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 . We extend the Secant map to the real projective plane . The line at infinity is invariant, and there is one (if k is odd) or two (if k is even) fixed points at . We show that these are of saddle type, and this allows us to better understand the dynamics of the Secant map near infinity.