Approximately Singular Systems and Ill-Conditioned Polynomial Systems
By “approximately singular system” we mean a system of multivariate polynomials the dimension of whose variety is increased by small amounts of perturbations. First, we give a necessary condition that the given system is approximately singular. Then, we classify polynomial systems which seems ill-conditioned to solve numerically into four types. Among these, the third one is approximately singular type. We give a simple well-conditioning method for the third type. We test the third type and its well-conditioned systems by various examples, from viewpoints of “global convergence”, “local convergence” and detail of individual computation. The results of experiments show that our well-conditioning method improves the global convergence largely.
Keywordsapproximate ideal approximately linear-dependent relation approximately singular system ill-conditioned polynomial system multivariate Newton’s method well-conditioning
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