Graphical representations for the homogeneous bivariate Newton’s method

Abstract

In this paper we propose a new and effective strategy to apply Newton’s method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid some numerical difficulties, such as divisions by values close to zero. In fact, we consider an iteration map defined on a real augmented projective plane. So, we obtain a global description of the basins of attraction of the fixed points associated to the intersection of the curves. As an application of our techniques, we can plot the basins of attraction of the roots in the following geometric models: hemisphere, hemicube, Möbius band, square and disk. We can also give local graphical representations on any rectangle of the plane.