题目：            A Numerical Method for Computing Border Curves of  Bi-parametric Real Polynomial Systems and Applications

报告人：       陈长波（中国科学院重庆绿色智能技术研究院）

时间地点：    2016.08.31  16:00pm  N420

摘要：            For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. The curve divides the rectangular region into connected subsets such that in each subset the real zero set of the polynomial system defines finitely many smooth functions, whose graphs are disjoint.  We propose a numerical method to compute the border curve, which is generated by the projection of a curve from a higher dimensional space and represented by polylines. A numerical error estimation of the computed border curve is provided. The border curve enables us to construct a so-called ``solution map", which can be used to describe the real solutions of a parametric system and count their numbers depending on different values of parameters. Moreover, for a given value $u$ of the parameters inside the rectangle but not on the border, the solution map tells the subset that $u$ belongs to together with a connected path from the corresponding sample point $w$ to $u$. Consequently, all the real solutions of the system at $u$ (which are isolated) can be obtained by tracking a real homotopy starting from all the real roots at $w$ throughout the path. The effectiveness of the proposed method is illustrated by some examples.