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An optimal fourth-order family of methods for multiple roots and its dynamics

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Abstract

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the principle focus of this paper is on developing a new fourth-order optimal family of iterative methods. From the computational point of view, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

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References

  1. Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013 (2013). Article ID 780153

  4. Devaney, R.L.: The mandelbrot set, the farey tree and the fibonacci sequence. Amer. Math. Monthly 106(4), 289–302 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, S., Liao, X., Cheng, L.: A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Neta, B., Scott, M., Chun, C.: Basins attractors for various methods for multiple roots. Appl. Math. Comput. 218, 5043–5066 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Petkovic, M.S., Neta, B., Petkovic, L.D., Dzunic, J.: Multipoint methods for solving nonlinear equations. Academic Press (2013)

  10. Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Schröder, E.: Über unendlichviele Algorithm zur Auffosung der Gleichungen. Math. Annal. 2, 317–365 (1870)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Sharifi, M., Babajee, D.K.R., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Traub, J.F.: Iterative Methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  15. 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 

  16. Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa

    Ramandeep Behl & Sandile S. Motsa

  2. Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022, Valencia, Spain

    Alicia Cordero & Juan R. Torregrosa

  3. University Institute of Engineering and Technology, Panjab University, Chandigarh, 160014, India

    Vinay Kanwar

Authors
  1. Ramandeep Behl
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  2. Alicia Cordero
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  3. Sandile S. Motsa
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  4. Juan R. Torregrosa
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  5. Vinay Kanwar
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Corresponding author

Correspondence to Juan R. Torregrosa.

Additional information

This research was partially supported by Ministerio de Economía y Competitividad MTM2014-52016-C02-02.

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Behl, R., Cordero, A., Motsa, S.S. et al. An optimal fourth-order family of methods for multiple roots and its dynamics. Numer Algor 71, 775–796 (2016). https://doi.org/10.1007/s11075-015-0023-5

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  • DOI: https://doi.org/10.1007/s11075-015-0023-5

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