X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ba5f51ea880ccdc2cc3344cb8b91022ff3e5b8cf;hb=e041595c10751828f196db2cda86bd0f15a81191;hp=60f9c34ec8bc271d65812859f51ca77636c8cbbc;hpb=874e3ce831e0b1901b3c280a32ffe18e36f54959;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 60f9c34..ba5f51e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -7,115 +7,404 @@ addsitedir(abspath('../../'))
 
 from sage.all import *
 
-def project_span(K):
+
+def drop_dependent(vs):
+    r"""
+    Return the largest linearly-independent subset of ``vs``.
+    """
+    result = []
+    m = matrix(vs).echelon_form()
+    for idx in range(0, m.nrows()):
+        if not m[idx].is_zero():
+            result.append(m[idx])
+
+    return result
+
+
+def basically_the_same(K1,K2):
+    r"""
+    ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+    otherwise.
+    """
+    if K1.lattice_dim() != K2.lattice_dim():
+        return False
+
+    if K1.nrays() != K2.nrays():
+        return False
+
+    if K1.dim() != K2.dim():
+        return False
+
+    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
+
+    if len(LL(K1)) != len(LL(K2)):
+        return False
+
+    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
+
+    if len(K1.facets()) != len(K2.facets()):
+        return False
+
+    return True
+
+
+
+def rho(K, K2=None):
     r"""
-    Project ``K`` into its own span.
+    Restrict ``K`` into its own span, or the span of another cone.
+
+    INPUT:
+
+    - ``K2`` -- another cone whose lattice has the same rank as this cone.
+
+    OUTPUT:
+
+    A new cone in a sublattice.
 
     EXAMPLES::
 
         sage: K = Cone([(1,)])
-        sage: project_span(K) == K
+        sage: rho(K) == K
         True
 
         sage: K2 = Cone([(1,0)])
-        sage: project_span(K2).rays()
+        sage: rho(K2).rays()
         N(1)
         in 1-d lattice N
         sage: K3 = Cone([(1,0,0)])
-        sage: project_span(K3).rays()
+        sage: rho(K3).rays()
         N(1)
         in 1-d lattice N
-        sage: project_span(K2) == project_span(K3)
+        sage: rho(K2) == rho(K3)
         True
 
     TESTS:
 
     The projected cone should always be solid::
 
-        sage: K = random_cone()
-        sage: K_S = project_span(K)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K)
         sage: K_S.is_solid()
         True
 
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
+
+        sage: set_random_seed()
+        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 should not affect the dimension of a cone::
+
+        sage: set_random_seed()
+        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 = 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 = 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: K = random_cone()
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
+        sage: set_random_seed()
+        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
 
-    """
-    F = K.lattice().base_field()
-    Q = K.lattice().quotient(K.sublattice_complement())
-    vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
+    ::
 
-    L = None
-    if len(vecs) == 0:
-        L = ToricLattice(0)
+        sage: set_random_seed()
+        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: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
 
-    return Cone(vecs, lattice=L)
+    ::
 
+        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
+
+    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 = 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: 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: 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: 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
 
-def rename_lattice(L,s):
-    r"""
-    Change all names of the given lattice to ``s``.
     """
-    L._name = s
-    L._dual_name = s
-    L._latex_name = s
-    L._latex_dual_name = s
+    if K2 is None:
+        K2 = K
+
+    # 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)
 
-def span_iso(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...
+    W_basis = drop_dependent(K2.rays())
+    W = V.subspace_with_basis(W_basis)
+
+    # 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() ]
+
+    L = ToricLattice(K2.dim())
+    return Cone(W_rays, lattice=L)
+
+
+
+def lineality(K):
     r"""
-    Return an isomorphism (and its inverse) that will send ``K`` into a
-    lower-dimensional space isomorphic to its span (and back).
+    Compute the lineality of this cone.
+
+    The lineality of a cone is the dimension of the largest linear
+    subspace contained in that cone.
+
+    OUTPUT:
+
+    A nonnegative integer; the dimension of the largest subspace
+    contained within this cone.
+
+    REFERENCES:
+
+    .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
+       University Press, Princeton, 1970.
 
