Conjugacies classes of some numerical methods
DOI:
https://doi.org/10.4067/S0716-09172001000100001Abstract
We study the dynamics of some numerical root finding methods such as the Newton, Halley, K¨onig and Schröder methods for three and four degree complex polynomials.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 discussion. 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).
[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 discussion. 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).
Published
2017-04-24
How to Cite
[1]
S. Plaza Salinas, “Conjugacies classes of some numerical methods”, Proyecciones (Antofagasta, On line), vol. 20, no. 1, pp. 01-17, Apr. 2017.
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