From: Michael Orlitzky Date: Fri, 16 Oct 2015 00:08:22 +0000 (-0400) Subject: Play around with positive operators and Z-transformations. Add a new test. X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=5b7044f0fad38851282ffdc07b55b98c11b7f78e;p=sage.d.git Play around with positive operators and Z-transformations. Add a new test. --- diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 49df3e9..f78c27e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -40,14 +40,14 @@ def is_lyapunov_like(L,K): The identity is always Lyapunov-like in a nontrivial space:: sage: set_random_seed() - sage: K = random_cone(min_ambient_dim = 1, max_rays = 8) + sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8) sage: L = identity_matrix(K.lattice_dim()) sage: is_lyapunov_like(L,K) True As is the "zero" transformation:: - sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8) sage: R = K.lattice().vector_space().base_ring() sage: L = zero_matrix(R, K.lattice_dim()) sage: is_lyapunov_like(L,K) @@ -56,7 +56,7 @@ def is_lyapunov_like(L,K): Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like on ``K``:: - sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) True @@ -122,7 +122,7 @@ def random_element(K): return v -def positive_operators(K): +def positive_operator_gens(K): r""" Compute generators of the cone of positive operators on this cone. @@ -139,17 +139,17 @@ def positive_operators(K): The trivial cone in a trivial space has no positive operators:: sage: K = Cone([], ToricLattice(0)) - sage: positive_operators(K) + sage: positive_operator_gens(K) [] Positive operators on the nonnegative orthant are nonnegative matrices:: sage: K = Cone([(1,)]) - sage: positive_operators(K) + sage: positive_operator_gens(K) [[1]] sage: K = Cone([(1,0),(0,1)]) - sage: positive_operators(K) + sage: positive_operator_gens(K) [ [1 0] [0 1] [0 0] [0 0] [0 0], [0 0], [1 0], [0 1] @@ -160,13 +160,13 @@ def positive_operators(K): sage: K = Cone([(1,),(-1,)]) sage: K.is_full_space() True - sage: positive_operators(K) + sage: positive_operator_gens(K) [[1], [-1]] sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) sage: K.is_full_space() True - sage: positive_operators(K) + 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] @@ -177,42 +177,49 @@ def positive_operators(K): A positive operator on a cone should send its generators into the cone:: sage: K = random_cone(max_ambient_dim = 6) - sage: pi_of_K = positive_operators(K) + 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()]) True + The dimension of the cone of positive operators is given by the + corollary in my paper:: + + sage: K = random_cone(max_ambient_dim = 6) + sage: n = K.lattice_dim() + sage: m = K.dim() + sage: l = K.lineality() + sage: pi_of_K = positive_operator_gens(K) + sage: actual = Cone([p.list() for p in pi_of_K]).dim() + sage: expected = n**2 - l*(n - l) - (n - m)*m + sage: actual == expected + True + """ - # 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. - V = K.lattice().vector_space() - W = VectorSpace(V.base_ring(), V.dimension()**2) + # 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() tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] + # Convert those tensor products to long vectors. + 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.. - L = ToricLattice(W.dimension()) - pi_dual = Cone(vectors, lattice=L) + pi_dual = Cone(vectors, ToricLattice(W.dimension())) # Now compute the desired cone from its dual... pi_cone = pi_dual.dual() # And finally convert its rays back to matrix representations. - M = MatrixSpace(V.base_ring(), V.dimension()) - + M = MatrixSpace(F, n) return [ M(v.list()) for v in pi_cone.rays() ] -def Z_transformations(K): +def Z_transformation_gens(K): r""" Compute generators of the cone of Z-transformations on this cone. @@ -230,13 +237,13 @@ def Z_transformations(K): That is, matrices whose off-diagonal elements are nonnegative:: sage: K = Cone([(1,0),(0,1)]) - sage: Z_transformations(K) + sage: Z_transformation_gens(K) [ [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0] [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1] ] sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]) - sage: all([ z[i][j] <= 0 for z in Z_transformations(K) + sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K) ....: for i in range(z.nrows()) ....: for j in range(z.ncols()) ....: if i != j ]) @@ -245,7 +252,7 @@ def Z_transformations(K): The trivial cone in a trivial space has no Z-transformations:: sage: K = Cone([], ToricLattice(0)) - sage: Z_transformations(K) + sage: Z_transformation_gens(K) [] Z-transformations on a subspace are Lyapunov-like and vice-versa:: @@ -254,7 +261,7 @@ def Z_transformations(K): sage: K.is_full_space() True sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) - sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ]) + sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ]) sage: zs == lls True @@ -264,7 +271,7 @@ def Z_transformations(K): sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 6) - sage: Z_of_K = Z_transformations(K) + sage: Z_of_K = Z_transformation_gens(K) sage: dcs = K.discrete_complementarity_set() sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K ....: for (x,s) in dcs]) @@ -275,32 +282,29 @@ def Z_transformations(K): sage: set_random_seed() sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) - sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ]) + sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ]) sage: z_cone.linear_subspace() == lls True """ - # 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. - V = K.lattice().vector_space() - W = VectorSpace(V.base_ring(), V.dimension()**2) + # 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() - C_of_K = K.discrete_complementarity_set() - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + # 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 ] # Create the *dual* cone of the cross-positive operators, # expressed as long vectors.. - L = ToricLattice(W.dimension()) - Sigma_dual = Cone(vectors, lattice=L) + Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension())) # Now compute the desired cone from its dual... Sigma_cone = Sigma_dual.dual() @@ -308,6 +312,5 @@ def Z_transformations(K): # 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(V.base_ring(), V.dimension()) - + M = MatrixSpace(F, n) return [ -M(v.list()) for v in Sigma_cone.rays() ]