Skip to main content
SpringerLink
Log in

Influence of the multiplicity of the roots on the basins of attraction of Newton’s method

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this work, we develop and implement two algorithms for plotting and computing the measure of the basins of attraction of rational maps defined on the Riemann sphere. These algorithms are based on the subdivisions of a cubical decomposition of a sphere and they have been made by using different computational environments. As an application, we study the basins of attraction of the fixed points of the rational functions obtained when Newton’s method is applied to a polynomial with two roots of multiplicities m and n. We focus our attention on the analysis of the influence of the multiplicities m and n on the measure of the two basins of attraction. As a consequence of the numerical results given in this work, we conclude that, if m > n, the probability that a point in the Riemann Sphere belongs to the basin of the root with multiplicity m is bigger than the other case. In addition, if n is fixed and m tends to infinity, the probability of reaching the root with multiplicity n tends to zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)

    MATH  MathSciNet  Google Scholar 

  2. Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)

    MATH  Google Scholar 

  3. Beardon, A.F.: Iteration of Rational Functions. Springer-Verlag, New York (2000)

    MATH  Google Scholar 

  4. García-Calcines, J.M., Hernández, L.J., Rivas, M.T.: Limit and end functors of dynamical systems via exterior spaces. In: To appear in Bulletin of the Belgium Math. Soc.- Simon Stevin, vol. 21 (2014)

  5. Gilbert, W.J.: The complex dynamics of Newton’s method for a double root. Comput. Math. Appl. 22(10), 115–119 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Górski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen, F.K., Reinecke, M., Bartelman, M.: HEALPix–a framework for high resolution discretization, and fast analysis of data distributed on the sphere. Astrophys. J. 622, 759–771 (2005)

    Article  Google Scholar 

  7. Hernández, L.J., Marañón, M., Rivas, M.T.: Plotting basins of end points of rational maps with Sage. Tbilisi Math. J. 5(2), 71–99 (2012)

    MATH  Google Scholar 

  8. Peitgen, H.O., Jürgens, H., Saupe, D.: Cayley’s problem and Julia sets. Math. Intell. 6(2), 11–20 (1984)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24(1), 37–46 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Xingyuan, W., Bo, L.: Julia sets of the Schr{”oder iteration functions of a class of one-parameter polynomials with high degree. Appl. Math. Comput. 178(2), 461–473 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Xingyuan, W., Wang, T.: Julia sets of generalized Newton’s method. Fractals 15(4), 323–336 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Xingyuan, W., Wei, L.: The Julia set of Newton’s method for multiple root. Appl. Math. Comput. 172(1), 101–110 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Xingyuan, W., Xuejing, Y.: Julia sets for the standard Newton’s method, Halley’s method, and Schröder’s method. Appl. Math. Comput. 189(2), 1186–1195 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Xingyuan, W., Xuejing, Y.: Julia set of the Newton transformation for solving some complex exponential equation. Fractals 17(2), 197–204 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Xingyuan, W., Li, Y.-K., Sun, Y.-Y., Song, J.-M., Ge, F.-D.: Julia sets of Newton’s method for a class of complex-exponential function F(z) = P(z)e Q(z). Nonlinear Dyn. 62(4), 955–966 (2010)

    Article  Google Scholar 

  17. Yang, W.: Symmetries in the Julia sets of Newton’s method for multiple roots. Appl. Math. Comput. 217, 2490–2494 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Yershova, A., Jain, S., LaValle, S.M., Mitchell, J.C.: Generating Uniform Incremental Grids on SO(3) using the Hopf Fibration. Int. J. Robot. Res. 29(7), 801–812 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas y Computación, Universidad de La Rioja, c/ Luis de Ulloa s/n, 26004, Logroño, Spain

    José M. Gutiérrez, Luis J. Hernández-Paricio, Miguel Marañón-Grandes & M. Teresa Rivas-Rodríguez

Authors
  1. José M. Gutiérrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis J. Hernández-Paricio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel Marañón-Grandes
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Teresa Rivas-Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José M. Gutiérrez.

Additional information

The authors acknowledge the financial help given by the projects: MTM2011-28636-C02-01 of the Spanish Ministry of Science and Technology and API12/10 of the University of La Rioja. The third author has been partially supported by a FPI grant from the Comunidad Autónoma de La Rioja.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gutiérrez, J.M., Hernández-Paricio, L.J., Marañón-Grandes, M. et al. Influence of the multiplicity of the roots on the basins of attraction of Newton’s method. Numer Algor 66, 431–455 (2014). https://doi.org/10.1007/s11075-013-9742-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9742-7

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation