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

Open Access
Research Article
  • 802 Downloads
Part of the following topical collections:
  1. Inequalities in the A-Harmonic Equations and the Related Topics

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 Open image in new window 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 Open image in new window , there are 8, 6, and 2 matrices remaing undetermined, respectively. In this paper we give two more algorithms for Open image in new window 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.

Keywords

Symmetric Matrix Random Matrice Symmetric Matrice Experimental Matrice Coordinate Vector 

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 Open image in new window , it cannot guarantee to produce an answer. Table Open image in new window of [1] shows that for 1000 symmetric random matrices of order Open image in new window 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 Open image in new window , there are 8, 6, and 2 matrices remaining undetermined when Open image in new window , respectively. In this paper we continue our study as in [1] and give two algorithms for Open image in new window 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. Open image in new window

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 Open image in new window be symmetric and be partitioned into

with Open image in new window , Open image in new window . As in [2], let

be the simplex of order Open image in new window ; and let

be the standard simplex of order Open image in new window whose vertices are all vertices of Open image in new window . It is proved in [2] that an Open image in new window symmetric matrix Open image in new window is copositive if and only if Open image in new window for all Open image in new window . Consider the polyhedron in Open image in new window is the given vector of dimension Open image in new window in (1.1)) which has some vertices being vertices of Open image in new window , and all the other vertices being in the hyperplane Open image in new window . It is known (see [2, Section Open image in new window and Lemma Open image in new window ]) that the polyhedron Open image in new window can be subdivided into Open image in new window simplices Open image in new window in Open image in new window such that Open image in new window , Open image in new window is a subsimplex of Open image in new window and Open image in new window if Open image in new window , and the vertices of Open image in new window are all vertices of Open image in new window . We mention this fact since that Open image in new window is subdivided into simplices Open image in new window .

Denote the vertices of Open image in new window by Open image in new window , then Open image in new window is a vertex of Open image in new window , or a common point of the line connecting two vertices of Open image in new window and the hyperplane Open image in new window and should be presented in the barycenter coordinates of Open image in new window . If Open image in new window is the Open image in new window th vertex of Open image in new window , then it is represented by the coordinate vector Open image in new window with a 1 in the Open image in new window th position and all 0's elsewhere; otherwise write Open image in new window to denote that it is the common point of line Open image in new window and the hyperplane Open image in new window . Each Open image in new window determines a matrix Open image in new window (see [2, Lemma Open image in new window ]), to simplify the notation we still write Open image in new window with Open image in new window or Open image in new window . For example, if Open image in new window share only one vertex Open image in new window with Open image in new window and the other vertices are Open image in new window , then

Lemma 1.1 (see [2]).

Let Open image in new window be symmetric and partitioned as in (1.1) with Open image in new window , Open image in new window being copositive and Open image in new window is subdivided into simplices Open image in new window which determine matrices Open image in new window . Then Open image in new window is copositive if and only if Open image in new window , Open image in new window are all copositive (see [2, Lemma  3.1]); Open image in new window is strictly copositive if and only if Open image in new window , Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive (see [1]).

It is noticed from [2] that if the polyhedron Open image in new window contains Open image in new window vertices (coordinate vectors of the standard simplex Open image in new window not in the hyperplane Open image in new window , then Open image in new window contains exact Open image in new window vertices in the hyperplane Open image in new window , and that Open image in new window can be subdivided into Open image in new window simplices Open image in new window of dimension Open image in new window such that Open image in new window is a simplex of dimension Open image in new window for Open image in new window and Open image in new window is a simplex of dimension Open image in new window when Open image in new window . Open image in new window

Lemma 1.2 (see [1]).

Let Open image in new window . If there are Open image in new window Open image in new window -triples of pairwise different vertices of Open image in new window satisfying the following two conditions:

(i)each Open image in new window contains at least one coordinate vector vertex;

(ii) Open image in new window has exactly Open image in new window vertices for Open image in new window , and Open image in new window has less than Open image in new window vertices when Open image in new window ,

then Open image in new window can be subdivided into Open image in new window simplices Open image in new window , where Open image in new window is the simplex whose vertices are the elements of Open image in new window .

These two lemmas are basic for proving Theorems Open image in new window , 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]. Open image in new window

Theorem 2.1.

Let Open image in new window be symmetric and be partitioned as in (1.1) and Open image in new window , then at least one of the following cases must happen:
  1. (a)

    If one Open image in new window principal submatrix of Open image in new window is not copositive, then Open image in new window is not copositive. Otherwise it holds that Open image in new window and Open image in new window is copositive.

