From 5559552f7e3ab5328a42359c2b4c6b238839b060 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Mon, 5 Oct 2015 01:05:47 -0400
Subject: [PATCH] Add Z_transformations() function.

---
 mjo/cone/cone.py | 81 ++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 81 insertions(+)

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 3f915ed..7d919e4 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -670,3 +670,84 @@ 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:
+
+    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: 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: 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 positive operators, expressed as
+    # long vectors..
+    L = ToricLattice(W.dimension())
+    Z_dual = Cone(vectors, lattice=L)
+
+    # Now compute the desired cone from its dual...
+    Z_cone = Z_dual.dual()
+
+    # And finally convert its rays back to matrix representations.
+    M = MatrixSpace(V.base_ring(), V.dimension())
+
+    return [ M(v.list()) for v in Z_cone.rays() ]
-- 
2.44.2