-Sanchez, M.: Improving order and efficiency: composition with a modified Newton’s method. J. Comput. Appl. Math. 231, 592–597 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Haijun, W.: On new third-order convergent iterative formulas. Numer. Algorithms 48, 317–325 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kou, J., Li, Y., Wang, X.: A family of fifth-order iterations composed of Newton and third-order methods. Appl. Math. Comput. 186, 1258–1262 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kyurkchiev, N., Iliev, A.: A note on the constructing of nonstationary methods for solving nonlinear equations with raised speed of convergence. Serdica J. Comput. 3, 47–74 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lancaster, P.: Error analysis for the Newton–Raphson method. Numer. Math. 9, 55–68 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Petkovic, M.S.: On a general class of multipoint root-finding methods of high computational efficiency. SIAM J. Numer. Anal. 47, 4402–4414 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Sauer, T.: Numerical Analysis. Pearson, Boston (2006)Google Scholar
  14. 14.
    Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)zbMATHGoogle Scholar
  16. 16.
    Wang, X., Kou, J., Li, Y.: Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wang, X., Liu, L.: Modified Ostrowski’s method with eighth-order convergence and high efficiency index. Appl. Math. Lett. 23, 549–554 (2010)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Yun, B.I., Petkovic, M.S.: Iterative methods based on the Signum function approach for solving nonlinear equations. Numer. Algorithms 52, 649–662 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Sistan and BaluchestanZahedanIran
  2. 2.Young Researchers Club, Zahedan BranchIslamic Azad UniversityZahedanIran

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