from sage.all import *
# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
-# have to explicitly mangle our sitedir here so that "mjo.cone"
-# resolves.
+# have to explicitly mangle our sitedir here so that our module names
+# resolve.
from os.path import abspath
from site import addsitedir
addsitedir(abspath('../../'))
-from mjo.cone.symmetric_psd import factor_psd
-
+from mjo.cone.symmetric_psd import factor_psd, is_symmetric_psd
+from mjo.matrix_vector import isomorphism
def is_doubly_nonnegative(A):
INPUT:
- - ``A`` - The matrix in question
+ - ``A`` - The matrix in question
OUTPUT:
"""
if A.base_ring() == SR:
- msg = 'The base ring of ``A`` cannot be the Symbolic Ring'
+ msg = 'The matrix ``A`` cannot be the symbolic.'
raise ValueError.new(msg)
- # First make sure that ``A`` is symmetric.
- if not A.is_symmetric():
- return False
-
# Check that all of the entries of ``A`` are nonnegative.
if not all([ a >= 0 for a in A.list() ]):
return False
- # If ``A`` is symmetric and non-negative, we only need to check
- # that it is positive semidefinite. For that we can consult its
- # minimum eigenvalue, which should be zero or greater. Since ``A``
- # is symmetric, its eigenvalues are guaranteed to be real.
- return min(A.eigenvalues()) >= 0
+ # It's nonnegative, so all we need to do is check that it's
+ # symmetric positive-semidefinite.
+ return is_symmetric_psd(A)
+
+
+
+def has_admissible_extreme_rank(A):
+ """
+ The extreme matrices of the doubly-nonnegative cone have some
+ restrictions on their ranks. This function checks to see whether or
+ not ``A`` could be extreme based on its rank.
+
+ INPUT:
+
+ - ``A`` - The matrix in question
+
+ OUTPUT:
+
+ ``False`` if the rank of ``A`` precludes it from being an extreme
+ matrix of the doubly-nonnegative cone, ``True`` otherwise.
+
+ REFERENCE:
+
+ Hamilton-Jester, Crista Lee; Li, Chi-Kwong. Extreme Vectors of
+ Doubly Nonnegative Matrices. Rocky Mountain Journal of Mathematics
+ 26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
+ http://projecteuclid.org/euclid.rmjm/1181071993.
+
+ EXAMPLES:
+
+ The zero matrix has rank zero, which is admissible::
+
+ sage: A = zero_matrix(QQ, 5, 5)
+ sage: has_admissible_extreme_rank(A)
+ True
+
+ """
+ if not A.is_symmetric():
+ # This function is more or less internal, so blow up if passed
+ # something unexpected.
+ raise ValueError('The matrix ``A`` must be symmetric.')
+
+ r = rank(A)
+ n = ZZ(A.nrows()) # Columns would work, too, since ``A`` is symmetric.
+
+ if r == 0:
+ # Zero is in the doubly-nonnegative cone.
+ return True
+
+ # See Theorem 3.1 in the cited reference.
+ if r == 2:
+ return False
+
+ if n.mod(2) == 0:
+ # n is even
+ return r <= max(1, n-3)
+ else:
+ # n is odd
+ return r <= max(1, n-2)
+
+
+def E(matrix_space, i,j):
+ """
+ Return the ``i``,``j``th element of the standard basis in
+ ``matrix_space``.
+
+ INPUT:
+
+ - ``matrix_space`` - The underlying matrix space of whose basis
+ the returned matrix is an element
+
+ - ``i`` - The row index of the single nonzero entry
+
+ - ``j`` - The column index of the single nonzero entry
+
+ OUTPUT:
+
+ A basis element of ``matrix_space``. It has a single \"1\" in the
+ ``i``,``j`` row,column and zeros elsewhere.
+
+ EXAMPLES::
+
+ sage: M = MatrixSpace(ZZ, 2, 2)
+ sage: E(M,0,0)
+ [1 0]
+ [0 0]
+ sage: E(M,0,1)
+ [0 1]
+ [0 0]
+ sage: E(M,1,0)
+ [0 0]
+ [1 0]
+ sage: E(M,1,1)
+ [0 0]
+ [0 1]
+ sage: E(M,2,1)
+ Traceback (most recent call last):
+ ...
+ IndexError: Index `i` is out of bounds.
+ sage: E(M,1,2)
+ Traceback (most recent call last):
+ ...
+ IndexError: Index `j` is out of bounds.
+
+ """
+ # We need to check these ourselves, see below.
+ if i >= matrix_space.nrows():
+ raise IndexError('Index `i` is out of bounds.')
+ if j >= matrix_space.ncols():
+ raise IndexError('Index `j` is out of bounds.')
