X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ba5f51ea880ccdc2cc3344cb8b91022ff3e5b8cf;hb=e041595c10751828f196db2cda86bd0f15a81191;hp=f5371d6d803cdbc97fe4ba41cb26f0ee691f1a2a;hpb=4418c497a443fb1f5cb068ced5a2ddd5a9a0ad05;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index f5371d6..ba5f51e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -12,18 +12,11 @@ def drop_dependent(vs):
     r"""
     Return the largest linearly-independent subset of ``vs``.
     """
-    if len(vs) == 0:
-        # ...for lazy enough definitions of linearly-independent
-        return vs
-
     result = []
-    old_V = VectorSpace(vs[0].parent().base_field(), 0)
-
-    for v in vs:
-        new_V = span(result + [v])
-        if new_V.dimension() > old_V.dimension():
-            result.append(v)
-            old_V = new_V
+    m = matrix(vs).echelon_form()
+    for idx in range(0, m.nrows()):
+        if not m[idx].is_zero():
+            result.append(m[idx])
 
     return result
 
@@ -66,88 +59,7 @@ def basically_the_same(K1,K2):
 
 
 
-def iso_space(K):
-    r"""
-    Construct the space `W \times W^{\perp}` isomorphic to the ambient space
-    of ``K`` where `W` is equal to the span of ``K``.
-    """
-    V = K.lattice().vector_space()
-
-    # Create the space W \times W^{\perp} isomorphic to V.
-    # First we get an orthogonal (but not normal) basis...
-    M = matrix(V.base_field(), K.rays())
-    W_basis = drop_dependent(K.rays())
-
-    W = V.subspace_with_basis(W_basis)
-    W_perp = W.complement()
-
-    return W.cartesian_product(W_perp)
-
-
-def ips_iso(K):
-    r"""
-    Construct the IPS isomorphism and its inverse from our paper.
-
-    Given a cone ``K``, the returned isomorphism will split its ambient
-    vector space `V` into a cartesian product `W \times W^{\perp}` where
-    `W` equals the span of ``K``.
-    """
-    V = K.lattice().vector_space()
-    V_iso = iso_space(K)
-    (W, W_perp) = V_iso.cartesian_factors()
-
-    # A space equivalent to V, but using our basis.
-    V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
-
-    def phi(v):
-        # Write v in terms of our custom basis, where the first dim(W)
-        # coordinates are for the W-part of the basis.
-        cs = V_user.coordinates(v)
-
-        w1 = sum([ V_user.basis()[idx]*cs[idx]
-                    for idx in range(0, W.dimension()) ])
-        w2 = sum([ V_user.basis()[idx]*cs[idx]
-                    for idx in range(W.dimension(), V.dimension()) ])
-
-        return V_iso( (w1, w2) )
-
-
-    def phi_inv( pair ):
-        # Crash if the arguments are in the wrong spaces.
-        V_iso(pair)
-
-        #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
-        #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
-        #             for idx in range(0,n-m) ])
-
-        return sum( pair.cartesian_factors() )
-
-
-    return (phi,phi_inv)
-
-
-
-def unrestrict_span(K, K2=None):
-    if K2 is None:
-        K2 = K
-
-    _,phi_inv = ips_iso(K2)
-    V_iso = iso_space(K2)
-    (W, W_perp) = V_iso.cartesian_factors()
-
-    rays = []
-    for r in K.rays():
-        w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ])
-        pair = V_iso( (w, W_perp.zero()) )
-        rays.append( phi_inv(pair) )
-
-    L = ToricLattice(W.dimension() + W_perp.dimension())
-
-    return Cone(rays, lattice=L)
-
-
-
-def restrict_span(K, K2=None):
+def rho(K, K2=None):
     r"""
     Restrict ``K`` into its own span, or the span of another cone.
 
