Iteration functions for pth roots of complex numbers

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.

[1]  Gareth E. Roberts,et al.  Newton's versus Halley's Method: a Dynamical Systems Approach , 2004, Int. J. Bifurc. Chaos.

[2]  Minghua Lin.  A residual recurrence for Halley’s method for the matrix pth root , 2010 .

[3]  NICHOLAS J. HIGHAM,et al.  A SCHUR–NEWTON METHOD FOR THE MATRIX PTH ROOT AND ITS INVERSE∗ , 2005 .

[4]  L. Comtet,et al.  Advanced Combinatorics: The Art of Finite and Infinite Expansions , 1974 .

[5]  Carlos Aguirre,et al.  A Panoramic View of Some perturbed Nonlinear Wave Equations , 2004, Int. J. Bifurc. Chaos.

[6]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[7]  G. Julia.  Mémoire sur l'itération des fonctions rationnelles , 1918 .

[8]  Nicholas J. Higham,et al.  Algorithms for the matrix pth root , 2005, Numerical Algorithms.

[9]  R. A. Smith,et al.  Infinite product expansions for matrix n-th roots , 1968, Journal of the Australian Mathematical Society.

[10]  Krystyna Zietak,et al.  A Padé family of iterations for the matrix sector function and the matrix pth root , 2009, Numer. Linear Algebra Appl..

[11]  Edward R Vrscay.  Julia sets and mandelbrot-like sets associated with higher order Schro¨der rational iteration functions: A computer assisted study , 1986 .

[12]  P. Fatou,et al.  Sur les équations fonctionnelles , 1920 .

[13]  Ernst Schröder,et al.  Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen , 1870 .

[14]  Sergio Amat,et al.  On the dynamics of some Newton's type iterative functions , 2009, Int. J. Comput. Math..

[15]  John Riordan,et al.  Inverse Relations and Combinatorial Identities , 1964 .

[16]  A. Householder.  The numerical treatment of a single nonlinear equation , 1970 .

[17]  Nicholas J. Higham,et al.  A Schur-Newton Method for the Matrix \lowercase{\boldmathp}th Root and its Inverse , 2006, SIAM J. Matrix Anal. Appl..

[18]  P. Blanchard.  Complex analytic dynamics on the Riemann sphere , 1984 .

[19]  Bruno Iannazzo,et al.  On the Newton Method for the Matrix P th Root , 2006, SIAM J. Matrix Anal. Appl..

[20]  Slobodan Lakić.  On the Computation of the Matrix k-th Root , 1998 .

[21]  Chun-Hua Guo,et al.  On Newton’s method and Halley’s method for the principal pth root of a matrix , 2010 .

[22]  Dennis Sullivan,et al.  On the iteration of a rational function: Computer experiments with Newton's method , 1983 .

[23]  Bruno Iannazzo,et al.  A Family of Rational Iterations and Its Application to the Computation of the Matrix pth Root , 2008, SIAM J. Matrix Anal. Appl..

[24]  Christian Henriksen,et al.  On König's root-finding algorithms* , 2003 .