+
+ # The basis here is returned as a one-dimensional list, so we need
+ # to compute the offset into it based on ``i`` and ``j``. Since we
+ # compute the index ourselves, we need to do bounds-checking
+ # manually. Otherwise for e.g. a 2x2 matrix space, the index (0,2)
+ # would be computed as offset 3 into a four-element list and we
+ # would succeed incorrectly.
+ idx = matrix_space.ncols()*i + j
+ return matrix_space.basis()[idx]
"""
Returns ``True`` if the given matrix is an extreme matrix of the
doubly-nonnegative cone, and ``False`` otherwise.
+
+ REFERENCES:
+
+ 1. Hamilton-Jester, Crista Lee; Li, Chi-Kwong. Extreme Vectors of
+ Doubly Nonnegative Matrices. Rocky Mountain Journal of Mathematics
+ 26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
+ http://projecteuclid.org/euclid.rmjm/1181071993.
+
+ 2. Berman, Abraham and Shaked-Monderer, Naomi. Completely Positive
+ Matrices. World Scientific, 2003.
+
+ EXAMPLES:
+
+ The zero matrix is an extreme matrix::
+
+ sage: A = zero_matrix(QQ, 5, 5)
+ sage: is_extreme_doubly_nonnegative(A)
+ True
+
+ Any extreme vector of the completely positive cone is an extreme
+ vector of the doubly-nonnegative cone::
+
+ sage: v = vector([1,2,3,4,5,6])
+ sage: A = v.column() * v.row()
+ sage: A = A.change_ring(QQ)
+ sage: is_extreme_doubly_nonnegative(A)
+ True
+
+ We should be able to generate the extreme completely positive
+ vectors randomly::
+
+ sage: v = vector(map(abs, random_vector(ZZ, 4)))
+ sage: A = v.column() * v.row()
+ sage: A = A.change_ring(QQ)
+ sage: is_extreme_doubly_nonnegative(A)
+ True
+ sage: v = vector(map(abs, random_vector(ZZ, 10)))
+ sage: A = v.column() * v.row()
+ sage: A = A.change_ring(QQ)
+ sage: is_extreme_doubly_nonnegative(A)
+ True
+
+ The following matrix is completely positive but has rank 3, so by a
+ remark in reference #1 it is not extreme::
+
+ sage: A = matrix(QQ, [[1,2,1],[2,6,3],[1,3,5]])
+ sage: is_extreme_doubly_nonnegative(A)
+ False
+
+ The following matrix is completely positive (diagonal) with rank 2,
+ so it is also not extreme::
+
+ sage: A = matrix(QQ, [[1,0,0],[2,0,0],[0,0,0]])
+ sage: is_extreme_doubly_nonnegative(A)
+ False
+
"""
- raise NotImplementedError()
+
+ if not A.base_ring().is_exact() and not A.base_ring() is SR:
+ msg = 'The base ring of ``A`` must be either exact or symbolic.'
+ raise ValueError(msg)
+
+ if not A.base_ring().is_field():
+ raise ValueError('The base ring of ``A`` must be a field.')
+
+ if not A.base_ring() is SR:
+ # Change the base field of ``A`` so that we are sure we can take
+ # roots. The symbolic ring has no algebraic_closure method.
+ A = A.change_ring(A.base_ring().algebraic_closure())
+
+ # Step 1 (see reference #1)
+ k = A.rank()
+
+ if k == 0:
+ # Short circuit, we know the zero matrix is extreme.
+ return True
+
+ if not is_symmetric_psd(A):
+ return False
+
+ # Step 1.5, appeal to Theorem 3.1 in reference #1 to short
+ # circuit.
+ if not has_admissible_extreme_rank(A):
+ return False
+
+ # Step 2
+ X = factor_psd(A)
+
+ # Step 3
+ #
+ # Begin with an empty spanning set, and add a new matrix to it
+ # whenever we come across an index pair `$(i,j)$` with
+ # `$A_{ij} = 0$`.
+ spanning_set = []
+ for j in range(0, A.ncols()):
+ for i in range(0,j):
+ if A[i,j] == 0:
+ M = A.matrix_space()
+ S = X.transpose() * (E(M,i,j) + E(M,j,i)) * X
+ spanning_set.append(S)
+
+ # The spanning set that we have at this point is of matrices. We
+ # only care about the dimension of the spanned space, and Sage
+ # can't compute the dimension of a set of matrices anyway, so we
+ # convert them all to vectors and just ask for the dimension of the
+ # resulting vector space.
+ (phi, phi_inverse) = isomorphism(A.matrix_space())
+ vectors = map(phi,spanning_set)
+
+ V = span(vectors, A.base_ring())
+ d = V.dimension()
+
+ # Needed to safely divide by two here (we don't want integer
+ # division). We ensured that the base ring of ``A`` is a field
+ # earlier.
+ two = A.base_ring()(2)
+ return d == (k*(k + 1)/two - 1)
projects / sage.d.git / blobdiff