X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b9e930e6819643710b82c06faa0b72b934298d96;hb=6bd30534d5aa984c73f511121efa8fda4386c51a;hp=f2e8b2e9ee104e6cbd216e7473fd65dd76947d98;hpb=44802773ad9e5151890ed37e7bb2463ff9fc4135;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index f2e8b2e..b9e930e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,106 +8,58 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
-def iso_space(K):
+def drop_dependent(vs):
     r"""
-    Construct the space `W \times W^{\perp}` isomorphic to the ambient space
-    of ``K`` where `W` is equal to the span of ``K``.
+    Return the largest linearly-independent subset of ``vs``.
     """
-    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,_ = M.gram_schmidt()
-
-    W = V.subspace_with_basis(W_basis)
-    W_perp = W.complement()
+    result = []
+    m = matrix(vs).echelon_form()
+    for idx in range(0, m.nrows()):
+        if not m[idx].is_zero():
+            result.append(m[idx])
 
-    return W.cartesian_product(W_perp)
+    return result
 
 
-def ips_iso(K):
+def basically_the_same(K1,K2):
     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``.
+    ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+    otherwise.
     """
-    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)
-
+    if K1.lattice_dim() != K2.lattice_dim():
+        return False
 
+    if K1.nrays() != K2.nrays():
+        return False
 
-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())
+    if K1.dim() != K2.dim():
+        return False
 
-    return Cone(rays, lattice=L)
+    if lineality(K1) != lineality(K2):
+        return False
 
+    if K1.is_solid() != K2.is_solid():
+        return False
 
+    if K1.is_strictly_convex() != K2.is_strictly_convex():
+        return False
 
-def intersect_span(K1, K2):
-    r"""
-    Return a new cone obtained by intersecting ``K1`` with the span of ``K2``.
-    """
-    L = K1.lattice()
+    if len(LL(K1)) != len(LL(K2)):
+        return False
 
-    if L.rank() != K2.lattice().rank():
-        raise ValueError('K1 and K2 must belong to lattices of the same rank.')
+    C_of_K1 = discrete_complementarity_set(K1)
+    C_of_K2 = discrete_complementarity_set(K2)
+    if len(C_of_K1) != len(C_of_K2):
+        return False
 
-    SL_gens = list(K2.rays())
-    span_K2_gens = SL_gens + [ -g for g in SL_gens ]
+    if len(K1.facets()) != len(K2.facets()):
+        return False
 
-    # The lattices have the same rank (see above) so this should work.
-    span_K2 = Cone(span_K2_gens, L)
-    return K1.intersection(span_K2)
+    return True
 
 
 
-def restrict_span(K, K2=None):
+def rho(K, K2=None):
     r"""
     Restrict ``K`` into its own span, or the span of another cone.
 
@@ -122,18 +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:
@@ -141,8 +93,8 @@ def restrict_span(K, K2=None):
     The projected cone should always be solid::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10)
-        sage: K_S = restrict_span(K)
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K)
         sage: K_S.is_solid()
         True
 
@@ -150,104 +102,143 @@ def restrict_span(K, K2=None):
     dimension as the space we restricted it to::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10)
-        sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() )
+        sage: K = random_cone(max_dim = 8)
+        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 = 10, 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 = 10)
-        sage: K.dim() == restrict_span(K).dim()
+        sage: K = random_cone(max_dim = 8)
+        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 = 10)
-        sage: lineality(K) == lineality(restrict_span(K))
+        sage: K = random_cone(max_dim = 8)
+        sage: lineality(K) == lineality(rho(K))
         True
 
     No matter which space we restrict to, the lineality should not
     increase::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10)
-        sage: J = intersect_span(K, K.dual())
-        sage: lineality(K) >= lineality(restrict_span(J, K.dual()))
+        sage: K = random_cone(max_dim = 8)
+        sage: lineality(K) >= lineality(rho(K))
+        True
+        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 = 10)
-        sage: K_S = restrict_span(K)
-        sage: P = restrict_span(K_S.dual()).dual()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
         sage: P.is_proper()
         True
 
