Add a test for the lineality spaces of Z/pi-star being equal.

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

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index f78c27e7e8ba748274b968601fff61c4701a98c1..702472c30466f7cba79d4a79f1bf47a552079364 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -176,7 +176,8 @@ def positive_operator_gens(K):
 
     A positive operator on a cone should send its generators into the cone::
 
-        sage: K = random_cone(max_ambient_dim = 6)
+        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*x) for p in pi_of_K for x in K.rays()])
         True
@@ -184,16 +185,41 @@ def positive_operator_gens(K):
     The dimension of the cone of positive operators is given by the
     corollary in my paper::
 
-        sage: K = random_cone(max_ambient_dim = 6)
+        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: actual = Cone([p.list() for p in pi_of_K]).dim()
-        sage: expected = n**2 - l*(n - l) - (n - m)*m
+        sage: L = ToricLattice(n**2)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+        sage: expected = n**2 - l*(m - l) - (n - m)*m
         sage: actual == expected
         True
 
+    The lineality of the cone of positive operators is given by the
+    corollary in my paper::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: n = K.lattice_dim()
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(n**2)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+        sage: expected = n**2 - K.dim()*K.dual().dim()
+        sage: actual == expected
+        True
+
+    The cone ``K`` is proper if and only if the cone of positive
+    operators on ``K`` is proper::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(K.lattice_dim()**2)
+        sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+        sage: K.is_proper() == pi_cone.is_proper()
+        True
     """
     # Matrices are not vectors in Sage, so we have to convert them
     # to vectors explicitly before we can find a basis. We need these
@@ -286,6 +312,27 @@ def Z_transformation_gens(K):
         sage: z_cone.linear_subspace() == lls
         True
 
+    And thus, the lineality of Z is the Lyapunov rank::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=6)
+        sage: Z_of_K = Z_transformation_gens(K)
+        sage: L = ToricLattice(K.lattice_dim()**2)
+        sage: z_cone  = Cone([ z.list() for z in Z_of_K ], lattice=L)
+        sage: z_cone.lineality() == K.lyapunov_rank()
+        True
+
+    The lineality spaces of pi-star and Z-star are equal:
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: Z_of_K = Z_transformation_gens(K)
+        sage: L = ToricLattice(K.lattice_dim()**2)
+        sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
+        sage: z_star  = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
+        sage: pi_star.linear_subspace() == z_star.linear_subspace()
+        True
     """
     # Matrices are not vectors in Sage, so we have to convert them
     # to vectors explicitly before we can find a basis. We need these
@@ -314,3 +361,18 @@ def Z_transformation_gens(K):
     # not cross-positive ones.
     M = MatrixSpace(F, n)
     return [ -M(v.list()) for v in Sigma_cone.rays() ]
+
+
+def Z_cone(K):
+    gens = Z_transformation_gens(K)
+    L = None
+    if len(gens) == 0:
+        L = ToricLattice(0)
+    return Cone([ g.list() for g in gens ], lattice=L)
+
+def pi_cone(K):
+    gens = positive_operator_gens(K)
+    L = None
+    if len(gens) == 0:
+        L = ToricLattice(0)
+    return Cone([ g.list() for g in gens ], lattice=L)