SciELO - Scientific Electronic Library Online

 
vol.20 número1A MULTIPLIER GLIDING HUMP PROPERTY FOR SEQUENCE SPACES índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

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        

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons