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A Family of Fifth-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

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Abstract

We present a family of fifth-order iterative methods for finding multiple roots of nonlinear equations. Numerical examples are considered to check the validity of the theoretical results. It is shown that the new methods are competitive with other methods for finding multiple roots. Basins of attraction for the new methods and some existing methods to observe the dynamics in the complex plane are drawn.

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REFERENCES

  1. Schröder, E., Über unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann., 1870, vol. 2, pp. 317–365.

    Article  MathSciNet  Google Scholar 

  2. Traub, J.F., Iterative Methods for the Solution of Equations, Englewood Cliffs: Prentice-Hall, 1964.

    MATH  Google Scholar 

  3. Biazar, J. and Ghanbari, B., A New Third-Order Family of Nonlinear Solvers for Multiple Roots, Comput. Math. Appl., 2010, vol. 59, no. 10, pp. 3315–3319.

    Article  MathSciNet  Google Scholar 

  4. Chun, C. and Neta, B., A Third-Order Modification of Newton’s Method for Multiple Roots, Appl. Math. Comput., 2009, vol. 211, no. 2, pp. 474–479.

    Article  MathSciNet  Google Scholar 

  5. Chun, C., Bae, H.J., and Neta, B., New Families of Nonlinear Third-Order Solvers for Finding Multiple Roots, Comput. Math. Appl., 2009, vol. 57, no. 9, pp. 1574–1582.

    Article  MathSciNet  Google Scholar 

  6. Hansen, E. and Patrick, M., A Family of Root Finding Methods,Num. Math., 1976, vol. 27, pp. 257–269.

    Article  MathSciNet  Google Scholar 

  7. Neta, B., New Third Order Nonlinear Solvers for Multiple Roots,Appl. Math. Comput., 2008, vol. 202, pp. 162–170.

    Article  MathSciNet  Google Scholar 

  8. Osada, N., An Optimal Multiple Root-Finding Method of Order Three,J. Comput. Appl. Math., 1994, vol. 51, no. 1, pp. 131–133.

    Article  MathSciNet  Google Scholar 

  9. Sharma, R. and Bahl, A., General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods, Adv. Num. An., 2014, vol. 2014, Article ID 963878; https://doi.org/10.1155/2014/963878.

    Article  MathSciNet  MATH  Google Scholar 

  10. Sharma, J.R. and Sharma, R., Modified Chebyshev–Halley Type Method and Its Variants for Computing Multiple Roots, Num. Algor., 2012, vol. 61, pp. 567–578.

    Article  MathSciNet  Google Scholar 

  11. Dong, C., A Basic Theorem of Constructing an Iterative Formula of the Higher Order for Computing Multiple Roots of an Equation,Math. Num. Sinica, 1982, vol. 11, pp. 445–450.

    MathSciNet  MATH  Google Scholar 

  12. Homeier, H.H.H., On Newton-Type Methods for Multiple Roots with Cubic Convergence, J. Comput. Appl. Math., 2009, vol. 231, no. 1, pp. 249–254.

    Article  MathSciNet  Google Scholar 

  13. Kumar, S., Kanwar, V., and Singh, S., On Some Modified Families of Multipoint Iterative Methods for Multiple Roots of Nonlinear Equations, Appl. Math. Comput., 2012, vol. 218, pp. 7382–7394.

    Article  MathSciNet  Google Scholar 

  14. Li, S., Li, H., and Cheng, L., Some Second-Derivative-Free Variants of Halley’s Method for Multiple Roots, Appl. Math. Comput., 2009, vol. 215, iss. 6, pp. 2192–2198.

    Article  MathSciNet  Google Scholar 

  15. Neta, B., New Third Order Nonlinear Solvers for Multiple Roots,Appl. Math. Comput., 2008, vol. 202, pp. 162–170.

    Article  MathSciNet  Google Scholar 

  16. Victory, H.D. and Neta, B., A Higher Order Method for Multiple Zeros of Nonlinear Functions, Int. J. Comput. Math., 1983, vol. 12, nos. 3/4, pp. 329–335.

    Article  MathSciNet  Google Scholar 

  17. Zhou, X., Chen, X., and Song, Y., Families of Third and Fourth Order Methods for Multiple Roots of Nonlinear Equations, Appl. Math. Comput., 2013, vol. 219, pp. 6030–6038.

    Article  MathSciNet  Google Scholar 

  18. Dong, C., A Family of Multipoint Iterative Functions for Finding Multiple Roots of Equations, Int. J. Comput. Math., 1987, vol. 21, nos. 3/4, pp. 363–367.

    Article  Google Scholar 

  19. Li, S.G., Cheng, L.Z., and Neta, B., Some Fourth-Order Nonlinear Solvers with Closed Formulae for Multiple Roots, Comput. Math. Appl., 2010, vol. 59, no. 1, pp. 126–135.

    Article  MathSciNet  Google Scholar 

  20. Li, S.G., Liao, X., and Cheng, L.Z., A New Fourth-Order Iterative Method for Finding Multiple Roots of Nonlinear Equations,Appl. Math. Comput., 2009, vol. 215, pp. 1288–1292.

    Article  MathSciNet  Google Scholar 

  21. Neta, B., Extension of Murakami’s High-Order Non-Linear Solver to Multiple Roots, Int. J. Comput. Math., 2010, vol. 87, no. 5, pp. 1023–1031.

    Article  MathSciNet  Google Scholar 

  22. Neta, B. and Johnson, A.N., High-Order Nonlinear Solver for Multiple Roots, Comput. Math. Appl., 2008, vol. 55, no. 9, pp. 2012–2017.

    Article  MathSciNet  Google Scholar 

  23. Ostrowski, A.M., Solution of Equations and Systems of Equations, New York: Academic Press, 1966.

    MATH  Google Scholar 

  24. Wolfram, S., The Mathematica Book, 5th ed., Wolfram Media, 2003.

    MATH  Google Scholar 

  25. Jay, L.O., A Note on Q-Order of Convergence, BIT Num. Math., 2001, vol. 41, pp. 422–429.

    Article  MathSciNet  Google Scholar 

  26. Scott, M., Neta, B., and Chun, C., Basin Attractors for Various Methods, Appl. Math. Comput., 2011, vol. 218, pp. 2584–2599.

    Article  MathSciNet  Google Scholar 

  27. Magreñan, Á.A., A New Tool to Study Real Dynamics: The Convergence Plane, Appl. Math. Comput., 2014, vol. 248, pp. 215–224.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, 148106, Longowal, Punjab, India

    J. R. Sharma

  2. Department of Mathematics, Guru Nanak Dev University, 143005, Amritsar, India

    H. Arora

Authors
  1. J. R. Sharma
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  2. H. Arora
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Correspondence to J. R. Sharma or H. Arora.

Additional information

Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2021, Vol. 24, No. 2, pp. 213–227.https://doi.org/10.15372/SJNM20210207.

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Sharma, J.R., Arora, H. A Family of Fifth-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations. Numer. Analys. Appl. 14, 186–199 (2021). https://doi.org/10.1134/S1995423921020075

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