@@ -162,18 +74,18 @@ def restrict_span(K, K2=None):
     EXAMPLES::
 
         sage: K = Cone([(1,)])
-        sage: restrict_span(K) == K
+        sage: rho(K) == K
         True
 
         sage: K2 = Cone([(1,0)])
-        sage: restrict_span(K2).rays()
+        sage: rho(K2).rays()
         N(1)
         in 1-d lattice N
         sage: K3 = Cone([(1,0,0)])
-        sage: restrict_span(K3).rays()
+        sage: rho(K3).rays()
         N(1)
         in 1-d lattice N
-        sage: restrict_span(K2) == restrict_span(K3)
+        sage: rho(K2) == rho(K3)
         True
 
     TESTS:
@@ -182,7 +94,7 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8)
-        sage: K_S = restrict_span(K)
+        sage: K_S = rho(K)
         sage: K_S.is_solid()
         True
 
@@ -191,30 +103,22 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8)
-        sage: K_S = restrict_span(K, K.dual() )
+        sage: K_S = rho(K, K.dual() )
         sage: K_S.lattice_dim() == K.dual().dim()
         True
 
-    This function has ``unrestrict_span()`` as its inverse::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_dim = 8, solid=True)
-        sage: J = restrict_span(K)
-        sage: K == unrestrict_span(J,K)
-        True
-
     This function should not affect the dimension of a cone::
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8)
-        sage: K.dim() == restrict_span(K).dim()
+        sage: K.dim() == rho(K).dim()
         True
 
     Nor should it affect the lineality of a cone::
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8)
-        sage: lineality(K) == lineality(restrict_span(K))
+        sage: lineality(K) == lineality(rho(K))
         True
 
     No matter which space we restrict to, the lineality should not
@@ -222,20 +126,20 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8)
-        sage: lineality(K) >= lineality(restrict_span(K))
+        sage: lineality(K) >= lineality(rho(K))
         True
-        sage: lineality(K) >= lineality(restrict_span(K, K.dual()))
+        sage: lineality(K) >= lineality(rho(K, K.dual()))
         True
 
     If we do this according to our paper, then the result is proper::
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
-        sage: K_S = restrict_span(K)
-        sage: P = restrict_span(K_S.dual()).dual()
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
         sage: P.is_proper()
         True
-        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P = rho(K_S, K_S.dual())
         sage: P.is_proper()
         True
 
@@ -243,11 +147,11 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
-        sage: K_S = restrict_span(K)
-        sage: P = restrict_span(K_S.dual()).dual()
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
         sage: P.is_proper()
         True
-        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P = rho(K_S, K_S.dual())
         sage: P.is_proper()
         True
 
@@ -255,11 +159,11 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
-        sage: K_S = restrict_span(K)
-        sage: P = restrict_span(K_S.dual()).dual()
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
         sage: P.is_proper()
         True
-        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P = rho(K_S, K_S.dual())
         sage: P.is_proper()
         True
 
@@ -267,11 +171,11 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
-        sage: K_S = restrict_span(K)
-        sage: P = restrict_span(K_S.dual()).dual()
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
         sage: P.is_proper()
         True
-        sage: P = restrict_span(K_S, K_S.dual())
+        sage: P = rho(K_S, K_S.dual())
         sage: P.is_proper()
         True
 
@@ -280,8 +184,8 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
-        sage: K_W = restrict_span(K, K.dual())
-        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
         sage: basically_the_same(K_W, K_star_W_star)
         True
 
@@ -289,8 +193,8 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
-        sage: K_W = restrict_span(K, K.dual())
-        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
         sage: basically_the_same(K_W, K_star_W_star)
         True
 
@@ -298,8 +202,8 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
-        sage: K_W = restrict_span(K, K.dual())
-        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
         sage: basically_the_same(K_W, K_star_W_star)
         True
 
@@ -307,8 +211,8 @@ def restrict_span(K, K2=None):
 
         sage: set_random_seed()
         sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
-        sage: K_W = restrict_span(K, K.dual())
-        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
         sage: basically_the_same(K_W, K_star_W_star)
         True
 
@@ -316,24 +220,25 @@ def restrict_span(K, K2=None):
     if K2 is None:
         K2 = K
 
-    phi,_ = ips_iso(K2)
-    (W, W_perp) = iso_space(K2).cartesian_factors()
-
-    ray_pairs = [ phi(r) for r in K.rays() ]
+    # First we project K onto the span of K2. This can be done with
+    # cones (i.e. without converting to vector spaces), but it's
+    # annoying to deal with lattice mismatches.
+    span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+    K = K.intersection(span_K2)
 
