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.
, it cannot guarantee to produce an answer. Table 
of [
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 remaining undetermined when 
, respectively. In this paper we continue our study as in [
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. 
be symmetric and be partitioned into
, 
. As in [
; and let
whose vertices are all vertices of 
. It is proved in [
symmetric matrix 
is copositive if and only if 
for all 
. Consider the polyhedron in 
is the given vector of dimension 
in (1.1)) which has some vertices being vertices of 
, and all the other vertices being in the hyperplane 
. It is known (see [
and Lemma 
]) that the polyhedron 
can be subdivided into 
simplices 
in 
such that 
, 
is a subsimplex of 
and 
if 
, and the vertices of 
are all vertices of 
. We mention this fact since that 
is subdivided into simplices 
.
by 
, then 
is a vertex of 
, or a common point of the line connecting two vertices of 
and the hyperplane 
and should be presented in the barycenter coordinates of 
. If 
is the 
th vertex of 
, then it is represented by the coordinate vector 
with a 1 in the 
th position and all 0's elsewhere; otherwise write 
to denote that it is the common point of line 
and the hyperplane 
. Each 
determines a matrix 
(see [
]), to simplify the notation we still write 
with 
or 
. For example, if 
share only one vertex 
with 
and the other vertices are 
, then
be symmetric and partitioned as in (1.1) with 
, 
being copositive and 
is subdivided into simplices 
which determine matrices 
. Then 
is copositive if and only if 
, 
are all copositive (see [
is strictly copositive if and only if 
, 
are all strictly copositive and 
and 
is strictly copositive (see [
contains 
vertices (coordinate vectors of the standard simplex 
not in the hyperplane 
, then 
contains exact 
vertices in the hyperplane 
, and that 
can be subdivided into 
simplices 
of dimension 
such that 
is a simplex of dimension 
for 
and 
is a simplex of dimension 
when 
.
. If there are 
-triples of pairwise different vertices of 
satisfying the following two conditions:
contains at least one coordinate vector vertex;
has exactly 
vertices for 
, and 
has less than 
vertices when 
,
can be subdivided into 
simplices 
, where 
is the simplex whose vertices are the elements of 
.
, 2.6, 2.7, and 2.8 in [
be symmetric and be partitioned as in (1.1) and 
, then at least one of the following cases must happen: 
principal submatrix of 
is not copositive, then 
is not copositive. Otherwise it holds that 
and 
is copositive.
then 
is copositive; if 
with 
and 
is strictly copositive, then 
is strictly copositive.
, then 
is copositive if and only if 
is copositive; 
is strictly copositive if and only if 
is strictly copositive and 
and 
is strictly copositive.
has exactly one negative entry: 
, then 
is copositive if and only if 
is copositive; 
is strictly copositive if and only if 
is strictly copositive, and 
and 
is strictly copositive, where
has exactly two negative entries:
, and 
, then 
is copositive if and only if 
and 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
has exactly three negative entries: 
and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
has exactly four negative entries: 
and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
has exactly five negative entries: 
and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
has exactly six negative entries: 
and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
]) that if 
is odd, then a copositive matrix 
must have a row with an even number of negative entries. In other words, if a symmetric matrix of odd order has row with an even number of negative entries, then some 
principal submatrices of it are not copositive. This fact will be used in Theorem 2.2. 
is symmetric, then at least one of the following cases must happen: 
principal submatrix of 
is not copositive, then 
is not copositive.
must have a row with an even number of negative entries and 
is copositive) find a row of 
which has exactly 
negative entries. If the 
th row does, then interchange the 
th row and column with the first row and column, and partition 
into (1.1) as in Theorem 2.1.
, then 
and 
is copositive; if 
with 
and 
is strictly copositive, then 
is strictly copositive.
, then 
, then 
is copositive if and only if 
is copositive; 
is strictly copositive if and only if 
is strictly copositive and 
and 
is strictly copositive.
, then 
has exactly two negative entries: 
, and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
, then 
has exactly four negative entries: 
and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
, then 
has exactly six negative entries: 
and 
, then 
is copositive if and only if 
are all copositive; 
is strictly copositive if and only if 
are all strictly copositive and 
and 
is strictly copositive, where
), Cha4(
), Cha5(
), Cha6(
), Cha7(
) and Cha(
), for determining the copositivity of symmetric matrices. Now we have made two more MATLAB functions of these type: Cha8(
) and Cha9(
) based on the two algorithms given by Theorems 2.1 and 2.2. The input of the functions is any 
or 
symmetric matrix 
and there are four possible return values: 
meaning "not copositive", "copositive (not strictly)" and "strictly copositive", "cannot determined," respectively. 