     
  2. (b)
     
  3. (c)

    If Open image in new window , then Open image in new window is copositive if and only if Open image in new window is copositive; Open image in new window is strictly copositive if and only if Open image in new window is strictly copositive and Open image in new window and Open image in new window is strictly copositive.

     
  4. (d)

    If Open image in new window has exactly one negative entry: Open image in new window , then Open image in new window is copositive if and only if Open image in new window is copositive; Open image in new window is strictly copositive if and only if Open image in new window is strictly copositive, and Open image in new window and Open image in new window is strictly copositive, where

     

 (e) If Open image in new window has exactly two negative entries: Open image in new window , and Open image in new window , then Open image in new window is copositive if and only if Open image in new window and Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

  (f) If Open image in new window has exactly three negative entries: Open image in new window and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

  (g) If Open image in new window has exactly four negative entries: Open image in new window and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

  (h) If Open image in new window has exactly five negative entries: Open image in new window and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

  (i) If Open image in new window has exactly six negative entries: Open image in new window and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

It is clear (see [1, Remark Open image in new window ]) that if Open image in new window is odd, then a copositive matrix Open image in new window 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 Open image in new window principal submatrices of it are not copositive. This fact will be used in Theorem 2.2. Open image in new window

Theorem 2.2.

If Open image in new window is symmetric, then at least one of the following cases must happen:
  1. (a)

    If one Open image in new window principal submatrix of Open image in new window is not copositive, then Open image in new window is not copositive.

     
Otherwise ( Open image in new window must have a row with an even number of negative entries and Open image in new window is copositive) find a row of Open image in new window which has exactly Open image in new window negative entries. If the Open image in new window th row does, then interchange the Open image in new window th row and column with the first row and column, and partition Open image in new window into (1.1) as in Theorem 2.1.
  1. (b)
     
  2. (c)

    If Open image in new window , then Open image in new window , then Open image in new window is copositive if and only if Open image in new window is copositive; Open image in new window is strictly copositive if and only if Open image in new window is strictly copositive and Open image in new window and Open image in new window is strictly copositive.

     
  3. (d)

    If Open image in new window , then Open image in new window has exactly two negative entries: Open image in new window , and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

     

  (e) If Open image in new window , then Open image in new window has exactly four negative entries: Open image in new window and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

  (f) If Open image in new window , then Open image in new window has exactly six negative entries: Open image in new window and Open image in new window , then Open image in new window is copositive if and only if Open image in new window are all copositive; Open image in new window is strictly copositive if and only if Open image in new window are all strictly copositive and Open image in new window and Open image in new window is strictly copositive, where

As mentioned in [1], we have made six MATLAB functions: Cha3( Open image in new window ), Cha4( Open image in new window ), Cha5( Open image in new window ), Cha6( Open image in new window ), Cha7( Open image in new window ) and Cha( Open image in new window ), for determining the copositivity of symmetric matrices. Now we have made two more MATLAB functions of these type: Cha8( Open image in new window ) and Cha9( Open image in new window ) based on the two algorithms given by Theorems 2.1 and 2.2. The input of the functions is any Open image in new window or Open image in new window symmetric matrix Open image in new window and there are four possible return values: Open image in new window meaning "not copositive", "copositive (not strictly)" and "strictly copositive", "cannot determined," respectively. Open image in new window

Main steps of Function Open image in new window

( Open image in new window ) Find out if Open image in new window has any Open image in new window principal submatrix which is not copositive. If so, then return with " Open image in new window " (Theorem 2.1(a)). Otherwise go to next step.

( Open image in new window ) Calculate the number Open image in new window of the negative entries of the first row of Open image in new window .

When Open image in new window use Theorem 2.1(b) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(c) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(d) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(e) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(f) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(g) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(h) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.1(i) to determine copositivity of Open image in new window and return. Open image in new window

Main steps of Function Open image in new window

( Open image in new window ) Find out if Open image in new window has any Open image in new window principal submatrix which is not copositive. If so, then return with " Open image in new window " (Theorem 2.2(a)). Otherwise Open image in new window must have some row containing exactly Open image in new window negative entries and go to the next step.

( Open image in new window ) Find out if Open image in new window has any row which has exactly Open image in new window negative entries. If the Open image in new window th row does, then interchange the Open image in new window th row and column of Open image in new window with the first row and column.

