if K1.is_strictly_convex() != K2.is_strictly_convex():
return False
- if len(K1.LL()) != len(K2.LL()):
+ if len(K1.lyapunov_like_basis()) != len(K2.lyapunov_like_basis()):
return False
C_of_K1 = K1.discrete_complementarity_set()
sage: actual == expected
True
- The Lyapunov rank of any cone is just the dimension of ``K.LL()``::
+ The Lyapunov rank of any cone is just the dimension of
+ ``K.lyapunov_like_basis()``::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
- sage: lyapunov_rank(K) == len(K.LL())
+ sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
True
We can make an imperfect cone perfect by adding a slack variable
# Non-pointed reduction lemma.
beta += l * m
- beta += len(K.LL())
+ beta += len(K.lyapunov_like_basis())
return beta
sage: is_lyapunov_like(L,K)
True
- Everything in ``K.LL()`` should be Lyapunov-like on ``K``::
+ Everything in ``K.lyapunov_like_basis()`` 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 K.LL()])
+ sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
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:
+
+ Z-transformations on the nonnegative orthant are just Z-matrices.
+ That is, matrices whose off-diagonal elements are nonnegative::
+
+ sage: K = Cone([(1,0),(0,1)])
+ sage: Z_transformations(K)
+ [
+ [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
+ [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
+ ]
+ sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
+ sage: all([ z[i][j] <= 0 for z in Z_transformations(K)
+ ....: for i in range(z.nrows())
+ ....: for j in range(z.ncols())
+ ....: if i != j ])
+ True
+
+ 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: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
+ sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ])
+ sage: zs == lls
+ True
+
+ TESTS:
+
+ The Z-property is possessed by every Z-transformation::
+
+ sage: set_random_seed()
+ 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: set_random_seed()
+ sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+ sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
+ sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ])
+ sage: z_cone.linear_subspace() == lls
+ 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 cross-positive operators,
+ # expressed as long vectors..
+ L = ToricLattice(W.dimension())
+ Sigma_dual = Cone(vectors, lattice=L)
+
+ # Now compute the desired cone from its dual...
+ Sigma_cone = Sigma_dual.dual()
+
+ # And finally convert its rays back to matrix representations.
+ # But first, make them negative, so we get Z-transformations and
+ # not cross-positive ones.
+ M = MatrixSpace(V.base_ring(), V.dimension())
+
+ return [ -M(v.list()) for v in Sigma_cone.rays() ]
projects / sage.d.git / blobdiff