Global dynamics of the real secant method

Abstract

We investigate the root finding algorithm given by the secant method applied to a real polynomial $p$ as a discrete dynamical system defined on $\mathbb R^2$. We study the shape and distribution of the basins of attraction associated to the roots of $p$, and we also show the existence of other stable dynamics that might affect the efficiency of the algorithm. Finally we extend the secant map to the punctured torus $\mathbb T^2_{\infty}$ which allow us to better understand the dynamics of the secant method near $\infty$ and facilitate the use of the secant map as a method to find all roots of a polynomial.