Open weak CAD and its applications. (May 2017)

• Record Type: Journal Article
• Authors: Han, Jingjun; Dai, Liyun; Hong, Hoon; Xia, Bican
• Is Part Of: Journal of symbolic computation. Volume 80:Part 3(2017)
• Page Start: 785
• Page End: 816
• Publication Date: 2017-05
• Subjects: Open weak CAD -- Open weak delineable -- CAD projection -- Semi-definiteness -- Copositivity; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.07.032
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The concept of open weak CAD is introduced. Every open CAD is an open weak CAD. On the contrary, an open weak CAD is not necessarily an open CAD. An algorithm for computing projection polynomials of open weak CADs is proposed. The key idea is to compute the intersection of projection factor sets produced by different projection orders. The resulting open weak CAD often has smaller number of sample points than open CADs. The algorithm can be used for computing sample points for all open connected components of f ≠ 0 for a given polynomial f . It can also be used for many other applications, such as testing semi-definiteness of polynomials and copositive problems. In fact, we solved several difficult semi-definiteness problems efficiently by using the algorithm. Furthermore, applying the algorithm to copositive problems, we find an explicit expression of the polynomials producing open weak CADs under some conditions, which significantly improves the efficiency of solving copositive problems.