X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=43ec8f7f18f031bea3e159f2f434b72b4d05909b;hb=75a1cc8b3affcc1ffcf4fa68572ede75873b9380;hp=eb5316330f6afefdc3b452482bfbd6d59deabbd2;hpb=263ce239a456e5e073a9a8835cdf6cf7e0dfaa98;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index eb53163..43ec8f7 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -118,7 +118,7 @@ def motzkin_decomposition(K): 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)) @@ -152,13 +152,15 @@ def motzkin_decomposition(K): 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) @@ -177,12 +179,6 @@ def positive_operator_gens(K): 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,)]) @@ -196,6 +192,27 @@ def positive_operator_gens(K): [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,)]) @@ -213,14 +230,93 @@ def positive_operator_gens(K): [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 @@ -238,6 +334,43 @@ def positive_operator_gens(K): 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:: @@ -275,7 +408,8 @@ def positive_operator_gens(K): 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... @@ -391,7 +525,8 @@ def Z_transformation_gens(K): 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...