Dynamics of the Secant map near infinity

Abstract

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.