SeMA Journal

, Volume 74, Issue 4, pp 361–369 | Cite as

Computing multiple zeros by using a parameter in Newton–Secant method

  • Massimiliano Ferrara
  • Somayeh Sharifi
  • Mehdi SalimiEmail author


In this paper, we modify the Newton–Secant method with third order of convergence for finding multiple roots of nonlinear equations. This method requires two evaluations of the function and one evaluation of its first derivative per iteration. This method has the efficiency index equal to \(3^{\frac{1}{3}}\approx 1.44225\). We describe the analysis of the proposed method along with numerical experiments including comparison with existing methods. Moreover, the attraction basins of the proposed method are shown and compared with other existing methods.


Multi-point iterative methods Newton–Secant method  Multiple roots Basin of attraction 

Mathematics Subject Classification



  1. 1.
    Amat, S., Busquier, S., Gutiérrez, J.M.: On the local convergence of secant-type methods. Int. J. Comput. Math. 81, 1153–1161 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. SCIENTIA Ser. A Math. Sci. 10, 3–35 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amat, S., Busquirer, S., Plaza, S.: Dynamics of a family of third-order iterative methods that do not require using second derivatives. Appl. Math. Comput. 154, 735–746 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Amat, S., Busquier, S.: On a higher order Secant method. Appl. Math. Comput. 141, 321–329 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Amat, S., Busquier, S.: A modified Secant method for semismooth equations. Appl. Math. Lett. 16, 877–881 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Amat, S., Busquier, S.: On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177, 819–823 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Amat, S., Busquier, S.: A two-step Steffensen’s method under modified convergence conditions. J. Math. Anal. Appl. 324, 1084–1092 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Argyros, I.K., George, S.: Ball comparison between two optimal eight-order methods under weak conditions. SeMA J. 72, 1–11 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On improved three-step schemes with high efficiency index and their dynamics. Numer. Algorithms 65, 153–169 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chun, C., Bae, H.J., Neta, B.: New families of nonlinear third-order solvers for finding multiple roots. Comput. Math. Appl. 57, 1574–1582 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ezquerro, J.A., Grau-Sánchez, M., Hernández-Verón, M.A., Noguera, M.: A study of optimization for Steffensen-type methods with frozen divided differences. SeMA J. 70, 23–46 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    García-Olivo, M., Gutiérrez, J.M., Magreãn, Á.A.: A complex dynamical approach of Chebyshev’s method. SeMA J. 71, 57–68 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hansen, E., Patrik, M.: A family of root finding methods. J. Numer. Math. 27, 257–269 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hazrat, R.: Mathematica: A Problem-Centered Approach. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Osada, N.: An optimal multiple root finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ostrowski, A.M.: Solution of Equations and Systems of Equations, 2nd edn. Academic, New York (1966)zbMATHGoogle Scholar
  20. 20.
    Petkovic, M.S., Neta, B., Petkovic, L.D., Dzunic, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Waltham (2013)zbMATHGoogle Scholar
  21. 21.
    Schroder, E.: Uber unendlich viele algorithmen zur Auflosung der Gliechungen. Mathematische Annalen 2, 317–365 (1870)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations by a derivative-free form of the King’s family with memory. Calcolo (2015). doi: 10.1007/s10092-015-0144-1 zbMATHGoogle 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 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

Copyright information

© Sociedad Española de Matemática Aplicada 2016

Authors and Affiliations

  1. 1.Department of Law and EconomicsUniversity Mediterranea of Reggio CalabriaReggio CalabriaItaly
  2. 2.MEDAlics, Research Centre at the University Dante AlighieriReggio CalabriaItaly
  3. 3.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany

Personalised recommendations