A Hybrid Procedure for Finding Real Points on a Real Algebraic Set

Abstract

Motivated by the idea of Shen, et al.’s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis, this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method first constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al. (2017). For each root returned by the homotopy continuation method, which is a complex approximation of some (complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and finally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples.