Subdivision Termination Criteria in Subdivision Multivariate Solvers

The need for robust solutions for sets of non-linear multivariate constraints or equations needs no motivation. Subdivision-based multivariate constraint solvers [1–3] typically employ the convex hull and subdivision/domain clipping properties of the Bézier/B-spline representation to detect all regions that may contain a feasible solution. Once such a region has been identified, a numerical improvement method is usually applied, which quickly converges to the root. Termination criteria for this subdivision/domain clipping approach are necessary so that, for example, no two roots reside in the same sub-domain (root isolation). This work presents two such termination criteria. The first theoretical criterion identifies sub-domains with at most a single solution. This criterion is based on the analysis of the normal cones of the multiviarates and has been known for some time [1]. Yet, a computationally tractable algorithm to examine this criterion has never been proposed. In this paper, we present such an algorithm for identifying sub-domains with at most a single solution that is based on a dual representation of the normal cones as parallel hyper-planes over the unit hyper-sphere. Further, we also offer a second termination criterion, based on the representation of bounding parallel hyper-plane pairs, to identify and reject sub-domains that contain no solution. We implemented both algorithms in the multivariate solver of the IRIT [4] solid modeling system and present examples using our implementation.