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

Abstract

Polynomial systems of equations frequently arise in many applications such as solid modelling, 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.