- V = K.lattice().vector_space()
-
- C_of_K = discrete_complementarity_set(K)
-
- matrices = [x.tensor_product(s) 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)
-
- def phi(m):
- r"""
- Convert a matrix to a vector isomorphically.
- """
- return W(m.list())
-
- vectors = [phi(m) for m in matrices]
-
- return (W.dimension() - W.span(vectors).rank())
+ if L.base_ring().is_exact() or L.base_ring() is SR:
+ # The "fast method" of creating a vector space based on a
+ # ``lyapunov_like_basis`` is actually slower than this.
+ return all([ s*(L*x) == 0
+ for (x,s) in K.discrete_complementarity_set() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
+
+def LL_cone(K):
+ gens = K.lyapunov_like_basis()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
+
+def Sigma_cone(K):
+ gens = K.cross_positive_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
+
+def Z_cone(K):
+ gens = K.Z_operators_gens()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
+
+def pi_cone(K1, K2=None):
+ if K2 is None:
+ K2 = K1
+ gens = K1.positive_operators_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)