Spherical projective path tracking for homotopy continuation methods

Abstract

Abstract. Solving systems of polynomial equations is an important problem in mathematics with a wide range of applications in many fields. The homotopy continuation method is a large class of reliable and efficient numerical methods for solving systems of polynomial equations. An essential component in the homotopy continuation method is the path tracking algorithm for tracking smooth paths of one real dimension. In this regard, "divergent paths" pose a tough challenge as the tracking of such paths is generally impossible. The existence of such paths is, in part, caused by C n , the space in which homotopy methods usually operate, being non-compact. A well known remedy is to operate inside the complex projective space CP n instead. Path tracking inside CP n is the focus of this article. Taking the Riemannian geometry of CP n into account, we derive the basic algorithm for projective path tracking using the sphere, S 2n+1 , as the model of computation. Remarkable results from numerical experiments using this method are presented.

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

“…Paper [12] introduces a collision-based homotopy continuation technique. Article [214] discusses the problem of divergent homotopy paths and proposes an algorithm which performs projective path tracking. Article [215] describes the software package PHCpack for solving polynomial systems using the homotopy method.…”

GPU-accelerated path tracker for polyhedral homotopy

“…We refer to Refs. [4,7,17,19] for discussions on this topic. 1 Another apparent downside is the use of potentially expensive exponential function.…”

“…To keep the test results consistent, a simplified "Euler-Newton" step is used, in which a single Euler prediction step is followed by a single Newton iteration. 7 In all runs, the batch size is set to 1 4 p, where p is the number of points to be processed simultaneously. This is likely far from optimal for hiding latency.…”

“…Moreover, using Intel MKL (math kernel library), the BLAS-based algorithm proposed here can also be easily ported back to modern CPU that support AVX-512 instructions. 9 Table 3 shows a four-way comparison of these implementations: 7 In actual path trackers, the number of Newton iterations depends on the quality of the prediction. Here, we fix the number of iterations to be exactly 1 so that the timing information from different runs can be compared.…”

Mixed cell computation in Hom4PS-3

“…There is a rich and growing body of works devoted to different aspects in the efficient and stable implementation of the polyhedral homotopy method and homotopy methods in general including tracking of homotopy paths (e.g. [5,6,8,9,14,52]), handling singular end points (e.g. [32,55,56]), certifying results (e.g.…”