Algorithms for determining the copositivity of a given symmetric matrix

Abstract Andersson et al. [L.E. Andersson, G. Chang, T. Elfving, Criteria for copositive matrices using simplices and barycentric coordinates, Linear Algebra Appl. 220 (1995) 9–30] gives necessary and sufficient conditions for symmetric matrices of order n ⩽ 5 to be copositive; and Yang and Li [S. Yang, X. Li, Some simple criteria for copositive matrices, in: Proceedings of the Seventh International Conference on Matrix Theory and Applications, Advances in Matrix Theory and Applications, World Academic Union, 2006] gives necessary and sufficient conditions for symmetric Z -matrices of any order to be (strictly) copositive; and some simpler sufficient conditions for symmetric matrices of any order to be copositive or strictly copositive or to be not copositive. Based on these known results we will present six algorithms of determining whether a given symmetric matrix is strictly copositive or copositive or not copositive. The algorithms for matrices of order 3, 4, 5, 6 or 7 are efficient, because they quickly give the complete answer for the test. The algorithm for matrices of order n ⩾ 8 is not guaranteed to produce an answer, but usually does produce one.