X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=6ad7c52904d6ba5e47ffe88f9d35b7934a857b2a;hb=5d4c579d47ef0c9228ae1d528850e92d77aaa9b0;hp=f6f378e61cd71c93eb4abf3de017272d11192e0f;hpb=8cc66a7a6e7ffbba7ff2f3894cc449ca3b7d566f;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index f6f378e..6ad7c52 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -637,13 +637,11 @@ 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: all([K.contains(p*x) for p in pi_of_k for x in K.rays()])
+        sage: pi_of_K = positive_operators(K)
+        sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
         True
 
     """
-    V = K.lattice().vector_space()
-
     # 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.
@@ -652,12 +650,10 @@ def positive_operators(K):
     # 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)
 
-    G1 = [ V(x) for x in K.rays() ]
-    G2 = [ V(s) for s in K.dual().rays() ]
-
-    tensor_products = [ s.tensor_product(x) for x in G1 for s in G2 ]
+    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 ]
@@ -674,3 +670,104 @@ def positive_operators(K):
     M = MatrixSpace(V.base_ring(), V.dimension())
 
     return [ M(v.list()) for v in pi_cone.rays() ]
+
+
+def Z_transformations(K):
+    r"""
+    Compute generators of the cone of Z-transformations on this cone.
+
+    OUTPUT:
+
+    A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
+    Each matrix ``L`` in the list should have the property that
+    ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
+    discrete complementarity set of ``K``. Moreover, any nonnegative
+    linear combination of these matrices shares the same property.
+
+    EXAMPLES:
+
+    Z-transformations on the nonnegative orthant are just Z-matrices.
+    That is, matrices whose off-diagonal elements are nonnegative::
+
+        sage: K = Cone([(1,0),(0,1)])
+        sage: Z_transformations(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)
+        ....:                    for i in range(z.nrows())
+        ....:                    for j in range(z.ncols())
+        ....:                    if i != j ])
+        True
+
+    The trivial cone in a trivial space has no Z-transformations::
+
+        sage: K = Cone([], ToricLattice(0))
+        sage: Z_transformations(K)
+        []
+
+    Z-transformations on a subspace are Lyapunov-like and vice-versa::
+
+        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+        sage: K.is_full_space()
+        True
+        sage: llvs = span([ vector(l.list()) for l in K.LL() ])
+        sage: zvs  = span([ vector(z.list()) for z in Z_transformations(K) ])
+        sage: zvs == llvs
+        True
+
+    TESTS:
+
+    The Z-property is possessed by every Z-transformation::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim = 6)
+        sage: Z_of_K = Z_transformations(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])
+        True
+
+    The lineality space of Z is LL::
+
+        sage: set_random_seed()
+        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+        sage: llvs = span([ vector(l.list()) for l in K.LL() ])
+        sage: z_cone  = Cone([ z.list() for z in Z_transformations(K) ])
+        sage: z_cone.linear_subspace() == llvs
+        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)
+
+    C_of_K = K.discrete_complementarity_set()
+    tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+
+    # Turn our matrices into long vectors...
+    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)
+
+    # Now compute the desired cone from its dual...
+    Sigma_cone = Sigma_dual.dual()
+
+    # 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())
+
+    return [ -M(v.list()) for v in Sigma_cone.rays() ]