-    If ``K`` is strictly convex, then both ``K_W`` and
-    ``K_star_W.dual()`` should equal ``K`` (after we unrestrict)::
+    ::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10, strictly_convex=True)
-        sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
-        sage: K_star_W_star = restrict_span(K.dual()).dual()
-        sage: j1 = unrestrict_span(K_W, K.dual())
-        sage: j2 = unrestrict_span(K_star_W_star, K.dual())
-        sage: j1 == j2
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
         True
-        sage: j1 == K
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
         True
-        sage: K; [ list(r) for r in K.rays() ]
 
     Test the proposition in our paper concerning the duals, where the
     subspace `W` is the span of `K^{*}`::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10, solid=False, strictly_convex=False)
-        sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
-        sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual()
-        sage: K_W.nrays() == K_star_W_star.nrays()
-        True
-        sage: K_W.dim() == K_star_W_star.dim()
+        sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+        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
-        sage: lineality(K_W) == lineality(K_star_W_star)
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+        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
-        sage: K_W.is_solid() == K_star_W_star.is_solid()
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+        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
-        sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex()
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+        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
 
     """
     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)
 
-    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)
 
 
 
@@ -304,7 +295,7 @@ def lineality(K):
     dimension of the ambient space, inclusive::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10)
+        sage: K = random_cone(max_dim = 8)
         sage: l = lineality(K)
         sage: l in ZZ
         True
@@ -314,7 +305,7 @@ def lineality(K):
     A strictly convex cone should have lineality zero::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim = 10, strictly_convex = True)
+        sage: K = random_cone(max_dim = 8, strictly_convex = True)
         sage: lineality(K)
         0
 
@@ -322,94 +313,6 @@ 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: codim(K)
-        0
-
-        sage: K = Cone([(0,)])
-        sage: codim(K)
-        1
-
-        sage: K = Cone([(0,0)])
-        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 = 10)
-        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 = 10, 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 = 10, 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.
@@ -543,6 +446,14 @@ def LL(K):
         [0 0 1]
         ]
 
+    If our cone is the entire space, then every transformation on it is
+    Lyapunov-like::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: M = MatrixSpace(QQ,2)
+        sage: M.basis() == LL(K)
+        True
+
     TESTS:
 
     The inner product `\left< L\left(x\right), s \right>` is zero for
@@ -550,7 +461,7 @@ def LL(K):
     of the cone::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=8, max_rays=10)
+        sage: K = random_cone(max_dim=8)
         sage: C_of_K = discrete_complementarity_set(K)
         sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
         sage: sum(map(abs, l))
@@ -562,7 +473,7 @@ def LL(K):
     \right)`
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=8, max_rays=10)
+        sage: K = random_cone(max_dim=8)
         sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
         sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
         sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
@@ -748,8 +659,8 @@ def lyapunov_rank(K):
     [Rudolf et al.]_::
 
         sage: set_random_seed()
-        sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True)
-        sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True)
         sage: K = K1.cartesian_product(K2)
         sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
         True
@@ -758,14 +669,35 @@ def lyapunov_rank(K):
     itself [Rudolf et al.]_::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=10, max_rays=10)
+        sage: K = random_cone(max_dim=8)
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
     Make sure we exercise the non-strictly-convex/non-solid case::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    Let's check the other permutations as well, just to be sure::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
@@ -776,7 +708,7 @@ def lyapunov_rank(K):
     the Lyapunov rank of the trivial cone will be zero::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
         sage: (n == 0 or 1 <= b) and b <= n
@@ -788,7 +720,7 @@ def lyapunov_rank(K):
     Lyapunov rank `n-1` in `n` dimensions::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=10)
+        sage: K = random_cone(max_dim=8)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
         sage: b == n-1
@@ -798,10 +730,10 @@ def lyapunov_rank(K):
     reduced to that of a proper cone [Orlitzky/Gowda]_::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=10)
+        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
@@ -811,7 +743,29 @@ def lyapunov_rank(K):
     The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    In fact the same can be said of any cone. These additional tests
+    just increase our confidence that the reduction scheme works::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
         sage: lyapunov_rank(K) == len(LL(K))
         True
 
@@ -825,7 +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
@@ -833,15 +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.dual()).dual()
-
-        #Ks = [ list(r) for r in sorted(K.rays()) ]
-        #Js = [ list(r) for r in sorted(J.rays()) ]
-
-        #if Ks != Js:
-        #    print [ list(r) for r in K_orig.rays() ]
+        # equivalent to K = rho(K.dual()).dual()
+        K = rho(K, K.dual())
 
         # Lemma 3
         beta += m * l