-"""
+r"""
The doubly-nonnegative cone in `S^{n}` is the set of all such matrices
that both,
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 our module names
-# resolve.
-from os.path import abspath
-from site import addsitedir
-addsitedir(abspath('../../'))
-from mjo.cone.symmetric_psd import factor_psd, is_symmetric_psd
-from mjo.matrix_vector import isomorphism
+from mjo.cone.symmetric_psd import (factor_psd,
+ random_symmetric_psd)
+from mjo.basis_repr import basis_repr
def is_doubly_nonnegative(A):
Either ``True`` if ``A`` is doubly-nonnegative, or ``False``
otherwise.
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import is_doubly_nonnegative
+
EXAMPLES:
Every completely positive matrix is doubly-nonnegative::
raise ValueError.new(msg)
# Check that all of the entries of ``A`` are nonnegative.
- if not all([ a >= 0 for a in A.list() ]):
+ if not all( a >= 0 for a in A.list() ):
return False
# It's nonnegative, so all we need to do is check that it's
# symmetric positive-semidefinite.
- return is_symmetric_psd(A)
+ return A.is_positive_semidefinite()
+
+
+
+def is_admissible_extreme_rank(r, n):
+ r"""
+ The extreme matrices of the doubly-nonnegative cone have some
+ restrictions on their ranks. This function checks to see whether the
+ rank ``r`` would be an admissible rank for an ``n``-by-``n`` matrix.
+
+ INPUT:
+
+ - ``r`` - The rank of the matrix.
+
+ - ``n`` - The dimension of the vector space on which the matrix acts.
+
+ OUTPUT:
+
+ Either ``True`` if a rank ``r`` matrix could be an extreme vector of
+ the doubly-nonnegative cone in `$\mathbb{R}^{n}$`, or ``False``
+ otherwise.
+
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import is_admissible_extreme_rank
+
+ EXAMPLES:
+
+ For dimension 5, only ranks zero, one, and three are admissible::
+
+ sage: is_admissible_extreme_rank(0,5)
+ True
+ sage: is_admissible_extreme_rank(1,5)
+ True
+ sage: is_admissible_extreme_rank(2,5)
+ False
+ sage: is_admissible_extreme_rank(3,5)
+ True
+ sage: is_admissible_extreme_rank(4,5)
+ False
+ sage: is_admissible_extreme_rank(5,5)
+ False
+
+ When given an impossible rank, we just return false::
+
+ sage: is_admissible_extreme_rank(100,5)
+ False
+
+ """
+ if r == 0:
+ # Zero is in the doubly-nonnegative cone.
+ return True
+ if r > n:
+ # Impossible, just return False
+ return False
+
+ # 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 has_admissible_extreme_rank(A):
26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
http://projecteuclid.org/euclid.rmjm/1181071993.
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import has_admissible_extreme_rank
+
EXAMPLES:
The zero matrix has rank zero, which is admissible::
- sage: A = zero_matrix(QQ, 5, 5)
- sage: has_admissible_extreme_rank(A)
- True
+ sage: A = zero_matrix(QQ, 5, 5)
+ sage: has_admissible_extreme_rank(A)
+ True
+
+ Likewise, rank one is admissible for dimension 5::
+
+ sage: v = vector(QQ, [1,2,3,4,5])
+ sage: A = v.column()*v.row()
+ sage: has_admissible_extreme_rank(A)
+ True
+
+ But rank 2 is never admissible::
+
+ sage: v1 = vector(QQ, [1,0,0,0,0])
+ sage: v2 = vector(QQ, [0,1,0,0,0])
+ sage: A = v1.column()*v1.row() + v2.column()*v2.row()
+ sage: has_admissible_extreme_rank(A)
+ False
+
+ In dimension 5, three is the only other admissible rank::
+
+ sage: v1 = vector(QQ, [1,0,0,0,0])
+ sage: v2 = vector(QQ, [0,1,0,0,0])
+ sage: v3 = vector(QQ, [0,0,1,0,0])
+ sage: A = v1.column()*v1.row()
+ sage: A += v2.column()*v2.row()
+ sage: A += v3.column()*v3.row()
+ sage: has_admissible_extreme_rank(A)
+ True
"""
if not A.is_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
+ return is_admissible_extreme_rank(r,n)
- # 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):
+def stdE(matrix_space, i,j):
"""
Return the ``i``,``j``th element of the standard basis in
``matrix_space``.
A basis element of ``matrix_space``. It has a single \"1\" in the
``i``,``j`` row,column and zeros elsewhere.
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import stdE
+
EXAMPLES::
sage: M = MatrixSpace(ZZ, 2, 2)
- sage: E(M,0,0)
+ sage: stdE(M,0,0)
[1 0]
[0 0]
- sage: E(M,0,1)
+ sage: stdE(M,0,1)
[0 1]
[0 0]
- sage: E(M,1,0)
+ sage: stdE(M,1,0)
[0 0]
[1 0]
- sage: E(M,1,1)
+ sage: stdE(M,1,1)
[0 0]
[0 1]
- sage: E(M,2,1)
+ sage: stdE(M,2,1)
Traceback (most recent call last):
...
IndexError: Index `i` is out of bounds.
- sage: E(M,1,2)
+ sage: stdE(M,1,2)
Traceback (most recent call last):
...
IndexError: Index `j` is out of bounds.
# 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]
+ return list(matrix_space.basis())[idx]
2. Berman, Abraham and Shaked-Monderer, Naomi. Completely Positive
Matrices. World Scientific, 2003.
