Play around with positive operators and Z-transformations. Add a new test.

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 49df3e9bc73b5071b4120f3ebfd41acb69921d01..f78c27e7e8ba748274b968601fff61c4701a98c1 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -40,14 +40,14 @@ def is_lyapunov_like(L,K):
     The identity is always Lyapunov-like in a nontrivial space::
 
         sage: set_random_seed()
-        sage: K = random_cone(min_ambient_dim = 1, max_rays = 8)
+        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
         sage: L = identity_matrix(K.lattice_dim())
         sage: is_lyapunov_like(L,K)
         True
 
     As is the "zero" transformation::
 
-        sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
+        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
         sage: R = K.lattice().vector_space().base_ring()
         sage: L = zero_matrix(R, K.lattice_dim())
         sage: is_lyapunov_like(L,K)
@@ -56,7 +56,7 @@ def is_lyapunov_like(L,K):
         Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
         on ``K``::
 
-        sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
+        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
         sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
         True
 
@@ -122,7 +122,7 @@ def random_element(K):
     return v
 
 
-def positive_operators(K):
+def positive_operator_gens(K):
     r"""
     Compute generators of the cone of positive operators on this cone.
 
@@ -139,17 +139,17 @@ def positive_operators(K):
     The trivial cone in a trivial space has no positive operators::
 
         sage: K = Cone([], ToricLattice(0))
-        sage: positive_operators(K)
+        sage: positive_operator_gens(K)
         []
 
     Positive operators on the nonnegative orthant are nonnegative matrices::
 
         sage: K = Cone([(1,)])
-        sage: positive_operators(K)
+        sage: positive_operator_gens(K)
         [[1]]
 
         sage: K = Cone([(1,0),(0,1)])
-        sage: positive_operators(K)
+        sage: positive_operator_gens(K)
         [
         [1 0]  [0 1]  [0 0]  [0 0]
         [0 0], [0 0], [1 0], [0 1]
@@ -160,13 +160,13 @@ def positive_operators(K):
         sage: K = Cone([(1,),(-1,)])
         sage: K.is_full_space()
         True
-        sage: positive_operators(K)
+        sage: positive_operator_gens(K)
         [[1], [-1]]
 
         sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
         sage: K.is_full_space()
         True
-        sage: positive_operators(K)
+        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]
@@ -177,42 +177,49 @@ def positive_operators(K):
     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: 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
 
+    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: 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: actual == expected
+        True
+
     """
-    # 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.
-    V = K.lattice().vector_space()
-    W = VectorSpace(V.base_ring(), V.dimension()**2)
+    # Matrices are not vectors in Sage, so we have to convert them
+    # to vectors explicitly before we can find a basis. We need these
+    # two values to construct the appropriate "long vector" space.
+    F = K.lattice().base_field()
+    n = K.lattice_dim()
 
     tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
 
-    # Turn our matrices into long vectors...
-    vectors = [ W(m.list()) for m in tensor_products ]
+    # Convert those tensor products to long vectors.
+    W = VectorSpace(F, n**2)
+    vectors = [ W(tp.list()) for tp 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)
+    pi_dual = Cone(vectors, ToricLattice(W.dimension()))
 
     # 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())
-
+    M = MatrixSpace(F, n)
     return [ M(v.list()) for v in pi_cone.rays() ]
 
 
-def Z_transformations(K):
+def Z_transformation_gens(K):
     r"""
     Compute generators of the cone of Z-transformations on this cone.
 
@@ -230,13 +237,13 @@ def Z_transformations(K):
     That is, matrices whose off-diagonal elements are nonnegative::
 
         sage: K = Cone([(1,0),(0,1)])
-        sage: Z_transformations(K)
+        sage: Z_transformation_gens(K)
         [
         [ 0 -1]  [ 0  0]  [-1  0]  [1 0]  [ 0  0]  [0 0]
         [ 0  0], [-1  0], [ 0  0], [0 0], [ 0 -1], [0 1]
         ]
         sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
-        sage: all([ z[i][j] <= 0 for z in Z_transformations(K)
+        sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
         ....:                    for i in range(z.nrows())
         ....:                    for j in range(z.ncols())
         ....:                    if i != j ])
@@ -245,7 +252,7 @@ def Z_transformations(K):
     The trivial cone in a trivial space has no Z-transformations::
 
         sage: K = Cone([], ToricLattice(0))
-        sage: Z_transformations(K)
+        sage: Z_transformation_gens(K)
         []
 
     Z-transformations on a subspace are Lyapunov-like and vice-versa::
@@ -254,7 +261,7 @@ def Z_transformations(K):
         sage: K.is_full_space()
         True
         sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
-        sage: zs  = span([ vector(z.list()) for z in Z_transformations(K) ])
+        sage: zs  = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
         sage: zs == lls
         True
 
@@ -264,7 +271,7 @@ def Z_transformations(K):
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim = 6)
-        sage: Z_of_K = Z_transformations(K)
+        sage: Z_of_K = Z_transformation_gens(K)
         sage: dcs = K.discrete_complementarity_set()
         sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
         ....:                                  for (x,s) in dcs])
@@ -275,32 +282,29 @@ def Z_transformations(K):
         sage: set_random_seed()
         sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
         sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
-        sage: z_cone  = Cone([ z.list() for z in Z_transformations(K) ])
+        sage: z_cone  = Cone([ z.list() for z in Z_transformation_gens(K) ])
         sage: z_cone.linear_subspace() == lls
         True
 
     """
-    # 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.
-    V = K.lattice().vector_space()
-    W = VectorSpace(V.base_ring(), V.dimension()**2)
+    # Matrices are not vectors in Sage, so we have to convert them
+    # to vectors explicitly before we can find a basis. We need these
+    # two values to construct the appropriate "long vector" space.
+    F = K.lattice().base_field()
+    n = K.lattice_dim()
 
-    C_of_K = K.discrete_complementarity_set()
-    tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+    # These tensor products contain generators for the dual cone of
+    # the cross-positive transformations.
+    tensor_products = [ s.tensor_product(x)
+                        for (x,s) in K.discrete_complementarity_set() ]
 
     # Turn our matrices into long vectors...
+    W = VectorSpace(F, n**2)
     vectors = [ W(m.list()) for m in tensor_products ]
 
     # Create the *dual* cone of the cross-positive operators,
     # expressed as long vectors..
-    L = ToricLattice(W.dimension())
-    Sigma_dual = Cone(vectors, lattice=L)
+    Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
 
     # Now compute the desired cone from its dual...
     Sigma_cone = Sigma_dual.dual()
@@ -308,6 +312,5 @@ def Z_transformations(K):
     # And finally convert its rays back to matrix representations.
     # But first, make them negative, so we get Z-transformations and
     # not cross-positive ones.
-    M = MatrixSpace(V.base_ring(), V.dimension())
-
+    M = MatrixSpace(F, n)
     return [ -M(v.list()) for v in Sigma_cone.rays() ]