When Open image in new window use Theorem 2.2(b) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.2(c) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.2(d) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.2(e) to determine copositivity of Open image in new window and return.

When Open image in new window use Theorem 2.2(f) to determine copositivity of Open image in new window and return.

3. Numerical Experiments and Discussion

Having all these eight functions: Cha3( Open image in new window ), Cha4( Open image in new window ), Open image in new window Cha8( Open image in new window ), Cha9( Open image in new window ), and Cha( Open image in new window ) we have performed the following experiments. Firstly, we use these functions to determine the copositivity of the Open image in new window symmetric matrix Open image in new window studied in [3], where Open image in new window satisfies Open image in new window ; Open image in new window only if Open image in new window and Open image in new window . When Open image in new window the experimental results obtained by old Cha( Open image in new window ) together with Cha3( Open image in new window ), Cha4( Open image in new window ), Open image in new window Cha7( Open image in new window ) are " Open image in new window " meaning "cannot be determined" and the experimental results by Cha9( Open image in new window ) are " Open image in new window " meaning "copositive but not strictly", which are the same results as obtained in [3]. Secondly we generate 1000 symmetric random matrices of order Open image in new window 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 Open image in new window , 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 Open image in new window , Open image in new window , Open image in new window , Open image in new window 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 Open image in new window ] proved that a symmetric matrix Open image in new window is copositive if and only if the minimum principal submatrix Open image in new window of Open image in new window which shares the maximum positive diagonal entries with Open image in new window is copositive and the matrix which is constructed from Open image in new window by replacing each entry of Open image in new window by 0 is nonnegative. To answer the third open problem of [4, 5], we proved that a symmetric matrix Open image in new window with unit diagonal is copositive if and only if the matrix constructed from Open image in new window by replacing each off-diagonal entry Open image in new window by Open image in new window 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 Open image in new window submatrix is copositive (Note that for a matrix with Open image in new window the chance that every principal Open image in new window submatrix is copositive is much less). In addition, the last line of Table 1 also holds for Open image in new window 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 Open image in new window 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 Open image in new window and so forth. which might be established by our method are not practically needed.

We surely believe that algorithm for Open image in new window will be tedious to describe and take more time to run because of its recurrent property. Open image in new window

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 Open image in new window . For instance, [3] proved the matrix Open image in new window mentioned at the beginning of this section is copositive (but not strictly) for any Open image in new window . Here we give another interesting example as follows. Open image in new window

Proposition 3.2.

Let Open image in new window be a symmetric matrix of any order, Open image in new window ; Open image in new window be the sum of all the negative entries of the Open image in new window th row of Open image in new window . Then Open image in new window is copositive if Open image in new window ; Open image in new window is strictly copositive if Open image in new window ; Open image in new window is irreducible and Open image in new window .

Proof.

Write Open image in new window , where Open image in new window and Open image in new window is the Open image in new window nonnegative matrix which shares all the negative (nonnegative) entries with Open image in new window and has the remained entries all being zero. Then Open image in new window , where Open image in new window and Open image in new window is a nonnegative matrix whose spectral radius Open image in new window if Open image in new window . Therefore, Open image in new window is an M-matrix if Open image in new window ; a nonsingular M-matrix if Open image in new window or Open image in new window is irreducible and Open image in new window , whence it is copositive, strictly copositive, respectively by [1, Theorem Open image in new window ]. Finally Open image in new window (as the sum of two copositive matrices) is copositive.

Notes

Acknowledgments

This work was supported by the NNSF China no. 10871230, NSF Zhejiang no. y607480, and Innovation Group Foundation of Anhui University

References

  1. 1.
    Yang S-J, Li X-X: Algorithms for determining the copositivity of a given symmetric matrix. Linear Algebra and Its Applications 2009, 430(2–3):609–618. 10.1016/j.laa.2008.07.028MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Johnson CR, Reams R: Constructing copositive matrices from interior matrices. Electronic Journal of Linear Algebra 2008, 17: 9–20.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Kaplan W: A copositivity probe. Linear Algebra and Its Applications 2001, 337: 237–251. 10.1016/S0024-3795(01)00351-2MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hogben L, Johnson CR, Reams R: The copositive completion problem. Linear Algebra and Its Applications 2005, 408: 207–211.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Yang Shang-jun et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.School of SciencesZhejiang Forestry UniversityHangzhouChina
  3. 3.Department of MathematicsChizhou InstituteChizhouChina

Personalised recommendations