-    # Shouldn't matter?
-    #
-    #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
-    #    msg = 'Cone has nonzero components in W-perp!'
-    #    raise ValueError(msg)
+    V = K.lattice().vector_space()
 
-    # Represent the cone in terms of a basis for W, i.e. with smaller
-    # vectors.
-    ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
+    # Create the space W \times W^{\perp} isomorphic to V.
+    # First we get an orthogonal (but not normal) basis...
+    W_basis = drop_dependent(K2.rays())
+    W = V.subspace_with_basis(W_basis)
 
-    L = ToricLattice(W.dimension())
+    # We've already intersected K with the span of K2, so every
+    # generator of K should belong to W now.
+    W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
 
-    return Cone(ws, lattice=L)
+    L = ToricLattice(K2.dim())
+    return Cone(W_rays, lattice=L)
 
 
 
@@ -408,6 +313,100 @@ def lineality(K):
     return K.linear_subspace().dimension()
 
 
+def codim(K):
+    r"""
+    Compute the codimension of this cone.
+
+    The codimension of a cone is the dimension of the space of all
+    elements perpendicular to every element of the cone. In other words,
+    the codimension is the difference between the dimension of the
+    ambient space and the dimension of the cone itself.
+
+    OUTPUT:
+
+    A nonnegative integer representing the dimension of the space of all
+    elements perpendicular to this cone.
+
+    .. seealso::
+
+        :meth:`dim`, :meth:`lattice_dim`
+
+    EXAMPLES:
+
+    The codimension of the nonnegative orthant is zero, since the span of
+    its generators equals the entire ambient space::
+
+        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: codim(K)
+        0
+
+    However, if we remove a ray so that the entire cone is contained
+    within the `x-y`-plane, then the resulting cone will have
+    codimension one, because the `z`-axis is perpendicular to every
+    element of the cone::
+
+        sage: K = Cone([(1,0,0), (0,1,0)])
+        sage: codim(K)
+        1
+
+    If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: codim(K)
+        0
+
+    And if the cone is trivial in any space, then its codimension is
+    equal to the dimension of the ambient space::
+
+        sage: K = Cone([], lattice=ToricLattice(0))
+        sage: K.lattice_dim()
+        0
+        sage: codim(K)
+        0
+
+        sage: K = Cone([(0,)])
+        sage: K.lattice_dim()
+        1
+        sage: codim(K)
+        1
+
+        sage: K = Cone([(0,0)])
+        sage: K.lattice_dim()
+        2
+        sage: codim(K)
+        2
+
+    TESTS:
+
+    The codimension of a cone should be an integer between zero and
+    the dimension of the ambient space, inclusive::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: c = codim(K)
+        sage: c in ZZ
+        True
+        sage: (0 <= c) and (c <= K.lattice_dim())
+        True
+
+    A solid cone should have codimension zero::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid = True)
+        sage: codim(K)
+        0
+
+    The codimension of a cone is equal to the lineality of its dual::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid = True)
+        sage: codim(K) == lineality(K.dual())
+        True
+
+    """
+    return (K.lattice_dim() - K.dim())
+
+
 def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
@@ -827,8 +826,8 @@ def lyapunov_rank(K):
         sage: set_random_seed()
         sage: K = random_cone(max_dim=8)
         sage: actual = lyapunov_rank(K)
-        sage: K_S = restrict_span(K)
-        sage: P = restrict_span(K_S.dual()).dual()
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
         sage: l = lineality(K)
         sage: c = codim(K)
         sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
@@ -874,7 +873,7 @@ def lyapunov_rank(K):
 
     if m < n:
         # K is not solid, project onto its span.
-        K = restrict_span(K)
+        K = rho(K)
 
         # Lemma 2
         beta += m*(n - m) + (n - m)**2
@@ -882,9 +881,8 @@ def lyapunov_rank(K):
     if l > 0:
         # K is not pointed, project its dual onto its span.
         # Uses a proposition from our paper, i.e. this is
-        # equivalent to K = restrict_span(K.dual()).dual()
-        #K = restrict_span(intersect_span(K,K.dual()), K.dual())
-        K = restrict_span(K, K.dual())
+        # equivalent to K = rho(K.dual()).dual()
+        K = rho(K, K.dual())
 
         # Lemma 3
         beta += m * l