On the role of singularities in Branin's method from dynamic and continuation perspectives

摘要

A global analysis of Branin's method (originally due to Davidenko) for finding all the real zeros of a vector function is carried out. The analysis is based on a global study of this method perceived of as a dynamical system. Since Branin's algorithm is closely related to homotopy methods, this paper sheds some light on the global performance of these methods when employed for locating all the zeros of a vector function. Following the dynamical system approach, the performance of Branin's algorithm is related to the existence of extraneous singularities as well as to the relative spatial distribution of the zeros of the vector function and singular manifolds. Branin's conjectures regarding the types and the role of extraneous singularities are examined and counter-examples are provided to disprove them. We conclude that the performance of Branin's method for locating all the zeros of a vector function is questionable even in the absence of extraneous singularities.