Numerical Algorithms

, Volume 57, Issue 3, pp 329–356 | Cite as

Iteration functions for pth roots of complex numbers

Original Paper

Abstract

A novel way of generating higher-order iteration functions for the computation of pth roots of complex numbers is the main contribution of the present work. The behavior of some of these iteration functions will be analyzed and the conditions on the starting values that guarantee the convergence will be stated. The illustration of the basins of attractions of the pth roots will be carried out by some computer generated plots. In order to compare the performance of the iterations some numerical examples will be considered.

Keywords

Basins of attraction Higher-order convergence Iteration function pth root Residual 

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References

  1. 1.
    Amat, S., Bermúdez, C., Busquier, S., Leauthier, P., Plaza, S.: On the dynamics of some Newton’s type iterative functions. Int. J. Comput. Math. 86(4), 631–639 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bini, D.A., Higham, N.J., Meini, B.: Algorithms for the matrix pth root. Numer. Algorithms 39, 349–378 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11, 85–141 (1984)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Buff, X., Henriksen, C.: On König’s root-finding algorithms. Nonlinearity 16, 989–1015 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht (1974)MATHGoogle Scholar
  6. 6.
    Curry, J.H., Garnett, L., Sullivan, D.: On the iteration of a rational function: computer experiments with Newton’s method. Commun. Math. Phys. 91, 267–277 (1983)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    di Bruno, F.: Traité de l’Élimination. Paris (1859)Google Scholar
  8. 8.
    Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919); Bull. Soc. Math. Fr. 48, 33–94, 208–314 (1920)MathSciNetGoogle Scholar
  9. 9.
    Guo, C.-H., Higham, N.J.: A Schur–Newton method for the matrix pth root and its inverse. SIAM J. Matrix Anal. Appl. 28, 788–804 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Guo, C.-H.: On Newton’s method and Halley’s method for the principal pth root of a matrix. Linear Algebra Appl. 432, 1905–1922 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Householder, A.S.: The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill, New York (1970)MATHGoogle Scholar
  12. 12.
    Iannazzo, B.: On the Newton method for the matrix pth root. SIAM J. Matrix Anal. Appl. 28, 503–523 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Iannazzo, B.: A family of rational iterations and its applications to the computation of the matrix pth root. SIAM J. Matrix Anal. Appl. 30, 1445–1462 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Julia, G.: Memoire sur l’iteration des fonctions rationelles. J. Math. Pures Appl. 8, 47–245 (1918)Google Scholar
  15. 15.
    Lakic̀, S.: On the computation of the matrix k-th root. Z. Angew. Math. Mech. 78, 167–172 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Laszkiewicz, B., Ziȩtak, K.: A Padé family of iterations for the matrix sector function and the matrix pth root. Numer. Linear Algebra Appl. 16, 951–970 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lin, M.: A residual recurrence for Halley’s method for the matrix pth. Linear Algebra Appl. 432, 2928–2930 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Riordan, J.: Combinatorial Identities. Wiley, New York (1968)MATHGoogle Scholar
  19. 19.
    Roberts, G.E., Kobelski, J.H.: Newton’s versus Halley’s method: an approach via complex dynamics. Int. J. Bifurc. Chaos 14, 1–17 (2004)CrossRefGoogle Scholar
  20. 20.
    Schröder, E.: Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870). English translation by G.W. Stewart: E. Schröder, On infinitely many algorithms for solving equations (1993)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Smith, R.A.: Infinite product expansions for matrix n-th roots. J. Aust. Math. Soc. 8, 242–249 (1968)MATHCrossRefGoogle Scholar
  22. 22.
    Traub, J.F.: Iterative Methods for Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)MATHGoogle Scholar
  23. 23.
    Vrscay, E.R.: Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iterative maps: a computer assisted study. Math. Comput. 46, 151–169 (1986)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Instituto Superior de Engenharia de CoimbraCoimbraPortugal
  2. 2.Institute of Systems and RoboticsUniversity of CoimbraCoimbraPortugal
  3. 3.Centro de Matemática da Universidade do PortoPortoPortugal

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