Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere

  • 104 Accesses

  • 2 Citations


In this paper, we present dynamical viewpoints under the Möbius conjugacy map on the Riemann sphere for a uniparametric family of optimal eighth-order multiple-root finders. Various conjugacy properties are investigated including the invariance of the fixed point and its multiplier, which enables the family of iterative methods to trace along the consistent orbits in position. The parameter spaces and dynamical planes are studied and illustrated to visualize the periodic components and their geometric properties. Both theoretical and computational analyses along with a numerical algorithm are carried out regarding the bifurcation points of satellite and primitive components.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    Ahlfors, L.V.: Complex Analysis. McGraw-Hill Book, Inc. (1979)

  2. 2.

    Beardon, A.F.: Iteration of Rational Functions. Springer, New York (1991)

  3. 3.

    Ainsworth, J., Dawson, M., Pianta, J., Warwick, J.: The Farey Sequence, (2012)

  4. 4.

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

  5. 5.

    Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aeq. Math. 69, 212–223 (2005)

  6. 6.

    Argyros, I.K., Magreñán, A.́A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)

  7. 7.

    Behl, R., Cordero, A., Motsa, S., Torregrosa, J.: On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 265, 520–532 (2015)

  8. 8.

    Behl, R., Cordero, A., Motsa, S., Torregrosa, J., Kanwar, V.: An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algor. 71, 775–796 (2016)

  9. 9.

    Blanchard, P.: The dynamics of Newton’s method. In: Proceedings of Symposia in Applied Mathematics, vol. 49, pp 139–154. American Mathematical Society (1994)

  10. 10.

    Campos, B., Cordero, A., Torregrosa, J., Vindel, P.: Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family. Numer. Algor. 73, 141–156 (2016)

  11. 11.

    Chicharro, F., Cordero, A., Gutiérrez, J., Torregrosa, J.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 7023–7035 (2013)

  12. 12.

    Chicharro, F., Cordero, A., Torregrosa, J.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013(780153), 1–11 (2013)

  13. 13.

    Chun, C., Lee, M.Y., Neta, B., Dz̆unić, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)

  14. 14.

    Chun, C., Neta, B., Kim, S.: On Jarratt’s family of optimal fourth-order iterative methods and their dynamics. Fractals 22, 1450013 (2014).

  15. 15.

    Cordero, A., García-Maimó, J., Torregrosa, J., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)

  16. 16.

    Devaney, R.L.: Complex dynamical systems: the mathematics behind the Mandelbrot and Julia sets. Proc. Symposia Appl. Math. ISSN 0160–7634(49), 1–29 (1994)

  17. 17.

    Geum, Y.H., Kim, Y.I.: Cubic convergence of parameter-controlled Newton-secant method for multiple zeros. J. Comput. Appl. Math. 233(4), 931–937 (2009)

  18. 18.

    Geum, Y.H., Kim, Y.I.: A two-parameter family of fourth-order iterative methods with optimal convergence for multiple zeros. J. Appl. Math. 2013(369067), 1–7 (2013)

  19. 19.

    Geum, Y.H., Kim, Y.I., Neta, B.: Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. J. Comput. Appl. Math. 333, 131–156 (2018)

  20. 20.

    Geum, Y.H., Kim, Y.I., Magreñán, Á.A.: A study of dynamics via Möbius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions. J. Comput. Appl. Math. 344, 608–623 (2018)

  21. 21.

    Gulick, D: Encounters with Chaos. McGraw-Hill Inc (1992)

  22. 22.

    Hinich, V.: Riemann surfaces,

  23. 23.

    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Publishing Company (1973)

  24. 24.

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

  25. 25.

    Lipschutz, S.: Theory and Problems of General Topology, Schaum’s Outline Series. McGraw-Hill Inc (1965)

  26. 26.

    Magreñán, A.́A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)

  27. 27.

    Magreñán, A.́ A., Gutíerrez, J.M.: Real dynamics for damped Newton’s method applied to cubic polynomials. J. Comput. Appl. Math. 275, 527–538 (2015)

  28. 28.

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

  29. 29.

    García-Olivo, M., Gutíerrez, J.M., Magreñán, A.́A.: A complex dynamical approach of Chebyshev’s method. SeMA 71, 57–68 (2015)

  30. 30.

    Peitgen, H., Richter, P: The Beauty of Fractals. Springer (1986)

  31. 31.

    Vazquez-Lozano, J.E., Cordero, A., Torregrosa, J.: Dynamical analysis on cubic polynomials of Damped Traubs method for approximating multiple roots. Appl. Math. Comput. 328, 82–99 (2018)

  32. 32.

    Wang, X., Zhang, T., Qinc, Y.: Efficient two-step derivative-free iterative methods with memory and their dynamics. Int. J. Comput. Math. 93, 1423–1446 (2016)

  33. 33.

    Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media (2003)

  34. 34.

    Zafar, F., Cordero, A., Sultana, S., Torregrosa, J.: Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics. J. Comput. Appl. Math. 342, 352–374 (2018)

  35. 35.

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

Download references

Author information

Correspondence to Young Ik Kim.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lee, M., Kim, Y.I. The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere. Numer Algor 83, 1063–1090 (2020).

Download citation


  • Parameter space
  • Möbius map
  • Bifurcation point
  • Multiple-root
  • Eighth-order
  • Conjugacy

Mathematics Subject Classification (2010)

  • 65H05
  • 65H99
  • 41A25
  • 65B99