Add random_doubly_nonnegative() and random_extreme_doubly_nonnegative() functions.

diff --git a/mjo/cone/doubly_nonnegative.py b/mjo/cone/doubly_nonnegative.py
index ebb5b685fd3c5e1843fb4379e7920316dc159a15..56a655e52c5cd60cbf1073c351e65833e5f54500 100644 (file)
--- a/mjo/cone/doubly_nonnegative.py
+++ b/mjo/cone/doubly_nonnegative.py
@@ -19,7 +19,7 @@ from sage.all import *
 from os.path import abspath
 from site import addsitedir
 addsitedir(abspath('../../'))
-from mjo.cone.symmetric_psd import factor_psd, is_symmetric_psd
+from mjo.cone.symmetric_psd import factor_psd, is_symmetric_psd, random_psd
 from mjo.matrix_vector import isomorphism
 
 
@@ -378,3 +378,141 @@ def is_extreme_doubly_nonnegative(A):
     # 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.
+
+    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_psd(V, accept_zero, rank)
+
+    while not all([ x >= 0 for x in A.list() ]):
+        A = random_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.
+
+    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 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