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

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Abstract

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).

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References

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chun, C., Neta, B.: A third-order modification of Newton’s method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  5. Dong, C.: A family of multiopoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)

    Article  MATH  Google Scholar 

  6. Ferrara, M., Sharifi, S., Salimi, M.: Computing multiple zeros by using a parameter in Newton–Secant method. SeMA J. 74, 361–369 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: On some computational orders of convergence. Appl. Math. Lett. 23, 472–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hazrat, R.: Mathematica: A Problem-Centered Approach. Springer, New York (2010)

    Book  MATH  Google Scholar 

  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. Homeier, H.H.H.: On Newton-type methods for multiple roots with cubic convergence. J. Comput. Appl. Math. 231, 249–254 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Comput. Appl. Math. 21, 643–651 (1974)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ostrowski, A.M.: Solution of Equations and Systems of Equations, vol. 9. Academic Press, London (2009)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schroder, E.: Uber unendlich viele algorithmen zur Auflosung der Gliechungen. Math. Ann. 2, 317365 (1870)

    Article  Google Scholar 

  20. Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  25. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)

    MATH  Google Scholar 

  26. Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24, 37–46 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Victory, H.D., Neta, B.: A higher order method for multiple zeros of nonlinear functions. Int. J. Comput. Math. 12, 329–335 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Mehdi Salimi.

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Jamaludin, N.A.A., Nik Long, N.M.A., Salimi, M. et al. Review of some iterative methods for solving nonlinear equations with multiple zeros. Afr. Mat. 30, 355–369 (2019). https://doi.org/10.1007/s13370-018-00650-3

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  • DOI: https://doi.org/10.1007/s13370-018-00650-3

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