A Note on Algorithms for Determining the Copositivity of a Given Symmetric Matrix

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

(1.1)

with , . As in [2], let

(1.2)

be the simplex of order ; and let

(1.3)

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

(1.4)

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.

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:

1. (a)

   If one principal submatrix of is not copositive, then is not copositive. Otherwise it holds that and is copositive.

    
2. (b)

   If then is copositive; if with and is strictly copositive, then is strictly copositive.

    
3. (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.

    
4. (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

    

(2.1)

 (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

(2.2)

  (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

(2.3)

  (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

(2.4)

  (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

(2.5)

  (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

(2.6)

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:

1. (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.

1. (b)

   If , then and is copositive; if with and is strictly copositive, then is strictly copositive.

    
2. (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.

    
3. (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

    

(2.7)

  (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

(2.8)

  (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

(2.9)

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.

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.

	%		%		%		%
8							0	0
9			0	0			0	0
10	0	0	0	0			0	0

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.