First Order Perturbation and Local Stability of Pa

Authors: Daniel Lichtblau

Publish Date: 2016/03/14

Volume: 10, Issue: 1, Pages: 143-163

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Abstract

A problem frequently encountered in geometric constraint solving and related settings is to ascertain sensitivity of solutions arising from a well constrained input configuration This is important for tolerancing and motion planning for example An example would be determining lines simultaneously tangent to four given spheres which originates as a lineofsight problem how much does a perturbation of the input affect the positioning of these lines Once translated to an algebraic setting one has a system of polynomial equations with some coefficients parametrized and wants to determine the solutions and a good approximation of their sensitivity to changes in the parameters We will compute this sensitivity from first order changes in an appropriate Gröbner basis We demonstrate the applicability on several examples We also discuss a more global form of stability wherein one wants to know about perturbations that might change the character of the solution space eg by having fewer than the generic number of solutions

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