from sage.all import *
+
def project_span(K):
r"""
Project ``K`` into its own span.
The projected cone should always be solid::
- sage: K = random_cone()
+ sage: K = random_cone(max_dim = 10)
sage: K_S = project_span(K)
sage: K_S.is_solid()
True
If we do this according to our paper, then the result is proper::
- sage: K = random_cone()
+ sage: K = random_cone(max_dim = 10)
sage: K_S = project_span(K)
sage: P = project_span(K_S.dual()).dual()
sage: P.is_proper()
True
"""
- F = K.lattice().base_field()
- Q = K.lattice().quotient(K.sublattice_complement())
+ L = K.lattice()
+ F = L.base_field()
+ Q = L.quotient(K.sublattice_complement())
vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
- L = None
+ newL = None
if len(vecs) == 0:
- L = ToricLattice(0)
-
- return Cone(vecs, lattice=L)
-
-
-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
-
-def span_iso(K):
- r"""
- Return an isomorphism (and its inverse) that will send ``K`` into a
- lower-dimensional space isomorphic to its span (and back).
-
- EXAMPLES:
-
- The inverse composed with the isomorphism should be the identity::
-
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi_inv(phi(K)) == K
- True
-
- The image of ``K`` under the isomorphism should have full dimension::
-
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi(K).dim() == phi(K).lattice_dim()
- True
-
- """
- phi_domain = K.sublattice().vector_space()
- phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
-
- # S goes from the new space to the cone space.
- S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
-
- # 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)
+ newL = ToricLattice(0)
- return Cone(newrays, lattice=L)
-
- 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())
-
-
- return (phi, phi_inverse)
+ return Cone(vecs, lattice=newL)
sage: lyapunov_rank(octant)
3
+ The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+ [Orlitzky/Gowda]_::
+
+ sage: R5 = VectorSpace(QQ, 5)
+ sage: gens = R5.basis() + [ -r for r in R5.basis() ]
+ sage: K = Cone(gens)
+ sage: lyapunov_rank(K)
+ 25
+
The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
[Rudolf et al.]_::
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: zero = (0,0,0,0,0)
+ sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero])
+ sage: lyapunov_rank(K)
+ 19
+ sage: K.lattice_dim()**2 - K.dim()*(K.lattice_dim() - K.dim())
+ 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)])
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: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True)
+ sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True)
sage: K = K1.cartesian_product(K2)
sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
True
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: K = random_cone(max_dim=10)
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: 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(J_T) + K.dim()*(l + codim) + codim**2
+ sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**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: K = random_cone(max_dim=10, 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
+ """
+ beta = 0
- 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
- True
+ m = K.dim()
+ n = K.lattice_dim()
+ l = K.linear_subspace().dimension()
- 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
- True
+ if m < n:
+ # K is not solid, project onto its span.
+ K = project_span(K)
- And test with the project_span function::
+ # Lemma 2
+ beta += m*(n - m) + (n - m)**2
- 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
- True
+ if l > 0:
+ # K is not pointed, project its dual onto its span.
+ K = project_span(K.dual()).dual()
- """
- return len(LL(K))
+ # Lemma 3
+ beta += m * l
+
+ beta += len(LL(K))
+ return beta
projects / sage.d.git / blobdiff