if K1.is_strictly_convex() != K2.is_strictly_convex():
return False
- if len(LL(K1)) != len(LL(K2)):
+ if len(K1.LL()) != len(K2.LL()):
return False
C_of_K1 = K1.discrete_complementarity_set()
return Cone(K_W_rays, lattice=L)
-def LL(K):
- r"""
- Compute a basis of Lyapunov-like transformations on this cone.
-
- OUTPUT:
-
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given cone.
-
- EXAMPLES:
-
- The trivial cone has no Lyapunov-like transformations::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
- []
-
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
-
- sage: K = Cone([(1,)])
- sage: LL(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
- [
- [1 0] [0 0]
- [0 0], [0 1]
- ]
-
- sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: LL(K)
- [
- [1 0 0] [0 0 0] [0 0 0]
- [0 0 0] [0 1 0] [0 0 0]
- [0 0 0], [0 0 0], [0 0 1]
- ]
-
- Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
- `L^{3}_{\infty}` cones [Rudolf et al.]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: LL(L31)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
-
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: LL(L3infty)
- [
- [1 0 0]
- [0 1 0]
- [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: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: C_of_K = K.discrete_complementarity_set()
- 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_ambient_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 = K.discrete_complementarity_set()
-
- 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
- # many are linearly-indepenedent.
- #
- # The space W has the same base ring as V, but dimension
- # dim(V)^2. So it has the same dimension as the space of linear
- # transformations on V. In other words, it's just the right size
- # to create an isomorphism between it and our matrices.
- W = VectorSpace(V.base_ring(), V.dimension()**2)
-
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
-
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
-
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
-
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
-
- return matrix_basis
-
-
-
def lyapunov_rank(K):
r"""
Compute the Lyapunov rank (or bilinearity rank) of this cone.
sage: actual == expected
True
- The Lyapunov rank of any cone is just the dimension of ``LL(K)``::
+ The Lyapunov rank of any cone is just the dimension of ``K.LL()``::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: lyapunov_rank(K) == len(K.LL())
True
We can make an imperfect cone perfect by adding a slack variable
# Non-pointed reduction lemma.
beta += l * m
- beta += len(LL(K))
+ beta += len(K.LL())
return beta
sage: is_lyapunov_like(L,K)
True
- Everything in ``LL(K)`` should be Lyapunov-like on ``K``::
+ Everything in ``K.LL()`` should be Lyapunov-like on ``K``::
sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
- sage: all([is_lyapunov_like(L,K) for L in LL(K)])
+ sage: all([is_lyapunov_like(L,K) for L in K.LL()])
True
"""
A positive operator on a cone should send its generators into the cone::
sage: K = random_cone(max_ambient_dim = 6)
- sage: pi_of_k = positive_operators(K)
- sage: all([K.contains(p*x) for p in pi_of_k for x in K.rays()])
+ sage: pi_of_K = positive_operators(K)
+ sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
True
"""
- V = K.lattice().vector_space()
-
# Sage doesn't think matrices are vectors, so we have to convert
# our matrices to vectors explicitly before we can figure out how
# many are linearly-indepenedent.
# dim(V)^2. So it has the same dimension as the space of linear
# transformations on V. In other words, it's just the right size
# to create an isomorphism between it and our matrices.
+ V = K.lattice().vector_space()
W = VectorSpace(V.base_ring(), V.dimension()**2)
- G1 = [ V(x) for x in K.rays() ]
- G2 = [ V(s) for s in K.dual().rays() ]
-
- tensor_products = [ s.tensor_product(x) for x in G1 for s in G2 ]
+ tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
# Turn our matrices into long vectors...
vectors = [ W(m.list()) for m in tensor_products ]
M = MatrixSpace(V.base_ring(), V.dimension())
return [ M(v.list()) for v in pi_cone.rays() ]
+
+
+def Z_transformations(K):
+ r"""
+ Compute generators of the cone of Z-transformations on this cone.
+
+ OUTPUT:
+
+ A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
+ Each matrix ``L`` in the list should have the property that
+ ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
+ discrete complementarity set of ``K``. Moreover, any nonnegative
+ linear combination of these matrices shares the same property.
+
+ EXAMPLES:
+
+ The trivial cone in a trivial space has no Z-transformations::
+
+ sage: K = Cone([], ToricLattice(0))
+ sage: Z_transformations(K)
+ []
+
+ Z-transformations on a subspace are Lyapunov-like and vice-versa::
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: llvs = span([ vector(l.list()) for l in K.LL() ])
+ sage: zvs = span([ vector(z.list()) for z in Z_transformations(K) ])
+ sage: zvs == llvs
+ True
+
+ TESTS:
+
+ The Z-property is possessed by every Z-transformation::
+
+ sage: K = random_cone(max_ambient_dim = 6)
+ sage: Z_of_K = Z_transformations(K)
+ sage: dcs = K.discrete_complementarity_set()
+ sage: all([z(x).inner_product(s) <= 0 for z in Z_of_K
+ ....: for (x,s) in dcs])
+ True
+
+ The lineality space of Z is LL::
+
+ sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+ sage: llvs = span([ vector(l.list()) for l in K.LL() ])
+ sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ])
+ sage: z_cone.linear_subspace() == llvs
+ True
+
+ """
+ # Sage doesn't think matrices are vectors, so we have to convert
+ # our matrices to vectors explicitly before we can figure out how
+ # many are linearly-indepenedent.
+ #
+ # The space W has the same base ring as V, but dimension
+ # dim(V)^2. So it has the same dimension as the space of linear
+ # transformations on V. In other words, it's just the right size
+ # to create an isomorphism between it and our matrices.
+ V = K.lattice().vector_space()
+ W = VectorSpace(V.base_ring(), V.dimension()**2)
+
+ C_of_K = K.discrete_complementarity_set()
+ tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
+
+ # Turn our matrices into long vectors...
+ vectors = [ W(m.list()) for m in tensor_products ]
+
+ # Create the *dual* cone of the positive operators, expressed as
+ # long vectors..
+ L = ToricLattice(W.dimension())
+ Z_dual = Cone(vectors, lattice=L)
+
+ # Now compute the desired cone from its dual...
+ Z_cone = Z_dual.dual()
+
+ # And finally convert its rays back to matrix representations.
+ M = MatrixSpace(V.base_ring(), V.dimension())
+
+ return [ M(v.list()) for v in Z_cone.rays() ]
projects / sage.d.git / blobdiff