X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=a1ded5270032de21f2647de8453384fde4804c72;hb=090b2c77aa4bd371d66885451f9df44c6b6d818f;hp=a5f5f2f4fcf1ac603a9d8d48a75a0924003bd8cf;hpb=cd746d047748a9aa56c51bb78c3db9882ea16a16;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index a5f5f2f..a1ded52 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -65,81 +65,95 @@ def is_lyapunov_like(L,K):
                 for (x,s) in K.discrete_complementarity_set()])
 
 
-def random_element(K):
+def motzkin_decomposition(K):
     r"""
-    Return a random element of ``K`` from its ambient vector space.
+    Return the pair of components in the motzkin decomposition of this cone.
 
-    ALGORITHM:
+    Every convex cone is the direct sum of a strictly convex cone and a
+    linear subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
+    strictly convex, ``S`` is a subspace, and ``K`` is the direct sum of
+    ``P`` and ``S``.
 
-    The cone ``K`` is specified in terms of its generators, so that
-    ``K`` is equal to the convex conic combination of those generators.
-    To choose a random element of ``K``, we assign random nonnegative
-    coefficients to each generator of ``K`` and construct a new vector
-    from the scaled rays.
+    OUTPUT:
 
-    A vector, rather than a ray, is returned so that the element may
-    have non-integer coordinates. Thus the element may have an
-    arbitrarily small norm.
+    An ordered pair ``(P,S)`` of closed convex polyhedral cones where
+    ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
+    direct sum of ``P`` and ``S``.
 
     EXAMPLES:
 
-    A random element of the trivial cone is zero::
+    The nonnegative orthant is strictly convex, so it is its own
+    strictly convex component and its subspace component is trivial::
 
-        sage: set_random_seed()
-        sage: K = Cone([], ToricLattice(0))
-        sage: random_element(K)
-        ()
-        sage: K = Cone([(0,)])
-        sage: random_element(K)
-        (0)
-        sage: K = Cone([(0,0)])
-        sage: random_element(K)
-        (0, 0)
-        sage: K = Cone([(0,0,0)])
-        sage: random_element(K)
-        (0, 0, 0)
-
-    A random element of the nonnegative orthant should have all
-    components nonnegative::
-
-        sage: set_random_seed()
         sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
-        sage: all([ x >= 0 for x in random_element(K) ])
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: K.is_equivalent(P)
+        True
+        sage: S.is_trivial()
+        True
+
+    Likewise, full spaces are their own subspace components::
+
+        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+        sage: K.is_full_space()
+        True
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: K.is_equivalent(S)
+        True
+        sage: P.is_trivial()
         True
 
     TESTS:
 
-    Any cone should contain a random element of itself::
+    A random point in the cone should belong to either the strictly
+    convex component or the subspace component. If the point is nonzero,
+    it cannot be in both::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
-        sage: K.contains(random_element(K))
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: x = K.random_element()
+        sage: P.contains(x) or S.contains(x)
+        True
+        sage: x.is_zero() or (P.contains(x) != S.contains(x))
         True
 
-    A strictly convex cone contains no lines, and thus no negative
-    multiples of any of its elements besides zero::
+    The strictly convex component should always be strictly convex, and
+    the subspace component should always be a subspace::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
-        sage: x = random_element(K)
-        sage: x.is_zero() or not K.contains(-x)
+        sage: K = random_cone(max_ambient_dim=8)
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: P.is_strictly_convex()
+        True
+        sage: S.lineality() == S.dim()
         True
 
-    The sum of random elements of a cone lies in the cone::
+    The generators of the strictly convex component are obtained from
+    the orthogonal projections of the original generators onto the
+    orthogonal complement of the subspace component::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
-        sage: K.contains(sum([random_element(K) for i in range(10)]))
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: S_perp = S.linear_subspace().complement()
+        sage: A = S_perp.matrix().transpose()
+        sage: proj = A * (A.transpose()*A).inverse() * A.transpose()
+        sage: expected = Cone([ proj*g for g in K ], K.lattice())
+        sage: P.is_equivalent(expected)
         True
-
     """
-    V = K.lattice().vector_space()
-    scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ]
+    linspace_gens  = [ copy(b) for b in K.linear_subspace().basis() ]
+    linspace_gens += [ -b for b in linspace_gens ]
+
+    S = Cone(linspace_gens, K.lattice())
 
-    # Make sure we return a vector. Without the coercion, we might
-    # return ``0`` when ``K`` has no rays.
-    return V(sum(scaled_gens))
+    # Since ``S`` is a subspace, its dual is its orthogonal complement
+    # (albeit in the wrong lattice).
+    S_perp = Cone(S.dual(), K.lattice())
+    P = K.intersection(S_perp)
 
+    return (P,S)
 
 def positive_operator_gens(K):
     r"""