From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 2 Oct 2015 14:50:46 +0000 (-0400)
Subject: Remove function LL() for Sage proper.
X-Git-Url: http://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=8cc66a7a6e7ffbba7ff2f3894cc449ca3b7d566f;p=sage.d.git

Remove function LL() for Sage proper.
---

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index f8b8791..f6f378e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -76,7 +76,7 @@ def _basically_the_same(K1, K2):
     if K1.is_strictly_convex() != K2.is_strictly_convex():
         return False
 
-    if len(LL(K1)) != len(LL(K2)):
+    if len(K1.LL()) != len(K2.LL()):
         return False
 
     C_of_K1 = K1.discrete_complementarity_set()
@@ -211,134 +211,6 @@ def _restrict_to_space(K, W):
     return Cone(K_W_rays, lattice=L)
 
 
-def LL(K):
-    r"""
-    Compute a basis of Lyapunov-like transformations on this cone.
-
-    OUTPUT:
-
-    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]
-        ]
-
-    If our cone is the entire space, then every transformation on it is
-    Lyapunov-like::
-
-        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
-        sage: M = MatrixSpace(QQ,2)
-        sage: M.basis() == LL(K)
-        True
-
-    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: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: C_of_K = K.discrete_complementarity_set()
-        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
-
-    The Lyapunov-like transformations on a cone and its dual are related
-    by transposition, but we're not guaranteed to compute transposed
-    elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
-    \right)`
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
-        sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
-        sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
-        sage: LL2_vecs = [ V(m.list()) for m in LL2   ]
-        sage: V.span(LL1_vecs) == V.span(LL2_vecs)
-        True
-
-    """
-    V = K.lattice().vector_space()
-
-    C_of_K = K.discrete_complementarity_set()
-
-    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
-    # 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)
-
-    # Turn our matrices into long vectors...
-    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).
-    LL_vector = W.span(vectors).complement()
-
-    # Now construct an ambient MatrixSpace in which to stick our
-    # transformations.
-    M = MatrixSpace(V.base_ring(), V.dimension())
-
-    matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
-
-    return matrix_basis
-
-
-
 def lyapunov_rank(K):
     r"""
     Compute the Lyapunov rank (or bilinearity rank) of this cone.
@@ -542,11 +414,11 @@ def lyapunov_rank(K):
         sage: actual == expected
         True
 
-    The Lyapunov rank of any cone is just the dimension of ``LL(K)``::
+    The Lyapunov rank of any cone is just the dimension of ``K.LL()``::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
-        sage: lyapunov_rank(K) == len(LL(K))
+        sage: lyapunov_rank(K) == len(K.LL())
         True
 
     We can make an imperfect cone perfect by adding a slack variable
@@ -584,7 +456,7 @@ def lyapunov_rank(K):
         # Non-pointed reduction lemma.
         beta += l * m
 
-    beta += len(LL(K))
+    beta += len(K.LL())
     return beta
 
 
@@ -642,10 +514,10 @@ def is_lyapunov_like(L,K):
         sage: is_lyapunov_like(L,K)
         True
 
-    Everything in ``LL(K)`` should be Lyapunov-like on ``K``::
+    Everything in ``K.LL()`` should be Lyapunov-like on ``K``::
 
         sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
-        sage: all([is_lyapunov_like(L,K) for L in LL(K)])
+        sage: all([is_lyapunov_like(L,K) for L in K.LL()])
         True
 
     """