Add a description for the pointed_decomposition() doctest.

[sage.d.git] / mjo / cone / cone.py

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 702472c30466f7cba79d4a79f1bf47a552079364..daeeb8936edb5416b269a131afe95c286da75598 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -40,14 +40,14 @@ def is_lyapunov_like(L,K):
     The identity is always Lyapunov-like in a nontrivial space::
 
         sage: set_random_seed()
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
         sage: L = identity_matrix(K.lattice_dim())
         sage: is_lyapunov_like(L,K)
         True
 
     As is the "zero" transformation::
 
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
         sage: R = K.lattice().vector_space().base_ring()
         sage: L = zero_matrix(R, K.lattice_dim())
         sage: is_lyapunov_like(L,K)
@@ -56,7 +56,7 @@ def is_lyapunov_like(L,K):
         Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
         on ``K``::
 
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
         sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
         True
 
@@ -99,28 +99,87 @@ def random_element(K):
         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) ])
+        True
+
     TESTS:
 
-    Any cone should contain an element of itself::
+    Any cone should contain a random element of itself::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_rays = 8)
+        sage: K = random_cone(max_ambient_dim=8)
         sage: K.contains(random_element(K))
         True
 
+    A strictly convex cone contains no lines, and thus no negative
+    multiples of any of its elements besides zero::
+
+        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)
+        True
+
+    The sum of random elements of a cone lies in the cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8)
+        sage: K.contains(sum([random_element(K) for i in range(10)]))
+        True
+
     """
     V = K.lattice().vector_space()
-    F = V.base_ring()
-    coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
-    vector_gens  = map(V, K.rays())
-    scaled_gens  = [ coefficients[i]*vector_gens[i]
-                         for i in range(len(vector_gens)) ]
+    scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ]
 
     # Make sure we return a vector. Without the coercion, we might
     # return ``0`` when ``K`` has no rays.
-    v = V(sum(scaled_gens))
-    return v
+    return V(sum(scaled_gens))
+
+
+def pointed_decomposition(K):
+    """
+    Every convex cone is the direct sum of a pointed cone and a linear
+    subspace. Return a pair ``(P,S)`` of cones such that ``P`` is
+    pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P``
+    and ``S``.
 
+    OUTPUT:
+
+    An ordered pair ``(P,S)`` of closed convex polyhedral cones where
+    ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum
+    of ``P`` and ``S``.
+
+    TESTS:
+
+    A random point in the cone should belong to either the pointed
+    subcone ``P`` or the subspace ``S``. If the point is nonzero, it
+    should lie in one but not both of them::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=8)
+        sage: (P,S) = pointed_decomposition(K)
+        sage: x = random_element(K)
+        sage: P.contains(x) or S.contains(x)
+        True
+        sage: x.is_zero() or (P.contains(x) != S.contains(x))
+        True
+    """
+    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())
+
+    # 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"""
@@ -186,7 +245,7 @@ def positive_operator_gens(K):
     corollary in my paper::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 5)
+        sage: K = random_cone(max_ambient_dim=5)
         sage: n = K.lattice_dim()
         sage: m = K.dim()
         sage: l = K.lineality()
@@ -296,7 +355,7 @@ def Z_transformation_gens(K):
     The Z-property is possessed by every Z-transformation::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 6)
+        sage: K = random_cone(max_ambient_dim=6)
         sage: Z_of_K = Z_transformation_gens(K)
         sage: dcs = K.discrete_complementarity_set()
         sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
@@ -306,7 +365,7 @@ def Z_transformation_gens(K):
     The lineality space of Z is LL::
 
         sage: set_random_seed()
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
         sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
         sage: z_cone  = Cone([ z.list() for z in Z_transformation_gens(K) ])
         sage: z_cone.linear_subspace() == lls