Abstract
The paper is devoted to the analysis of certain dynamical properties of a family of iterative Newton type methods used to find roots of non-linear equations. We present a procedure for constructing polynomials in such a way that superattracting cycles of any prescribed length occur when these iterative methods are applied. This paper completes the study begun in Amat, Bermúclez, Busquier, et al., (2009).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amat S, Bermúdez C, Busquier S, et al. On the dynamics of some Newton’s type iterative functions. Int J Comput Math, 2009, 86: 631–639
Barna B. Über die Divergenspunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischen Gleichungen II. Publ Math Debrecen, 1956, 4: 384–397
Bergweiler W. Iteration of meromorphics functions. Bull Amer Math Soc, 1993, 29: 151–188
Blanchard P. Complex analytic dynamics on the Riemann sphere. Bull Amer Math Soc, 1984, 11: 85–141
Cayley A. The Newton-Fourier imaginary problem. Amer J Math, 1879, 2: 97
Cayley A. On the Newton-Fourier imaginary problem. Proc Cambridge Phil Soc, 1880, 3: 231–232
Cosnard M, Masse C. Convergence presque partout de la methode de Newton. C R Math Acad Sci Paris, 1983, 297: 549–552
Curry J H, Garnett L, Sullivan D. On the iteration of a rational function: computer experiment with Newton’s method. Comm Math Phys, 1983, 91: 267–277
Ezquerro J A, Salanova M A. A note on a family of Newton type iterative processes. Int J Comput Math, 1996, 62: 223–232
Hernández M A, Salanova M A. A family of Newton type iterative processes. Int J Comput Math, 1994, 51: 205–214
Hurley M. Attracting orbits in Newton’s method. Trans Amer Math Soc, 1986, 237: 143–158
Peitgen H O, ed. Newton’s Method and Dynamical Systems. Dordrecht: Kluwer Academic Publishers, 1989
Plaza S, Vergara V. Existence of attracting cycles for Newton’s method. Sci Ser A Math Sci (N S), 2001, 7: 31–36
Vrscay E. Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study. Math Comp, 1986, 46: 151–169
Vrscay E, Gilbert G. Extraneous fixed points, basin boundary and chaotic dynamics for Schröder and König rational iteration functions. Numer Math, 1988, 52: 1–16
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amat, S., Busquier, S., Navarro, E. et al. Superattracting cycles for some Newton type iterative methods. Sci. China Math. 54, 539–544 (2011). https://doi.org/10.1007/s11425-010-4141-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-4141-1