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A Note on Algorithms for Determining the Copositivity of a Given Symmetric Matrix
Journal of Inequalities and Applications volume 2010, Article number: 498631 (2009)
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.
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 , it cannot guarantee to produce an answer. Table
of [1] shows that for 1000 symmetric random matrices of order
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 [1] and give two 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.
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.
Let be symmetric and be partitioned into

with ,
. As in [2], let

be the simplex of order ; and let

be the standard simplex of order whose vertices are all vertices of
. It is proved in [2] that an
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 [2, Section
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
.
Denote the vertices of 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 [2, Lemma
]), to simplify the notation we still write
with
or
. For example, if
share only one vertex
with
and the other vertices are
, then

Lemma 1.1 (see [2]).
Let 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 [2, Lemma  3.1]);
is strictly copositive if and only if
,
are all strictly copositive and
and
is strictly copositive (see [1]).
It is noticed from [2] that if the polyhedron 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
.
Lemma 1.2 (see [1]).
Let . If there are
-triples of pairwise different vertices of
satisfying the following two conditions:
(i)each contains at least one coordinate vector vertex;
(ii) has exactly
vertices for
, and
has less than
vertices when
,
then can be subdivided into
simplices
, where
is the simplex whose vertices are the elements of
.
These two lemmas are basic for proving Theorems , 2.6, 2.7, and 2.8 in [1]; they are also basic for proving Theorems 2.1 and 2.2 of this paper.
2. Main Theorems and Algorithms
The following two theorems give two algorithms for determining the copositivity of a given symmetric matrix of order 8 or 9. These two theorems can be proved by Lemma 1.1 and Lemma 1.2 following the same pattern as in [1].
Theorem 2.1.
Let be symmetric and be partitioned as in (1.1) and
, then at least one of the following cases must happen:
-
(a)
If one
principal submatrix of
is not copositive, then
is not copositive. Otherwise it holds that
and
is copositive.
-
(b)
If
then
is copositive; if
with
and
is strictly copositive, then
is strictly copositive.
-
(c)
If
, then
is copositive if and only if
is copositive;
is strictly copositive if and only if
is strictly copositive and
and
is strictly copositive.
-
(d)
If
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

 (e) If 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

  (f) If 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

  (g) If 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

  (h) If 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

  (i) If 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


It is clear (see [1, Remark ]) 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.
Theorem 2.2.
If is symmetric, then at least one of the following cases must happen:
-
(a)
If one
principal submatrix of
is not copositive, then
is not copositive.
Otherwise ( 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.
-
(b)
If
, then
and
is copositive; if
with
and
is strictly copositive, then
is strictly copositive.
-
(c)
If
, 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.
-
(d)
If
, 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

  (e) If , 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

  (f) If , 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


As mentioned in [1], we have made six MATLAB functions: Cha3(), 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.
Main steps of Function
() 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
.
When use Theorem 2.1(b) to determine copositivity of
and return.
When use Theorem 2.1(c) to determine copositivity of
and return.
When use Theorem 2.1(d) to determine copositivity of
and return.
When use Theorem 2.1(e) to determine copositivity of
and return.
When use Theorem 2.1(f) to determine copositivity of
and return.
When use Theorem 2.1(g) to determine copositivity of
and return.
When use Theorem 2.1(h) to determine copositivity of
and return.
When use Theorem 2.1(i) to determine copositivity of
and return.
Main steps of Function
() 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.
When use Theorem 2.2(b) to determine copositivity of
and return.
When use Theorem 2.2(c) to determine copositivity of
and return.
When use Theorem 2.2(d) to determine copositivity of
and return.
When use Theorem 2.2(e) to determine copositivity of
and return.
When use Theorem 2.2(f) to determine copositivity of
and return.
3. Numerical Experiments and Discussion
Having all these eight functions: Cha3(), 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 [3], where
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 [3]. Secondly we generate 1000 symmetric random matrices of order
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 1, where
,
,
,
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.
Kaplan [4, Theorem ] 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 [4, 5], we proved that a symmetric matrix
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 1 also holds for
because of the fact that a symmetric matrix is not copositive if any of its principal submatrix is not copositive. Table 1 does give us some noticeable information as follows.
Remark 3.1.
For 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.

We surely believe that algorithm for will be tedious to describe and take more time to run because of its recurrent property.
Since there is almost always no symmetric copositive matrix of order larger than 9, the interest of researchers may concentrate on sufficient conditions for copositive matrices of larger orders, or of general order . For instance, [3] proved the matrix
mentioned at the beginning of this section is copositive (but not strictly) for any
. Here we give another interesting example as follows.
Proposition 3.2.
Let 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
.
Proof.
Write , 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 [1, Theorem
]. Finally
(as the sum of two copositive matrices) is copositive.
References
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Andersson L-E, Chang G, Elfving T: Criteria for copositive matrices using simplices and barycentric coordinates. Linear Algebra and Its Applications 1995, 220: 9–30.
Johnson CR, Reams R: Constructing copositive matrices from interior matrices. Electronic Journal of Linear Algebra 2008, 17: 9–20.
Kaplan W: A copositivity probe. Linear Algebra and Its Applications 2001, 337: 237–251. 10.1016/S0024-3795(01)00351-2
Hogben L, Johnson CR, Reams R: The copositive completion problem. Linear Algebra and Its Applications 2005, 408: 207–211.
Acknowledgments
This work was supported by the NNSF China no. 10871230, NSF Zhejiang no. y607480, and Innovation Group Foundation of Anhui University
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Shang-jun, Y., Chang-qing, X. & Xiao-xin, L. A Note on Algorithms for Determining the Copositivity of a Given Symmetric Matrix. J Inequal Appl 2010, 498631 (2009). https://doi.org/10.1155/2010/498631
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DOI: https://doi.org/10.1155/2010/498631
Keywords
- Symmetric Matrix
- Random Matrice
- Symmetric Matrice
- Experimental Matrice
- Coordinate Vector