From db016cc07b5f0cb45a96022eb468e3080786078b Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Thu, 1 Oct 2015 09:43:22 -0400
Subject: [PATCH] Add the positive_operators() function.

---
 mjo/cone/cone.py | 94 ++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 94 insertions(+)

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index bf121cb..b4d2be0 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -806,3 +806,97 @@ def random_element(K):
     # return ``0`` when ``K`` has no rays.
     v = V(sum(scaled_gens))
     return v
+
+
+def positive_operators(K):
+    r"""
+    Compute generators of the cone of positive operators on this cone.
+
+    OUTPUT:
+
+    A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
+    Each matrix ``P`` in the list should have the property that ``P*x``
+    is an element of ``K`` whenever ``x`` is an element of
+    ``K``. Moreover, any nonnegative linear combination of these
+    matrices shares the same property.
+
+    EXAMPLES:
+
+    The trivial cone in a trivial space has no positive operators::
+
+        sage: K = Cone([], ToricLattice(0))
+        sage: positive_operators(K)
+        []
+
+    Positive operators on the nonnegative orthant are nonnegative matrices::
+
+        sage: K = Cone([(1,)])
+        sage: positive_operators(K)
+        [[1]]
+
+        sage: K = Cone([(1,0),(0,1)])
+        sage: positive_operators(K)
+        [
+        [1 0]  [0 1]  [0 0]  [0 0]
+        [0 0], [0 0], [1 0], [0 1]
+        ]
+
+    Every operator is positive on the ambient vector space::
+
+        sage: K = Cone([(1,),(-1,)])
+        sage: K.is_full_space()
+        True
+        sage: positive_operators(K)
+        [[1], [-1]]
+
+        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+        sage: K.is_full_space()
+        True
+        sage: positive_operators(K)
+        [
+        [1 0]  [-1  0]  [0 1]  [ 0 -1]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
+        [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
+        ]
+
+    TESTS:
+
+    A positive operator on a cone should send its generators into the cone::
+
+        sage: K = random_cone(max_ambient_dim = 6)
+        sage: pi_of_k = positive_operators(K)
+        sage: all([K.contains(p*x) for p in pi_of_k for x in K.rays()])
+        True
+
+    """
+    V = K.lattice().vector_space()
+
+    # Sage doesn't think matrices are vectors, so we have to convert
+    # our matrices to vectors explicitly before we can figure out how
+    # many are linearly-indepenedent.
+    #
+    # The space W has the same base ring as V, but dimension
+    # dim(V)^2. So it has the same dimension as the space of linear
+    # transformations on V. In other words, it's just the right size
+    # to create an isomorphism between it and our matrices.
+    W = VectorSpace(V.base_ring(), V.dimension()**2)
+
+    G1 = [ V(x) for x in K.rays() ]
+    G2 = [ V(s) for s in K.dual().rays() ]
+
+    tensor_products = [ s.tensor_product(x) for x in G1 for s in G2 ]
+
+    # Turn our matrices into long vectors...
+    vectors = [ W(m.list()) for m in tensor_products ]
+
+    # Create the *dual* cone of the positive operators, expressed as
+    # long vectors..
+    L = ToricLattice(W.dimension())
+    pi_dual = Cone(vectors, lattice=L)
+
+    # Now compute the desired cone from its dual...
+    pi_cone = pi_dual.dual()
+
+    # And finally convert its rays back to matrix representations.
+    M = MatrixSpace(V.base_ring(), V.dimension())
+
+    return [ M(v.list()) for v in pi_cone.rays() ]
-- 
2.44.2