4. Conclusion and future work
In this paper, we have presented a general method for detecting the (zero dimensional set of) critical points in the parameter space of a (nonlinear) algebraic equation system F (x¯) = 0, ¯ where F : D ⊂ Rn → Rk, k ≤ n. The method assumes no knowledge of the geometry that gave rise to the equation system, and addresses only the geometry of the zero set of F : the intersection of k hyper-surfaces, given in implicit form by each of the scalar components of F = (f1,..., fk). One potential drawback of the method is that in many applications the parameter space of F is different than the underlying problem’s objective (or model) space: for example, the equations represented by F may have a parameter space that consists of parameters of several geometric objects, as well as other auxiliary variables that measure distances/time/control variables, etc. This means that solution points belonging to the zero set of F are mapped (in a post process) by some “application specific” function, to the actual solution points of interest to the user. It is not uncommon that this post process mapping has degeneracies of its own, and the presented method cannot possibly detect them. Nevertheless, the presented method proved useful in detecting points where a general k × n matrix with multivariate function elements has non-maximal rank. The proper exploitation of these singularity detections in subdivision solvers are to be investigated. First, consider locating all critical points, as a pre-process step of any general equation solving problem. As a result, it is possible to subdivide the problem in a manner that guarantees two types of sub-domains: sub-domains that contain the critical locations, and a set of sub-problems that are strictly regular: admitting no critical points, not even at domain boundaries, bounded away from the singularity. Next, when k < n, correct tessellation of the solution manifold (in general dimension), at the above mentioned singular sub-domains, is a very challenging problem, solution of which shall result in topologically correct “stitching” of the regular areas. Such results are known for the cases of f (x, y) = 0 and f (x, y, z) = 0, as in Bloomenthal and Ferguson (1995), Burr et al. (2008), Hass et al. (2007), Stander and Hart (2005).
