A Note on Algorithms for Determining the Copositivity of a Given Symmetric Matrix
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Research Article A Note on Algorithms for Determining the Copositivity of a Given Symmetric Matrix Yang Shang-jun,1 Xu Chang-qing,2 and Li Xiao-xin3 1
School of Mathematical Sciences, Anhui University, Hefei, Anhui, China School of Sciences, Zhejiang Forestry University, Hangzhou, ZheJiang 311300, China 3 Department of Mathematics, Chizhou Institute, Chizhou, Anhui, China 2
Correspondence should be addressed to Xu Chang-qing, [email protected] Received 5 October 2009; Accepted 10 November 2009 Academic Editor: Shusen Ding Copyright q 2010 Yang Shang-jun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the previous paper by the first and the third authors, we present six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive but not strictly, or not copositive. The algorithms for matrices of order n ≥ 8 are not guaranteed to produce an answer. It also shows that for 1000 symmetric random matrices of order 8, 9, and 10 with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to −1, there are 8, 6, and 2 matrices remaing undetermined, respectively. In this paper we give two more algorithms for n 8, 9 and our experiment shows that no such matrix of order 8 or 9 remains undetermined; and almost always no such matrix of order 10 remains undetermined. We also do some discussion based on our experimental results.
1. Introduction Reference 1 gives six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive but not strictly, or not copositive. The algorithms for matrices of order 3, 4, 5, 6 or 7 are efficient. But for matrices of order n ≥ 8, it cannot guarantee to produce an answer. Table 1 of 1 shows that for 1000 symmetric random matrices of order n with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to −1, there are 8, 6, and 2 matrices remaining undetermined when n 8, 9, and 10, respectively. In this paper we continue our study as in 1 and give two algorithms for n 8, 9, and our experiment shows that no such matrix of order 8 or 9 remains undetermined; and almost always no such matrix of order 10 remains undetermined. We also do some discussion based on our experimental results. In this paper we use all the concepts and notations of 1, 2 without explanation. Our main theorems will give the necessary and sufficient conditions for symmetric matrices of order 8 or 9 to be strictly copositive.
2
Journal of Inequalities and Applications Let A ∈ Rn×n be symmetric and be partitioned into A
a11 αT α A2
,
1.1
with a11 ≥ 0, B a11 A2 − ααT . As in 2, let T
n
U
u ∈ R : u u1 , u2 , . . . , un ≥ 0,
n
ui 1
1.2
1
be the simplex of order n − 1; and let T
n−1
u∈R
T
: u
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