Review of some iterative methods for solving nonlinear equations with multiple zeros

  • N. A. A. Jamaludin
  • N. M. A. Nik Long
  • Mehdi SalimiEmail author
  • Somayeh Sharifi


In this paper, some iterative methods with third order convergence for solving the nonlinear equation were reviewed and analyzed. The purpose is to find the best iteration schemes that have been formulated thus far. Hence, some numerical experiments and basin of attractions were performed and presented graphically. Based on the five test functions it was found that the best method is D87a due Dong’s Family method (Int J Comput Math 21:363–367, 1987).


Multi-point iterative methods Multiple roots Basin of attraction 

Mathematics Subject Classification




  1. 1.
    Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Scientia 10, 3–35 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: Improved three-step schemes with high efficiency index and their dynamics. Numer. Algorithms 65, 153–169 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chun, C., Neta, B.: A third-order modification of Newton’s method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dong, C.: A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation. Math. Numer. Sin. 11, 445–450 (1982)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dong, C.: A family of multiopoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ferrara, M., Sharifi, S., Salimi, M.: Computing multiple zeros by using a parameter in Newton–Secant method. SeMA J. 74, 361–369 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: On some computational orders of convergence. Appl. Math. Lett. 23, 472–478 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hazrat, R.: Mathematica: A Problem-Centered Approach. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Heydari, M., Hosseini, S.M., Loghmani, G.B.: Convergence of a family of third-order methods free from second derivatives for finding multiple roots of nonlinear equations. World Appl. Sci. J. 11, 507–512 (2010)Google Scholar
  10. 10.
    Homeier, H.H.H.: On Newton-type methods for multiple roots with cubic convergence. J. Comput. Appl. Math. 231, 249–254 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Comput. Appl. Math. 21, 643–651 (1974)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S.: A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algorithms 68, 261–288 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Matthies, G., Salimi, M., Sharifi, S., Varona, J.L.: An optimal eighth-order iterative method with its dynamics. Jpn. J. Ind. Appl. Math. 33, 751–766 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nik Long, N.M.A., Salimi, M., Sharifi, S., Ferrara, M.: Developing a new family of Newton–Secant method with memory based on a weight function. SeMA J. 74, 503–512 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ostrowski, A.M.: Solution of Equations and Systems of Equations, vol. 9. Academic Press, London (2009)Google Scholar
  17. 17.
    Salimi, M., Lotfi, T., Sharifi, S., Siegmund, S.: Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics. Int. J. Comput. Math. 94, 1759–1777 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Salimi, M., Nik Long, N.M.A., Sharifi, S., Pansera, B.A.: A multi-point iterative method for solving nonlinear equations with optimal order of convergence. Jpn. J. Ind. Appl. Math. 35, 497–509 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schroder, E.: Uber unendlich viele algorithmen zur Auflosung der Gliechungen. Math. Ann. 2, 317365 (1870)CrossRefGoogle Scholar
  20. 20.
    Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sharifi, S., Ferrara, M., Nik Long, N.M.A., Salimi, M.: Modified Potra–Pták method to determine the multiple zeros of nonlinear equations. arXiv:1510.00319 (2015)
  22. 22.
    Sharifi, S., Ferrara, M., Salimi, M., Siegmund, S.: New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations. Open Math. 14, 443–451 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations by a derivative-free form of the Kings family with memory. Calcolo 53, 201–215 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sharifi, S., Salimi, M., Siegmund, S., Lotfi, T.: A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations. Math. Comput. Simul. 119, 69–90 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)zbMATHGoogle Scholar
  26. 26.
    Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24, 37–46 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Victory, H.D., Neta, B.: A higher order method for multiple zeros of nonlinear functions. Int. J. Comput. Math. 12, 329–335 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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
  29. 29.
    Zhou, X., Chen, X., Song, Y.: Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversiti Putra MalaysiaSerdangMalaysia
  2. 2.Institute for Mathematical ResearchUniversiti Putra MalaysiaSerdangMalaysia
  3. 3.Center for Dynamics and Institute for Analysis, Department of MathematicsTechnische Universität DresdenDresdenGermany
  4. 4.MEDAlicsResearch Centre at the University Dante AlighieriReggio CalabriaItaly

Personalised recommendations