From: Michael Orlitzky <michael@orlitzky.com>
Date: Mon, 5 Oct 2015 19:30:08 +0000 (-0400)
Subject: Fix Z_transformations() implementation and tests.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=a77c200cedbebe92a7cf4b412c0c7e0e405c57af;p=sage.d.git

Fix Z_transformations() implementation and tests.
---

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 7d919e4..6ad7c52 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -686,6 +686,22 @@ def Z_transformations(K):
 
     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))
@@ -706,15 +722,17 @@ def Z_transformations(K):
 
     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])
+        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) ])
@@ -739,15 +757,17 @@ def Z_transformations(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..
+    # Create the *dual* cone of the cross-positive operators,
+    # expressed as long vectors..
     L = ToricLattice(W.dimension())
-    Z_dual = Cone(vectors, lattice=L)
+    Sigma_dual = Cone(vectors, lattice=L)
 
     # Now compute the desired cone from its dual...
-    Z_cone = Z_dual.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 Z_cone.rays() ]
+    return [ -M(v.list()) for v in Sigma_cone.rays() ]