X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=21f9862c24a9e9e3d9cffc52a1e789018f59a153;hb=c9ab469a53dc691d2646e22bdbc0ea0b3f3961c1;hp=eac86b374c131f4cad05dc27c7ec0eb7060deca2;hpb=c9c16b8aa4a4a6959b988d5c51e24c0fbe07e3dd;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index eac86b3..21f9862 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -135,6 +135,14 @@ def motzkin_decomposition(K): sage: S.lineality() == S.dim() True + A strictly convex cone should be equal to its strictly convex component:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8, strictly_convex=True) + sage: (P,_) = motzkin_decomposition(K) + sage: K.is_equivalent(P) + True + The generators of the components are obtained from orthogonal projections of the original generators [Stoer-Witzgall]_:: @@ -155,11 +163,11 @@ def motzkin_decomposition(K): # 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()) + S = Cone(S_gens, K.lattice(), check=False) # Since ``S`` is a subspace, the rays of its dual generate its # orthogonal complement. - S_perp = Cone(S.dual(), K.lattice()) + S_perp = Cone(S.dual(), K.lattice(), check=False) P = K.intersection(S_perp) return (P,S) @@ -411,8 +419,8 @@ def positive_operator_gens(K): 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).lineality() - sage: actual == n^2 + sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L) + sage: pi_cone.lineality() == n^2 True sage: K = Cone([(1,0),(0,1),(0,-1)]) sage: pi_of_K = positive_operator_gens(K) @@ -433,6 +441,25 @@ def positive_operator_gens(K): ....: check=False) sage: K.is_proper() == pi_cone.is_proper() True + + The positive operators of a permuted cone can be obtained by + conjugation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() + sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) + sage: pi_of_pK = positive_operator_gens(pK) + sage: actual = Cone([t.list() for t in pi_of_pK], + ....: lattice=L, + ....: check=False) + sage: pi_of_K = positive_operator_gens(K) + sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual.is_equivalent(expected) + True """ # Matrices are not vectors in Sage, so we have to convert them # to vectors explicitly before we can find a basis. We need these @@ -446,18 +473,27 @@ def positive_operator_gens(K): W = VectorSpace(F, n**2) vectors = [ W(tp.list()) for tp in tensor_products ] - # Create the *dual* cone of the positive operators, expressed as - # long vectors. WARNING: check=True is necessary even though it - # makes Cone() take forever. For an example take - # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]). - pi_dual = Cone(vectors, ToricLattice(W.dimension())) + 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 positive operators, expressed as + # long vectors. + pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check) # Now compute the desired cone from its dual... pi_cone = pi_dual.dual() # And finally convert its rays back to matrix representations. M = MatrixSpace(F, n) - return [ M(v.list()) for v in pi_cone.rays() ] + return [ M(v.list()) for v in pi_cone ] def Z_transformation_gens(K): @@ -496,6 +532,33 @@ def Z_transformation_gens(K): sage: Z_transformation_gens(K) [] + Every operator is a Z-transformation on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: Z_transformation_gens(K) + [[-1], [1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: Z_transformation_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] + ] + + A non-obvious application is to find the Z-transformations on the + right half-plane:: + + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_transformation_gens(K) + [ + [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0] + [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1] + ] + Z-transformations on a subspace are Lyapunov-like and vice-versa:: sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) @@ -518,32 +581,37 @@ def Z_transformation_gens(K): ....: for (x,s) in dcs]) True - The lineality space of Z is LL:: + The lineality space of the cone of Z-transformations is the space of + Lyapunov-like transformations:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) sage: L = ToricLattice(K.lattice_dim()**2) - sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ], + sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ], ....: lattice=L, ....: check=False) sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ] sage: lls = L.vector_space().span(ll_basis) - sage: z_cone.linear_subspace() == lls + sage: Z_cone.linear_subspace() == lls True - And thus, the lineality of Z is the Lyapunov rank:: + The lineality of the Z-transformations on a cone is the Lyapunov + rank of that cone:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) sage: Z_of_K = Z_transformation_gens(K) sage: L = ToricLattice(K.lattice_dim()**2) - sage: z_cone = Cone([ z.list() for z in Z_of_K ], + sage: Z_cone = Cone([ z.list() for z in Z_of_K ], ....: lattice=L, ....: check=False) - sage: z_cone.lineality() == K.lyapunov_rank() + sage: Z_cone.lineality() == K.lyapunov_rank() True - The lineality spaces of pi-star and Z-star are equal: + The lineality spaces of the duals of the positive operator and + Z-transformation cones are equal. From this it follows that the + dimensions of the Z-transformation cone and positive operator cone + are equal:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) @@ -553,13 +621,65 @@ def Z_transformation_gens(K): sage: pi_cone = Cone([p.list() for p in pi_of_K], ....: lattice=L, ....: check=False) - sage: pi_star = pi_cone.dual() - sage: z_cone = Cone([ z.list() for z in Z_of_K], + sage: Z_cone = Cone([ z.list() for z in Z_of_K], ....: lattice=L, ....: check=False) - sage: z_star = z_cone.dual() + sage: pi_cone.dim() == Z_cone.dim() + True + sage: pi_star = pi_cone.dual() + sage: z_star = Z_cone.dual() sage: pi_star.linear_subspace() == z_star.linear_subspace() 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: Z_of_K = Z_transformation_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = Z_cone.dim() + sage: actual == n^2 + True + sage: K = K.dual() + sage: K.is_full_space() + True + sage: Z_of_K = Z_transformation_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = Z_cone.dim() + sage: actual == n^2 + True + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_of_K = Z_transformation_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False) + sage: Z_cone.dim() == 3 + True + + The Z-transformations of a permuted cone can be obtained by + conjugation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() + sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) + sage: Z_of_pK = Z_transformation_gens(pK) + sage: actual = Cone([t.list() for t in Z_of_pK], + ....: lattice=L, + ....: check=False) + sage: Z_of_K = Z_transformation_gens(K) + sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: actual.is_equivalent(expected) + True """ # Matrices are not vectors in Sage, so we have to convert them # to vectors explicitly before we can find a basis. We need these @@ -576,11 +696,20 @@ def Z_transformation_gens(K): W = VectorSpace(F, n**2) vectors = [ W(m.list()) for m in tensor_products ] - # Create the *dual* cone of the cross-positive operators, - # expressed as long vectors. WARNING: check=True is necessary - # even though it makes Cone() take forever. For an example take - # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]). - Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension())) + 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() @@ -589,7 +718,7 @@ def Z_transformation_gens(K): # 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.rays() ] + return [ -M(v.list()) for v in Sigma_cone ] def Z_cone(K):