Efficient endgames and path tracking methods

Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. This article considers two ways in which path tracking can be made more efficient. First, a novel parallel algorithm for performing endgames at the end of homotopy paths, based on the Cauchy endgame, is presented, along with some heuristics useful in its implementation. Second, the effects of the choice of predictor method on the performance of a tracker is analyzed, along with notes on how to employ Runge-Kutta methods in conjunction with adaptive precision. These ideas have been implemented in the Bertini software package with several examples provided.