Add more positive operator examples from the paper.

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

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 84c3adb57d8396735460f5aefe5d669416e18ff1..ae3ec48cddc9700d4f63ae378fc01b178dee6e3b 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -83,7 +83,7 @@ def motzkin_decomposition(K):
     REFERENCES:
 
     .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
-       Optimization in Finite Dimensions I.  Springer-Verlag, New
+       Optimization in Finite Dimensions I. Springer-Verlag, New
        York, 1970.
 
     EXAMPLES:
@@ -152,18 +152,19 @@ def motzkin_decomposition(K):
         sage: S.is_equivalent(expected_S)
         True
     """
-    linspace_gens  = [ copy(b) for b in K.linear_subspace().basis() ]
-    linspace_gens += [ -b for b in linspace_gens ]
+    # The lines() method only returns one generator per line. For a true
+    # line, we also need a generator pointing in the opposite direction.
+    S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
+    S = Cone(S_gens, K.lattice())
 
-    S = Cone(linspace_gens, K.lattice())
-
-    # Since ``S`` is a subspace, its dual is its orthogonal complement
-    # (albeit in the wrong lattice).
+    # Since ``S`` is a subspace, the rays of its dual generate its
+    # orthogonal complement.
     S_perp = Cone(S.dual(), K.lattice())
     P = K.intersection(S_perp)
 
     return (P,S)
 
+
 def positive_operator_gens(K):
     r"""
     Compute generators of the cone of positive operators on this cone.
@@ -178,12 +179,6 @@ def positive_operator_gens(K):
 
     EXAMPLES:
 
-    The trivial cone in a trivial space has no positive operators::
-
-        sage: K = Cone([], ToricLattice(0))
-        sage: positive_operator_gens(K)
-        []
-
     Positive operators on the nonnegative orthant are nonnegative matrices::
 
         sage: K = Cone([(1,)])
@@ -197,6 +192,27 @@ def positive_operator_gens(K):
         [0 0], [0 0], [1 0], [0 1]
         ]
 
+    The trivial cone in a trivial space has no positive operators::
+
+        sage: K = Cone([], ToricLattice(0))
+        sage: positive_operator_gens(K)
+        []
+
+    Every operator is positive on the trivial cone::
+
+        sage: K = Cone([(0,)])
+        sage: positive_operator_gens(K)
+        [[1], [-1]]
+
+        sage: K = Cone([(0,0)])
+        sage: K.is_trivial()
+        True
+        sage: positive_operator_gens(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]
+        ]
+
     Every operator is positive on the ambient vector space::
 
         sage: K = Cone([(1,),(-1,)])
@@ -214,14 +230,58 @@ def positive_operator_gens(K):
         [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
         ]
 
+    A non-obvious application is to find the positive operators on the
+    right half-plane::
+
+        sage: K = Cone([(1,0),(0,1),(0,-1)])
+        sage: positive_operator_gens(K)
+        [
+        [1 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::
+    Each positive operator generator should send the generators of the
+    cone into the cone::
 
         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()])
+        sage: all([ K.contains(P*x) for P in pi_of_K for x in K ])
+        True
+
+    Each positive operator generator should send a random element of the
+    cone into the cone::
+
+        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 ])
+        True
+
+    A random element of the positive operator cone should send the
+    generators of the cone into the cone::
+
+        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([ g.list() for g in pi_of_K ], lattice=L)
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
+        sage: all([ K.contains(P*x) for x in K ])
+        True
+
+    A random element of the positive operator cone should send a random
+    element of the cone into the cone::
+
+        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([ 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())
         True
 
     The dimension of the cone of positive operators is given by the