     EXAMPLES:
 
-    The inverse composed with the isomorphism should be the identity::
+    The lineality of the nonnegative orthant is zero, since it clearly
+    contains no lines::
 
-        sage: K = random_cone(max_dim=10)
-        sage: (phi, phi_inv) = span_iso(K)
-        sage: phi_inv(phi(K)) == K
-        True
+        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: lineality(K)
+        0
+
+    However, if we add another ray so that the entire `x`-axis belongs
+    to the cone, then the resulting cone will have lineality one::
+
+        sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
+        sage: lineality(K)
+        1
+
+    If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
+    to the dimension of the ambient space (i.e. two)::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: lineality(K)
+        2
+
+    Per the definition, the lineality of the trivial cone in a trivial
+    space is zero::
 
-    The image of ``K`` under the isomorphism should have full dimension::
+        sage: K = Cone([], lattice=ToricLattice(0))
+        sage: lineality(K)
+        0
+
+    TESTS:
 
-        sage: K = random_cone(max_dim=10)
-        sage: (phi, phi_inv) = span_iso(K)
-        sage: phi(K).dim() == phi(K).lattice_dim()
+    The lineality 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: l = lineality(K)
+        sage: l in ZZ
+        True
+        sage: (0 <= l) and (l <= K.lattice_dim())
         True
 
+    A strictly convex cone should have lineality zero::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex = True)
+        sage: lineality(K)
+        0
+
     """
-    phi_domain = K.sublattice().vector_space()
-    phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
+    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.
 
-    # S goes from the new space to the cone space.
-    S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
+    OUTPUT:
+
+    A nonnegative integer representing the dimension of the space of all
+    elements perpendicular to this cone.
+
+    .. seealso::
+
+        :meth:`dim`, :meth:`lattice_dim`
+
+    EXAMPLES:
 
-    # phi goes from the cone space to the new space.
-    def phi(J_orig):
-        r"""
-        Takes a cone ``J`` and sends it into the new space.
-        """
-        newrays = map(S.inverse(), J_orig.rays())
-        L = None
-        if len(newrays) == 0:
-            L = ToricLattice(0)
+    The codimension of the nonnegative orthant is zero, since the span of
+    its generators equals the entire ambient space::
 
-        return Cone(newrays, lattice=L)
+        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: codim(K)
+        0
 
-    def phi_inverse(J_sub):
-        r"""
-        The inverse to phi which goes from the new space to the cone space.
-        """
-        newrays = map(S, J_sub.rays())
-        return Cone(newrays, lattice=K.lattice())
+    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
 
-    return (phi, phi_inverse)
+    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):
@@ -172,11 +461,12 @@ def discrete_complementarity_set(K):
     The complementarity set of the dual can be obtained by switching the
     components of the complementarity set of the original cone::
 
-        sage: K1 = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K1 = random_cone(max_dim=6)
         sage: K2 = K1.dual()
         sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
         sage: actual = discrete_complementarity_set(K1)
-        sage: actual == expected
+        sage: sorted(actual) == sorted(expected)
         True
 
     """
@@ -250,24 +540,47 @@ 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
     every pair `\left( x,s \right)` in the discrete complementarity set
     of the cone::
 
-        sage: K = random_cone(max_dim=8, max_rays=10)
+        sage: set_random_seed()
+        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))
         0
 
+    The Lyapunov-like transformations on a cone and its dual are related
+    by transposition, but we're not guaranteed to compute transposed
+    elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
+    \right)`
+
+        sage: set_random_seed()
+        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) ]
+        sage: LL2_vecs = [ V(m.list()) for m in LL2   ]
+        sage: V.span(LL1_vecs) == V.span(LL2_vecs)
+        True
+
     """
     V = K.lattice().vector_space()
 
     C_of_K = discrete_complementarity_set(K)
 
-    tensor_products = [s.tensor_product(x) for (x,s) in C_of_K]
+    tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
 
     # Sage doesn't think matrices are vectors, so we have to convert
     # our matrices to vectors explicitly before we can figure out how
@@ -365,6 +678,15 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(octant)
         3
 
