Advertisement

Numerical Algorithms

, Volume 73, Issue 1, pp 141–156 | Cite as

Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family

  • Beatriz Campos
  • Alicia Cordero
  • Juan R. Torregrosa
  • Pura VindelEmail author
Original Paper

Abstract

The study of the dynamical behaviour of the operators defined by iterative methods help us to know more deeply the regions where these methods have a good performance. In this paper, we follow the dynamical study of a multipoint variant of the known Chebyshev-Halley’s family, showing the existence of attractive periodic orbits of period 2 for some values of the parameter.

Keywords

Iterative methods Complex dynamics Chebyshev-Halley’s family 2-periodic orbits 2-bulbs 

Mathematics Subject Classification (2010)

37F10 65H05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beardon, A.F.: Iteration of Rational Functions, Graduate Texts in Mathematics. Springer-Verlag, New York (1991)CrossRefGoogle Scholar
  3. 3.
    Behl, R., Kanwar, V.: Variants of Chebyshev’s method with optimal order of convergence. Tamsui Oxf. J. Inf. Math. Sci. 29(1), 39–53 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Campos, B., Cordero, A., Magreñan, A., Torregrosa, J.R., Vindel, P.: Study of a bi-parametric family of iterative methods. Abstr. Appl. Anal. 2014. Art. ID 141643, 12 ppGoogle Scholar
  5. 5.
    Campos, B., Cordero, A., Torregrosa, J.R., Vindel, P.: Bifurcations in the dynamics of a variant of Chebyshev method. In: Proceedings of the 15th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2015, ISBN 978-84-617-2230-3, pp. 291–299 (2015)Google Scholar
  6. 6.
    Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. The Scientific World Journal Volume 2013 Article ID 780153Google Scholar
  7. 7.
    Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intelligencer 24, 37–46 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cordero, A., Torregrosa, J.P., Vindel, P.: Dynamics of a family of Chebyshev-Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233, 2688–2695 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equation. App. Math. Comput. 227, 567–592 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Instituto de Matemáticas y Aplicaciones de CastellónUniversitat Jaume ICastellónSpain
  2. 2.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValènciaSpain

Personalised recommendations