- # Matrices are not vectors in Sage, so we have to convert them
- # to vectors explicitly before we can find a basis. We need these
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- # These tensor products contain generators for the dual cone of
- # the cross-positive transformations.
- tensor_products = [ s.tensor_product(x)
- for (x,s) in K.discrete_complementarity_set() ]
-
- # Turn our matrices into long vectors...
- W = VectorSpace(F, n**2)
- vectors = [ W(m.list()) for m in tensor_products ]
-
- check = True
- if K.is_solid() or K.is_strictly_convex():
- # The lineality space of either ``K`` or ``K.dual()`` is
- # trivial and it's easy to show that our generating set is
- # minimal. I would love a proof that this works when ``K`` is
- # neither pointed nor solid.
- #
- # Note that in that case we can get *duplicates*, since the
- # tensor product of (x,s) is the same as that of (-x,-s).
- check = False
-
- # Create the dual cone of the cross-positive operators,
- # expressed as long vectors.
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check)
-
- # 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(F, n)
- return [ -M(v.list()) for v in Sigma_cone ]
+ 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)