Characterization of basin boundaries in Bairstows iterative methods
Abstract
In this paper we consider a two-dimensional map with a denominator which can vanish, obtained by applying Bairstow's method, an iterative algorithms to find the real roots of a polynomial based on Newton's method. The complex structure of the basins of attraction of the fixed points is related to the existence of singularities specific to maps with a vanishing denominator, such as sets of non definition, focal points and prefocal curves.
References
1. L. Bairstow, Investigations related to the stability of the aeroplane, Repts. & Memo. No. 154 Advis. Comm. Aeronaut. (1914).
2. L. Billings, J.H. Curry, On noninvertible maps of the plane: Eruptions, Chaos 6 (1996), 108-119.
3. L. Billings, J.H. Curry, B. Phipps, Lyapunov exponents, singularities, and a riddling bifurcation, Phys. Rew. Letters 79 (1997), 1018-1021.
4. G.I. Bischi, L. Gardini, Basin fractalization due to focal points in a class of triangular maps, International Journal of Bifurcation & Chaos 7 (1997), 1555-1577.
5. G.I. Bischi, L. Gardini, Focal points and basin fractalization in some rational maps, Grazer Math. Ber. (to appear).
6. G.I. Bischi, L. Gardini, C. Mira, Maps with denominator. Part 1: some generic properties, International Journal of Bifurcation & Chaos 9 (1999), 119-153.
7. G.I. Bischi, L. Gardini, C. Mira, New phenomena related to the presence of focal points in two-dimensional maps, this issue, 81-89.
8. S.L. Blish, J.H. Curry, On the geometry of Factorization algorithms, Lectures Notes in Applied Mathematics 26 (1990), 47-60.
9. D.W. Boyd, Nonconvergence in Bairstow's method, SIAM J. of Nonlinear Analysis 14 (1977), 571-574.
10. L. Gardini, G.I. Bischi, Focal points and focal values in rational maps, Grazer Math. Ber. (to appear).
11. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, Singapore 1996.
12. C. Mira, Some properties of two-dimensional (Z_0 - Z_2) maps not defined in the whole plane, Grazer Math. Ber. (to appear).
13. J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Pub. Co., New York 1982.
2. L. Billings, J.H. Curry, On noninvertible maps of the plane: Eruptions, Chaos 6 (1996), 108-119.
3. L. Billings, J.H. Curry, B. Phipps, Lyapunov exponents, singularities, and a riddling bifurcation, Phys. Rew. Letters 79 (1997), 1018-1021.
4. G.I. Bischi, L. Gardini, Basin fractalization due to focal points in a class of triangular maps, International Journal of Bifurcation & Chaos 7 (1997), 1555-1577.
5. G.I. Bischi, L. Gardini, Focal points and basin fractalization in some rational maps, Grazer Math. Ber. (to appear).
6. G.I. Bischi, L. Gardini, C. Mira, Maps with denominator. Part 1: some generic properties, International Journal of Bifurcation & Chaos 9 (1999), 119-153.
7. G.I. Bischi, L. Gardini, C. Mira, New phenomena related to the presence of focal points in two-dimensional maps, this issue, 81-89.
8. S.L. Blish, J.H. Curry, On the geometry of Factorization algorithms, Lectures Notes in Applied Mathematics 26 (1990), 47-60.
9. D.W. Boyd, Nonconvergence in Bairstow's method, SIAM J. of Nonlinear Analysis 14 (1977), 571-574.
10. L. Gardini, G.I. Bischi, Focal points and focal values in rational maps, Grazer Math. Ber. (to appear).
11. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, Singapore 1996.
12. C. Mira, Some properties of two-dimensional (Z_0 - Z_2) maps not defined in the whole plane, Grazer Math. Ber. (to appear).
13. J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Pub. Co., New York 1982.
GardiniL., BischiG.-I., & Fournier-PrunaretD. (1999). Characterization of basin boundaries in Bairstows iterative methods. Annales Mathematicae Silesianae, 13, 119-130. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14142
Laura Gardini
gardini@econ.uniurb.it
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Gian-Italo Bischi
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Daniele Fournier-Prunaret
GESNLA-LESIA-INSA, Complexe Scientifique de Rangueil, France France
GESNLA-LESIA-INSA, Complexe Scientifique de Rangueil, France France
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
