Dynamics of the Secant map near infinity
We investigate the root finding algorithm given by the secant method applied to a real polynomial $p$ of degree $k$ as a discrete dynamical system defined on $\mathbb R^2$. We extend the secant map to the real projective plane $\mathbb {R P}^2$. The line at infinity $\ell_{\infty}$ is invariant, and...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/189352 |
| Acceso en línea: | https://hdl.handle.net/2445/189352 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria de la bifurcació Sistemes dinàmics diferenciables Anàlisi numèrica Bifurcation theory Differentiable dynamical systems Numerical analysis |
| Sumario: | We investigate the root finding algorithm given by the secant method applied to a real polynomial $p$ of degree $k$ as a discrete dynamical system defined on $\mathbb R^2$. We extend the secant map to the real projective plane $\mathbb {R P}^2$. The line at infinity $\ell_{\infty}$ is invariant, and there is one (if $k$ is odd) or two (if $k$ is even) fixed points at $\ell_{\infty}$. We show that these are of saddle type, and this allows us to better understand the dynamics of the secant map near infinity. |
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