]> gitweb.michael.orlitzky.com - sage.d.git/commitdiff
In the middle of mangling things.
authorMichael Orlitzky <michael@orlitzky.com>
Fri, 12 Jun 2015 23:13:31 +0000 (19:13 -0400)
committerMichael Orlitzky <michael@orlitzky.com>
Fri, 12 Jun 2015 23:13:31 +0000 (19:13 -0400)
mjo/cone/cone.py

index 2d84337fda1cb5275add8171ba8e982559dc1853..b9e930e6819643710b82c06faa0b72b934298d96 100644 (file)
@@ -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
 
@@ -66,66 +59,6 @@ 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 rho(K, K2=None):
     r"""
     Restrict ``K`` into its own span, or the span of another cone.
@@ -141,18 +74,18 @@ def rho(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:
@@ -161,7 +94,7 @@ def rho(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
 
@@ -170,7 +103,7 @@ def rho(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
 
@@ -178,14 +111,14 @@ def rho(K, K2=None):
 
         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
@@ -193,20 +126,20 @@ def rho(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
 
@@ -214,11 +147,11 @@ def rho(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
 
@@ -226,11 +159,11 @@ def rho(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
 
@@ -238,11 +171,11 @@ def rho(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
 
@@ -251,8 +184,8 @@ def rho(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
 
@@ -260,8 +193,8 @@ def rho(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
 
@@ -269,8 +202,8 @@ def rho(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
 
@@ -278,8 +211,8 @@ def rho(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
 
@@ -287,24 +220,25 @@ def rho(K, K2=None):
     if K2 is None:
         K2 = K
 
-    phi,_ = ips_iso(K2)
-    (W, W_perp) = iso_space(K2).cartesian_factors()
+    # 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)
 
-    ray_pairs = [ phi(r) for r in K.rays() ]
-
-    # 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)
 
 
 
@@ -798,8 +732,8 @@ 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
@@ -845,7 +779,7 @@ 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
@@ -853,9 +787,8 @@ 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