Abstract
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 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 , there are 8, 6, and 2 matrices remaing undetermined, respectively. In this paper we give two more algorithms for 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.
Highlights
Reference 1 gives six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive but not strictly, or not copositive
We do some discussion based on our experimental results
To answer the third open problem of 4, 5, we proved that a symmetric matrix A with unit diagonal is copositive if and only if the matrix constructed from A by replacing each off-diagonal entry aij by min{aij, 1} is copositive
Summary
Reference 1 gives six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive but not strictly , or not copositive. 1.3 be the standard simplex of order n − 1 whose vertices are all vertices of U It is proved in 2 that an n × n symmetric matrix A is copositive if and only if uT Au ≥ 0 for all u ∈ U. L are all strictly copositive and a11 > 0 and A2 is strictly copositive (see [1]) It is noticed from 2 that if the polyhedron T − ⊆ Rn−1 contains f ≤ n − 1 vertices coordinate vectors of the standard simplex T not in the hyperplane Π {u ∈ Rn−1 : αT u 0}, T − contains exact g f n − 1 − f vertices in the hyperplane Π, and that T − can be subdivided into l f g − n 2 simplices {S1, S2, . These two lemmas are basic for proving Theorems 2.5, 2.6, 2.7, and 2.8 in 1 ; they are basic for proving Theorems 2.1 and 2.2 of this paper
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