Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube

Polynomial systems of equations frequently arise in many applications such as solid modeling, robotics, computer vision, chemistry, chemical engineering, and mechanical engineering . Locally convergent iterative methods such as quasi-Newton methods may diverge or fail to find all meaningful solutions of a polynomial system. Recently a homotopy algorithm has been proposed for polynomial systems that is guaranteed globally convergent (always converges from an arbitrary starting point) with probability one, finds all solutions to the polynomial system, and has a large amount of inherent parallelism. There are several ways the homotopy algorithms can be decomposed to run on a hypercube. The granularity of a decomposition has a profound effect on the performance of the algorithm. The results of decompositions with two different granularities are presented. The experiments were conducted on an iPSC-16 hypercube using actual industrial problems

Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces

In this paper, we apply a homotopy algorithm to the problem of finding points in a moving body that lie on specific algebraic surfaces for a given set of spatial configurations of the body. This problem is a generalization of Burmester's determination of points in a body that lie on a circle for five planar positions. We focus on seven surfaces that we term "reachable" because they correspond to serial chains with two degree-of-freedom positioning structures combined with a three degree-of-freedom spherical wrist. A homotopy algorithm based on generalized linear products is used to provide a convenient estimate of the number of solutions of these polynomial systems. A parallelized version of this algorithm was then used to numerically determine all of the solutions