sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8)
sage: (P,S) = motzkin_decomposition(K)
- sage: x = K.random_element()
+ sage: x = K.random_element(ring=QQ)
sage: P.contains(x) or S.contains(x)
True
sage: x.is_zero() or (P.contains(x) != S.contains(x))
sage: S.is_equivalent(expected_S)
True
"""
- # The lines() method only gives us one generator for each line,
- # so we negate the result and combine everything for the full set.
- S = Cone([p*l for p in [1,-1] for l in K.lines()], K.lattice())
+ # The lines() method only returns one generator per line. For a true
+ # line, we also need a generator pointing in the opposite direction.
+ S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
+ S = Cone(S_gens, K.lattice())
# Since ``S`` is a subspace, the rays of its dual generate its
# orthogonal complement.
- P = K.intersection( Cone(S.dual(), K.lattice()) )
+ S_perp = Cone(S.dual(), K.lattice())
+ P = K.intersection(S_perp)
return (P,S)
EXAMPLES:
- The trivial cone in a trivial space has no positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operator_gens(K)
- []
-
Positive operators on the nonnegative orthant are nonnegative matrices::
sage: K = Cone([(1,)])
[0 0], [0 0], [1 0], [0 1]
]
+ The trivial cone in a trivial space has no positive operators::
+
+ sage: K = Cone([], ToricLattice(0))
+ sage: positive_operator_gens(K)
+ []
+
+ Every operator is positive on the trivial cone::
+
+ sage: K = Cone([(0,)])
+ sage: positive_operator_gens(K)
+ [[1], [-1]]
+
+ sage: K = Cone([(0,0)])
+ sage: K.is_trivial()
+ True
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
+
Every operator is positive on the ambient vector space::
sage: K = Cone([(1,),(-1,)])
[0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
]
+ A non-obvious application is to find the positive operators on the
+ right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
+
TESTS:
- A positive operator on a cone should send its generators into the cone::
+ Each positive operator generator should send the generators of the
+ cone into the cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: all([ K.contains(P*x) for P in pi_of_K for x in K ])
+ True
+
+ Each positive operator generator should send a random element of the
+ cone into the cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
+ True
+
+ A random element of the positive operator cone should send the
+ generators of the cone into the cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+ sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+ sage: all([ K.contains(P*x) for x in K ])
+ True
+
+ A random element of the positive operator cone should send a random
+ element of the cone into the cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+ sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+ sage: K.contains(P*K.random_element(ring=QQ))
+ True
+
+ The lineality space of the dual of the cone of positive operators
+ can be computed from the lineality spaces of the cone and its dual::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+ sage: actual = pi_cone.dual().linear_subspace()
+ sage: U1 = [ vector((s.tensor_product(x)).list())
+ ....: for x in K.lines()
+ ....: for s in K.dual() ]
+ sage: U2 = [ vector((s.tensor_product(x)).list())
+ ....: for x in K
+ ....: for s in K.dual().lines() ]
+ sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
+ sage: actual == expected
+ True
+
+ The lineality of the dual of the cone of positive operators
+ is known from its lineality space::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=5)
+ sage: n = K.lattice_dim()
+ sage: m = K.dim()
+ sage: l = K.lineality()
sage: pi_of_K = positive_operator_gens(K)
- sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: actual = pi_cone.dual().lineality()
+ sage: expected = l*(m - l) + m*(n - m)
+ sage: actual == expected
True
The dimension of the cone of positive operators is given by the
sage: actual == expected
True
+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K]).dim()
+ sage: actual == 3
+ True
+
+ The cone of positive operators is solid when the original cone is proper::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=5,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: pi_cone.is_solid()
+ True
+
The lineality of the cone of positive operators is given by the
corollary in my paper::
vectors = [ W(tp.list()) for tp in tensor_products ]
# Create the *dual* cone of the positive operators, expressed as
- # long vectors..
+ # long vectors. WARNING: takes forever unless we pass check=False
+ # to Cone().
pi_dual = Cone(vectors, ToricLattice(W.dimension()))
# Now compute the desired cone from its dual...
vectors = [ W(m.list()) for m in tensor_products ]
# Create the *dual* cone of the cross-positive operators,
- # expressed as long vectors..
+ # expressed as long vectors. WARNING: takes forever unless we pass
+ # check=False to Cone().
Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
# Now compute the desired cone from its dual...
projects / sage.d.git / blobdiff