) Find out if 
has any 
principal submatrix which is not copositive. If so, then return with "
" (Theorem 2.1(a)). Otherwise go to next step.
) Calculate the number 
of the negative entries of the first row of 
.
use Theorem 2.1(b) to determine copositivity of 
and return.
use Theorem 2.1(c) to determine copositivity of 
and return.
use Theorem 2.1(d) to determine copositivity of 
and return.
use Theorem 2.1(e) to determine copositivity of 
and return.
use Theorem 2.1(f) to determine copositivity of 
and return.
use Theorem 2.1(g) to determine copositivity of 
and return.
use Theorem 2.1(h) to determine copositivity of 
and return.
use Theorem 2.1(i) to determine copositivity of 
and return.
) Find out if 
has any 
principal submatrix which is not copositive. If so, then return with "
" (Theorem 2.2(a)). Otherwise 
must have some row containing exactly 
negative entries and go to the next step.
) Find out if 
has any row which has exactly 
negative entries. If the 
th row does, then interchange the 
th row and column of 
with the first row and column.
use Theorem 2.2(b) to determine copositivity of 
and return.
use Theorem 2.2(c) to determine copositivity of 
and return.
use Theorem 2.2(d) to determine copositivity of 
and return.
use Theorem 2.2(e) to determine copositivity of 
and return.
use Theorem 2.2(f) to determine copositivity of 
and return.
), Cha4(
), 
Cha8(
), Cha9(
), and Cha(
) we have performed the following experiments. Firstly, we use these functions to determine the copositivity of the 
symmetric matrix 
studied in [
satisfies 
; 
only if 
and 
. When 
the experimental results obtained by old Cha(
) together with Cha3(
), Cha4(
), 
Cha7(
) are "
" meaning "cannot be determined" and the experimental results by Cha9(
) are "
" meaning "copositive but not strictly", which are the same results as obtained in [
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 
, and then use our MATLAB functions to determine the copositivity of these matrices. The main numerical result of the experiments is given in Table 
, 
, 
, 
undeter denote the number of strictly copositive matrices, the number of copositive (but not strictly) matrices, the number of noncopositive matrices, and the number of the remained matrices whose copositivity could not be determined by our algorithms, respectively. 
] proved that a symmetric matrix 
is copositive if and only if the minimum principal submatrix 
of 
which shares the maximum positive diagonal entries with 
is copositive and the matrix which is constructed from 
by replacing each entry of 
by 0 is nonnegative. To answer the third open problem of [
with unit diagonal is copositive if and only if the matrix constructed from 
by replacing each off-diagonal entry 
by 
is copositive. These two results make it reasonable that for determining copositivity we can restrict our attention only to symmetric matrices with unit diagonal and with positive entries all being less than or equal to 1, and our experimental matrices are all of this type. Furthermore, each of the test matrices is required that every of its principal 
submatrix is copositive (Note that for a matrix with 
the chance that every principal 
submatrix is copositive is much less). In addition, the last line of Table 
because of the fact that a symmetric matrix is not copositive if any of its principal submatrix is not copositive. Table 
almost always no random matrix is copositive, in other words, there is almost always no matrix remaining undetermined by our algorithms including the new ones developed in this paper. Therefore, the algorithms for 
and so forth. which might be established by our method are not practically needed.
will be tedious to describe and take more time to run because of its recurrent property.
. For instance, [
mentioned at the beginning of this section is copositive (but not strictly) for any 
. Here we give another interesting example as follows.
be a symmetric matrix of any order, 
; 
be the sum of all the negative entries of the 
th row of 
. Then 
is copositive if 
; 
is strictly copositive if 
; 
is irreducible and 
.
, where 
and 
is the 
nonnegative matrix which shares all the negative (nonnegative) entries with 
and has the remained entries all being zero. Then 
, where 
and 
is a nonnegative matrix whose spectral radius 
if 
. Therefore, 
is an M-matrix if 
; a nonsingular M-matrix if 
or 
is irreducible and 
, whence it is copositive, strictly copositive, respectively by [
]. Finally 
(as the sum of two copositive matrices) is copositive.