From: Michael Orlitzky <michael@orlitzky.com>
Date: Thu, 20 Dec 2018 01:13:34 +0000 (-0500)
Subject: mjo/cone/completely_positive.py: add completely_positive_operators_gens().
X-Git-Url: http://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=3d6ec3f4f138a6278be4f393b491aa1ac2bbffa8;p=sage.d.git

mjo/cone/completely_positive.py: add completely_positive_operators_gens().

Add a new function to return (as matrices) the generators of the
completely-positive cone of K, where K is some other given cone.
---

diff --git a/mjo/cone/completely_positive.py b/mjo/cone/completely_positive.py
index 8638a63..15be558 100644
--- a/mjo/cone/completely_positive.py
+++ b/mjo/cone/completely_positive.py
@@ -195,3 +195,57 @@ def is_extreme_completely_positive(A):
     # factorization into `$XX^{T}$` may not be unique!
     raise ValueError('Unable to determine extremity of ``A``.')
 
+
+
+def completely_positive_operators_gens(K):
+    r"""
+    Return a list of generators (matrices) for the completely-positive
+    cone of ``K``.
+
+    INPUT:
+
+    - ``K`` -- a closed convex rational polyhedral cone.
+
+    OUTPUT:
+
+    A list of matrices, the conic hull of which is the
+    completely-positive cone of ``K``.
+
+    SETUP::
+
+        sage: from mjo.cone.completely_positive import (
+        ....:   completely_positive_operators_gens,
+        ....:   is_completely_positive )
+        sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
+        sage: from mjo.matrix_vector import isomorphism
+
+    EXAMPLES::
+
+        sage: K = nonnegative_orthant(2)
+        sage: completely_positive_operators_gens(K)
+        [
+        [1 0]  [0 0]
+        [0 0], [0 1]
+        ]
+        sage: all( is_completely_positive(M)
+        ....:      for M in completely_positive_operators_gens(K) )
+        True
+
+    TESTS:
+
+    The completely-positive cone of ``K`` is subdual::
+
+       sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+       sage: cp_gens = completely_positive_operators_gens(K)
+       sage: n = K.lattice_dim()
+       sage: M = MatrixSpace(QQ, n, n)
+       sage: (p, p_inv) = isomorphism(M)
+       sage: L = ToricLattice(n**2)
+       sage: cp_cone = Cone( (p(m) for m in cp_gens), lattice=L )
+       sage: copos_cone = Cone(cp_cone.dual().rays(), lattice=L )
+       sage: all( x in copos_cone for x in cp_cone )
+       True
+
+    """
+    return [ x.tensor_product(x) for x in K ]
+