Servicios Personalizados
Revista
Articulo
Indicadores
-
Citado por SciELO
-
Accesos
Links relacionados
Compartir
Proyecciones (Antofagasta)
versión impresa ISSN 0716-0917
Proyecciones (Antofagasta) v.20 n.1 Antofagasta mayo 2001
http://dx.doi.org/10.4067/S0716-09172001000100001
CONJUGACIES CLASSES OF SOME
NUMERICAL METHODS*
SERGIO PLAZA
Universidad de Santiago, Santiago-Chile
Abstract
We study the dinamics of some numerical root finding methods such as the Newton, Halley, König and Schröder methods for three and four degree complex polynomials.
__________________
* Part of this work was supported by FONDECYT Grants #1970720 and #1961212, and DICYT Grant #9733 P.S.
References
[1] Alexander, D.S. A history of complex dynamics: from Schröder to Fatou and Julia. Vieweg, Aspects of Mathematics (1994).
[2] Argiropoulos, N., Drakopoulos, V., Böhm, A., Julia and Mandelbrot-like sets for higher order König rational iteration functions. Fractal Frontier, M.M. Novak and T.G. Dewey, eds. World Scientific, Singapore, 169-178 (1997).
[3] Argiropoulos, N., Drakopoulos, V., Böhm, A., Generalized computation of Schröder iteration functions to motivate families of Julia and Mandelbrot-like sets. SIAM J. Numer. Anal., Vol. 36, Nº 2, pp. 417-435, (1999).
[4] Arney, D.C. Robinson, B.T. Exhibiting chaos and fractals with a microcomputer. Comput. Math. Applic. Vol 19 (3), pp. 1-11, (1990).
[5] Ben-Israel, A. Newton's method with modified functions. Contemporary Mathematics 204, pp. 39-50, (1997).
[6] Ben-Israel A., Yau, L. The Newton and Halley method for complex roots. The American Mathematical Monthly 105, pp. 806-818, (1998).
[7] Blanchard, P. Complex Analytic Dynamics on the Riemann sphere. Bull. of AMS (new series) Vol. 11, number 1, July, pp. 85-141, (1984).
[8] Blanchard, P., Chiu, A. Complex Dynamics: an informal discusion. Fractal Geometry and Analysis. Eds. J. Bélair & S. Dubuc. Kluwer Academic Publishers, pp. 45-98, (1991).
[9] Cayley, A. The Newton-Fourier Imaginary Problem. Amer. J. Math. 2, 97, (1879).
[10] Cayley, A. On the Newton-Fourier Imaginary Problem. Proc. Cambridge Phil. Soc. 3, pp. 231-232, (1880).
[11] Curry, J.H., Garnett, L., Sullivan, D. On the iteration of a rational function: computer experiment with Newton method. Comm. Math. Phys. 91, pp. 267-277, (1983).
[12] Drakopoulos, V. On the additional fixed points of Schröder iteration function associated with a one-parameter family of cubic polynomilas. Comput. and Graphics, Vol. 22 (5), pp. 629-634, (1998).
[13] Douady A., Hubbard, J.H. On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Norm. Sup. (Paris) 18 (1985), 287-343.
[14] Gilbert, W. Newton's method for multiple roots. Comput. and Graphics, Vol. 18 (2), pp. 227-229, (1994).
[15] Gilbert, W. The complex dynamics of Newton's method for a double root. Computers Math. Applic., Vol. 22 (10), pp. 115-119, (1991).
[16] Emerenko, A., Lyubich, M., Y. The Dynamics of Analytic Transformations. Leningrad Math. J., Vol. 1 (3), pp. 563-634, (1990).
[17] Henrici, P. Applied and Computational Compex Analysis. Wiley, (1974).
[18] Milnor, J. Dynamics in One Complex Dimension: Introductory Lectures. Preprint #1990/5, SUNY StonyBrook, Institute for Mathematical Sciences.
[19] Peitgen, Heinz - Otto, (Ed.) Newton's Method and Dynamical Systems. Kluwer Academic Publishers, (1989).
[20] Schröder, E.O. On infinitely many algorithms for solving equations. Math. Ann. 2 (1870), pp. 317-265. Translated by G. W. Stewart, 1992 (these report is available by anonymous ftp from thales.cs.umd.edu in the directory pub/reports).
[21] Vrscay, E.R. Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study. Mathematics of Computation, Vol. 46 (173), pp. 151-169, (1986).
[22] Vrscay, E.R., Gilbert W.J. Extraneous fixed points, basin boundary and chaotic dynamics for Schröder and König rational iteration functions. Numer. Math. 52, pp. 1-16, (1988).
Received : February 2001.
Sergio Plaza Salinas
Departamento de Matemáticas y Cs. de la Computación
Universidad de Santiago de Chile
Casilla 307
Correo 2
Santiago
Chile
e-mail: splaza@fermat.usach.cl