Dynamics of the Secant map near infinity

Abstract

IWe 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 (Formula presented.). We extend the Secant map to the real projective plane (Formula presented.). The line at infinity (Formula presented.) is invariant, and there is one (if k is odd) or two (if k is even) fixed points at (Formula presented.). We show that these are of saddle type, and this allows us to better understand the dynamics of the Secant map near infinity. © 2022 Informa UK Limited, trading as Taylor & Francis Group.