Give an example where check=True is necessary for the pi/Z cones.

[sage.d.git] / mjo / cone / cone.py

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 8b07f86b329e191e105994e83676f3e30d8c4220..8790c30673a25af96c29bf22ff9e84458516da09 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -118,7 +118,7 @@ def motzkin_decomposition(K):
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
         sage: (P,S) = motzkin_decomposition(K)
-        sage: x = K.random_element()
+        sage: x = K.random_element(ring=QQ)
         sage: P.contains(x) or S.contains(x)
         True
         sage: x.is_zero() or (P.contains(x) != S.contains(x))
@@ -257,7 +257,7 @@ def positive_operator_gens(K):
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
         sage: pi_of_K = positive_operator_gens(K)
-        sage: all([ K.contains(P*K.random_element()) for P in pi_of_K ])
+        sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
         True
 
     A random element of the positive operator cone should send the
@@ -268,7 +268,7 @@ def positive_operator_gens(K):
         sage: pi_of_K = positive_operator_gens(K)
         sage: L = ToricLattice(K.lattice_dim()**2)
         sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
-        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
         sage: all([ K.contains(P*x) for x in K ])
         True
 
@@ -280,8 +280,8 @@ def positive_operator_gens(K):
         sage: pi_of_K = positive_operator_gens(K)
         sage: L = ToricLattice(K.lattice_dim()**2)
         sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
-        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
-        sage: K.contains(P*K.random_element())
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+        sage: K.contains(P*K.random_element(ring=QQ))
         True
 
     The lineality space of the dual of the cone of positive operators
@@ -303,6 +303,22 @@ def positive_operator_gens(K):
         sage: actual == expected
         True
 
+    The lineality of the dual of the cone of positive operators
+    is known from its lineality space::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: n = K.lattice_dim()
+        sage: m = K.dim()
+        sage: l = K.lineality()
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(n**2)
+        sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+        sage: actual = pi_cone.dual().lineality()
+        sage: expected = l*(m - l) + m*(n - m)
+        sage: actual == expected
+        True
+
     The dimension of the cone of positive operators is given by the
     corollary in my paper::
 
@@ -318,8 +334,33 @@ def positive_operator_gens(K):
         sage: actual == expected
         True
 
-    The lineality of the cone of positive operators is given by the
-    corollary in my paper::
+    The trivial cone, full space, and half-plane all give rise to the
+    expected dimensions::
+
+        sage: n = ZZ.random_element().abs()
+        sage: K = Cone([[0] * n], ToricLattice(n))
+        sage: K.is_trivial()
+        True
+        sage: L = ToricLattice(n^2)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+        sage: actual == n^2
+        True
+        sage: K = K.dual()
+        sage: K.is_full_space()
+        True
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+        sage: actual == n^2
+        True
+        sage: K = Cone([(1,0),(0,1),(0,-1)])
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K]).dim()
+        sage: actual == 3
+        True
+
+    The lineality of the cone of positive operators follows from the
+    description of its generators::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
@@ -331,8 +372,33 @@ def positive_operator_gens(K):
         sage: actual == expected
         True
 
-    The cone ``K`` is proper if and only if the cone of positive
-    operators on ``K`` is proper::
+    The trivial cone, full space, and half-plane all give rise to the
+    expected linealities::
+
+        sage: n = ZZ.random_element().abs()
+        sage: K = Cone([[0] * n], ToricLattice(n))
+        sage: K.is_trivial()
+        True
+        sage: L = ToricLattice(n^2)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+        sage: actual == n^2
+        True
+        sage: K = K.dual()
+        sage: K.is_full_space()
+        True
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+        sage: actual == n^2
+        True
+        sage: K = Cone([(1,0),(0,1),(0,-1)])
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K]).lineality()
+        sage: actual == 2
+        True
+
+    A cone is proper if and only if its cone of positive operators
+    is proper::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
@@ -355,7 +421,9 @@ def positive_operator_gens(K):
     vectors = [ W(tp.list()) for tp in tensor_products ]
 
     # Create the *dual* cone of the positive operators, expressed as
-    # long vectors..
+    # long vectors. WARNING: check=True is necessary even though it
+    # makes Cone() take forever. For an example take
+    # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]).
     pi_dual = Cone(vectors, ToricLattice(W.dimension()))
 
     # Now compute the desired cone from its dual...
@@ -471,7 +539,9 @@ def Z_transformation_gens(K):
     vectors = [ W(m.list()) for m in tensor_products ]
 
     # Create the *dual* cone of the cross-positive operators,
-    # expressed as long vectors..
+    # expressed as long vectors. WARNING: check=True is necessary
+    # even though it makes Cone() take forever. For an example take
+    # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]).
     Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
 
     # Now compute the desired cone from its dual...