+    The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+    [Orlitzky/Gowda]_::
+
+        sage: R5 = VectorSpace(QQ, 5)
+        sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+        sage: K = Cone(gs)
+        sage: lyapunov_rank(K)
+        25
+
     The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
     [Rudolf et al.]_::
 
@@ -378,7 +700,30 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(L3infty)
         1
 
-    The Lyapunov rank should be additive on a product of cones
+    A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
+    + 1` [Orlitzky/Gowda]_::
+
+        sage: K = Cone([(1,0,0,0,0)])
+        sage: lyapunov_rank(K)
+        21
+        sage: K.lattice_dim()**2 - K.lattice_dim() + 1
+        21
+
+    A subspace (of dimension `m`) in `n` dimensions should have a
+    Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
+
+        sage: e1 = (1,0,0,0,0)
+        sage: neg_e1 = (-1,0,0,0,0)
+        sage: e2 = (0,1,0,0,0)
+        sage: neg_e2 = (0,-1,0,0,0)
+        sage: z = (0,0,0,0,0)
+        sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
+        sage: lyapunov_rank(K)
+        19
+        sage: K.lattice_dim()**2 - K.dim()*codim(K)
+        19
+
+    The Lyapunov rank should be additive on a product of proper cones
     [Rudolf et al.]_::
 
         sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
@@ -404,11 +749,12 @@ def lyapunov_rank(K):
 
     TESTS:
 
-    The Lyapunov rank should be additive on a product of cones
+    The Lyapunov rank should be additive on a product of proper cones
     [Rudolf et al.]_::
 
-        sage: K1 = random_cone(max_dim=10, max_rays=10)
-        sage: K2 = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        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
@@ -416,7 +762,36 @@ def lyapunov_rank(K):
     The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
     itself [Rudolf et al.]_::
 
-        sage: K = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        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=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
 
@@ -426,7 +801,8 @@ def lyapunov_rank(K):
     trivial cone in a trivial space as well. However, in zero dimensions,
     the Lyapunov rank of the trivial cone will be zero::
 
-        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: set_random_seed()
+        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
@@ -437,7 +813,8 @@ def lyapunov_rank(K):
     In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
     Lyapunov rank `n-1` in `n` dimensions::
 
-        sage: K = random_cone(max_dim=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
         sage: b == n-1
@@ -446,79 +823,69 @@ def lyapunov_rank(K):
     The calculation of the Lyapunov rank of an improper cone can be
     reduced to that of a proper cone [Orlitzky/Gowda]_::
 
-        sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
         sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
+        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
         sage: actual == expected
         True
 
-    Repeat the previous test with different ``random_cone()`` params::
+    The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
 
-        sage: K = random_cone(max_dim=15, solid=False, strictly_convex=True)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
         True
 
-        sage: K = random_cone(max_dim=15, solid=True, strictly_convex=False)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
-        True
+    In fact the same can be said of any cone. These additional tests
+    just increase our confidence that the reduction scheme works::
 
-        sage: K = random_cone(max_dim=15, solid=True, strictly_convex=True)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
+        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: K = random_cone(max_dim=15)
-        sage: actual = lyapunov_rank(K)
-        sage: (phi1, _) = span_iso(K)
-        sage: K_S = phi1(K)
-        sage: (phi2, _) = span_iso(K_S.dual())
-        sage: J_T = phi2(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
         True
 
-    And test with the project_span function::
+    ::
 
-        sage: K = random_cone(max_dim=15)
-        sage: actual = lyapunov_rank(K)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2
-        sage: actual == expected
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == len(LL(K))
         True
 
     """
-    return len(LL(K))
+    K_orig = K
+    beta = 0
+
+    m = K.dim()
+    n = K.lattice_dim()
+    l = lineality(K)
+
+    if m < n:
+        # K is not solid, project onto its span.
+        K = rho(K)
+
+        # Lemma 2
+        beta += m*(n - m) + (n - m)**2
+
+    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 = rho(K.dual()).dual()
+        K = rho(K, K.dual())
+
+        # Lemma 3
+        beta += m * l
+
+    beta += len(LL(K))
+    return beta