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
-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.
- 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):
+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)
+ if K1.lattice_dim() != K2.lattice_dim():
+ return False
- 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()) ])
+ if K1.nrays() != K2.nrays():
+ return False
- return V_iso( (w1, w2) )
+ if K1.dim() != K2.dim():
+ return False
+ if lineality(K1) != lineality(K2):
+ return False
- def phi_inv( pair ):
- # Crash if the arguments are in the wrong spaces.
- V_iso(pair)
+ if K1.is_solid() != K2.is_solid():
+ return False
- #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) ])
+ if K1.is_strictly_convex() != K2.is_strictly_convex():
+ return False
- return sum( pair.cartesian_factors() )
+ 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
- return (phi,phi_inv)
-
-
-
-def unrestrict_span(K, K2=None):
- if K2 is None:
- K2 = K
+ if len(K1.facets()) != len(K2.facets()):
+ return False
- _,phi_inv = ips_iso(K2)
- V_iso = iso_space(K2)
- (W, W_perp) = V_iso.cartesian_factors()
+ return True
- 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 intersect_span(K1, K2):
- r"""
- Return a new cone obtained by intersecting ``K1`` with the span of ``K2``.
- """
- L = K1.lattice()
-
- if L.rank() != K2.lattice().rank():
- raise ValueError('K1 and K2 must belong to lattices of the same rank.')
-
- SL_gens = list(K2.rays())
- span_K2_gens = SL_gens + [ -g for g in SL_gens ]
-
- # The lattices have the same rank (see above) so this should work.
- span_K2 = Cone(span_K2_gens, L)
- return K1.intersection(span_K2)
-
-
-
-def restrict_span(K, K2=None):
+def rho(K, K2=None):
r"""
Restrict ``K`` into its own span, or the span of another cone.
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:
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
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()
+ 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: K_W.dim() == K_star_W_star.dim()
- 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)
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
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
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.
[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
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))
\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) ]
[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
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
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
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
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
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
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
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
projects / sage.d.git / blobdiff