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import is_extreme_doubly_nonnegative
+
EXAMPLES:
The zero matrix is an extreme matrix::
# Short circuit, we know the zero matrix is extreme.
return True
- if not is_symmetric_psd(A):
+ if not A.is_positive_semidefinite():
return False
# Step 1.5, appeal to Theorem 3.1 in reference #1 to short
# 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):
+ for j in range(A.ncols()):
+ for i in range(j):
if A[i,j] == 0:
M = A.matrix_space()
- S = X.transpose() * (E(M,i,j) + E(M,j,i)) * X
+ S = X.transpose() * (stdE(M,i,j) + stdE(M,j,i)) * X
spanning_set.append(S)
# The spanning set that we have at this point is of matrices. We
# 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())
+ (phi, phi_inverse) = basis_repr(A.matrix_space())
vectors = map(phi,spanning_set)
V = span(vectors, A.base_ring())
# earlier.
two = A.base_ring()(2)
return d == (k*(k + 1)/two - 1)
+
+
+def random_doubly_nonnegative(V, accept_zero=True, rank=None):
+ """
+ Generate a random doubly nonnegative matrix over the vector
+ space ``V``. That is, the returned matrix will be a linear
+ transformation on ``V``, with the same base ring as ``V``.
+
+ We take a very loose interpretation of "random," here. Otherwise we
+ would never (for example) choose a matrix on the boundary of the
+ cone.
+
+ INPUT:
+
+ - ``V`` - The vector space on which the returned matrix will act.
+
+ - ``accept_zero`` - Do you want to accept the zero matrix (which
+ is doubly nonnegative)? Default to ``True``.
+
+ - ``rank`` - Require the returned matrix to have the given rank
+ (optional).
+
+ OUTPUT:
+
+ A random doubly nonnegative matrix, i.e. a linear transformation
+ from ``V`` to itself.
+
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import (is_doubly_nonnegative,
+ ....: random_doubly_nonnegative)
+
+ EXAMPLES:
+
+ Well, it doesn't crash at least::
+
+ sage: V = VectorSpace(QQ, 2)
+ sage: A = random_doubly_nonnegative(V)
+ sage: A.matrix_space()
+ Full MatrixSpace of 2 by 2 dense matrices over Rational Field
+ sage: is_doubly_nonnegative(A)
+ True
+
+ A matrix with the desired rank is returned::
+
+ sage: V = VectorSpace(QQ, 5)
+ sage: A = random_doubly_nonnegative(V,False,1)
+ sage: A.rank()
+ 1
+ sage: A = random_doubly_nonnegative(V,False,2)
+ sage: A.rank()
+ 2
+ sage: A = random_doubly_nonnegative(V,False,3)
+ sage: A.rank()
+ 3
+ sage: A = random_doubly_nonnegative(V,False,4)
+ sage: A.rank()
+ 4
+ sage: A = random_doubly_nonnegative(V,False,5)
+ sage: A.rank()
+ 5
+
+ """
+
+ # Generate random symmetric positive-semidefinite matrices until
+ # one of them is nonnegative, then return that.
+ A = random_symmetric_psd(V, accept_zero, rank)
+
+ while not all( x >= 0 for x in A.list() ):
+ A = random_symmetric_psd(V, accept_zero, rank)
+
+ return A
+
+
+
+def random_extreme_doubly_nonnegative(V, accept_zero=True, rank=None):
+ """
+ Generate a random extreme doubly nonnegative matrix over the
+ vector space ``V``. That is, the returned matrix will be a linear
+ transformation on ``V``, with the same base ring as ``V``.
+
+ We take a very loose interpretation of "random," here. Otherwise we
+ would never (for example) choose a matrix on the boundary of the
+ cone.
+
+ INPUT:
+
+ - ``V`` - The vector space on which the returned matrix will act.
+
+ - ``accept_zero`` - Do you want to accept the zero matrix
+ (which is extreme)? Defaults to ``True``.
+
+ - ``rank`` - Require the returned matrix to have the given rank
+ (optional). WARNING: certain ranks are not possible
+ in any given dimension! If an impossible rank is
+ requested, a ValueError will be raised.
+
+ OUTPUT:
+
+ A random extreme doubly nonnegative matrix, i.e. a linear
+ transformation from ``V`` to itself.
+
+ SETUP::
+
+ sage: from mjo.cone.doubly_nonnegative import (is_extreme_doubly_nonnegative,
+ ....: random_extreme_doubly_nonnegative)
+
+ EXAMPLES:
+
+ Well, it doesn't crash at least::
+
+ sage: V = VectorSpace(QQ, 2)
+ sage: A = random_extreme_doubly_nonnegative(V)
+ sage: A.matrix_space()
+ Full MatrixSpace of 2 by 2 dense matrices over Rational Field
+ sage: is_extreme_doubly_nonnegative(A)
+ True
+
+ Rank 2 is never allowed, so we expect an error::
+
+ sage: V = VectorSpace(QQ, 5)
+ sage: A = random_extreme_doubly_nonnegative(V, False, 2)
+ Traceback (most recent call last):
+ ...
+ ValueError: Rank 2 not possible in dimension 5.
+
+ Rank 4 is not allowed in dimension 5::
+
+ sage: V = VectorSpace(QQ, 5)
+ sage: A = random_extreme_doubly_nonnegative(V, False, 4)
+ Traceback (most recent call last):
+ ...
+ ValueError: Rank 4 not possible in dimension 5.
+
+ """
+
+ if rank is not None and not is_admissible_extreme_rank(rank, V.dimension()):
+ msg = 'Rank %d not possible in dimension %d.'
+ raise ValueError(msg % (rank, V.dimension()))
+
+ # Generate random doubly-nonnegative matrices until
+ # one of them is extreme, then return that.
+ A = random_doubly_nonnegative(V, accept_zero, rank)
+
+ while not is_extreme_doubly_nonnegative(A):
+ A = random_doubly_nonnegative(V, accept_zero, rank)
+
+ return A
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