X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=4b0193692edd7655f2880408d143bf141e6d567c;hb=3a312d50b5aba08e72039f1ebcde7b12c62a1e9f;hp=a5482b3aa95f7198938007c4be615c4e7a97e17d;hpb=b97553aaaf9734644bee13bf484014f817456b26;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index a5482b3..4b01936 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -87,15 +87,77 @@ def LL(K):
 
     OUTPUT:
 
-    A ``MatrixSpace`` object `M` such that every matrix `L \in M` is
-    Lyapunov-like on this cone.
+    A list of matrices forming a basis for the space of all
+    Lyapunov-like transformations on the given cone.
+
+    EXAMPLES:
+
+    The trivial cone has no Lyapunov-like transformations::
+
+        sage: L = ToricLattice(0)
+        sage: K = Cone([], lattice=L)
+        sage: LL(K)
+        []
+
+    The Lyapunov-like transformations on the nonnegative orthant are
+    simply diagonal matrices::
+
+        sage: K = Cone([(1,)])
+        sage: LL(K)
+        [[1]]
+
+        sage: K = Cone([(1,0),(0,1)])
+        sage: LL(K)
+        [
+        [1 0]  [0 0]
+        [0 0], [0 1]
+        ]
+
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        sage: LL(K)
+        [
+        [1 0 0]  [0 0 0]  [0 0 0]
+        [0 0 0]  [0 1 0]  [0 0 0]
+        [0 0 0], [0 0 0], [0 0 1]
+        ]
+
+    Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
+    `L^{3}_{\infty}` cones [Rudolf et al.]_::
+
+        sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
+        sage: LL(L31)
+        [
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
+        ]
+
+        sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
+        sage: LL(L3infty)
+        [
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
+        ]
+
+    TESTS:
+
+    The inner product `\left< L\left(x\right), s \right>` is zero for
+    every pair `\left( x,s \right)` in the discrete complementarity set
+    of the cone::
+
+        sage: K = random_cone(max_dim=8, max_rays=10)
+        sage: C_of_K = discrete_complementarity_set(K)
+        sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
+        sage: sum(map(abs, l))
+        0
 
     """
     V = K.lattice().vector_space()
 
     C_of_K = discrete_complementarity_set(K)
 
-    matrices = [x.tensor_product(s) for (x,s) in C_of_K]
+    tensor_products = [s.tensor_product(x) for (x,s) in C_of_K]
 
     # Sage doesn't think matrices are vectors, so we have to convert
     # our matrices to vectors explicitly before we can figure out how
@@ -108,7 +170,7 @@ def LL(K):
     W = VectorSpace(V.base_ring(), V.dimension()**2)
 
     # Turn our matrices into long vectors...
-    vectors = [ W(m.list()) for m in matrices ]
+    vectors = [ W(m.list()) for m in tensor_products ]
 
     # Vector space representation of Lyapunov-like matrices
     # (i.e. vec(L) where L is Luapunov-like).
@@ -118,9 +180,9 @@ def LL(K):
     # transformations.
     M = MatrixSpace(V.base_ring(), V.dimension())
 
-    matrices = [ M(v.list()) for v in LL_vector.basis() ]
+    matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
 
-    return matrices
+    return matrix_basis
 
 
 
@@ -247,40 +309,17 @@ def lyapunov_rank(K):
 
     The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
     be any number between `1` and `n` inclusive, excluding `n-1`
-    [Gowda/Tao]_ (by accident, this holds for the trivial cone in a
-    trivial space as well)::
+    [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
+    trivial cone in a trivial space as well. However, in zero dimensions,
+    the Lyapunov rank of the trivial cone will be zero::
 
         sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
-        sage: 1 <= b and b <= n
+        sage: (n == 0 or 1 <= b) and b <= n
         True
         sage: b == n-1
         False
 
     """
-    V = K.lattice().vector_space()
-
-    C_of_K = discrete_complementarity_set(K)
-
-    matrices = [x.tensor_product(s) for (x,s) in C_of_K]
-
-    # 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.
-    W = VectorSpace(V.base_ring(), V.dimension()**2)
-
-    def phi(m):
-        r"""
-        Convert a matrix to a vector isomorphically.
-        """
-        return W(m.list())
-
-    vectors = [phi(m) for m in matrices]
-
-    return (W.dimension() - W.span(vectors).rank())
+    return len(LL(K))