X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=c71a24cbee0c9d0857fd3c8d8fe2ecbb0042238c;hb=a4109ab945f5d3ed94207c936e60b5b187ae450b;hp=8790c30673a25af96c29bf22ff9e84458516da09;hpb=744bc420b47449500f0365e093467ffd268aeda2;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 8790c30..c71a24c 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -10,6 +10,12 @@ def is_lyapunov_like(L,K): ``K``. It is known [Orlitzky]_ that this property need only be checked for generators of ``K`` and its dual. + There are faster ways of checking this property. For example, we + could compute a `lyapunov_like_basis` of the cone, and then test + whether or not the given matrix is contained in the span of that + basis. The value of this function is that it works on symbolic + matrices. + INPUT: - ``L`` -- A linear transformation or matrix. @@ -65,117 +71,37 @@ def is_lyapunov_like(L,K): for (x,s) in K.discrete_complementarity_set()]) -def motzkin_decomposition(K): +def positive_operator_gens(K1, K2 = None): r""" - Return the pair of components in the Motzkin decomposition of this cone. + Compute generators of the cone of positive operators on this cone. A + linear operator on a cone is positive if the image of the cone under + the operator is a subset of the cone. This concept can be extended + to two cones, where the image of the first cone under a positive + operator is a subset of the second cone. + + INPUT: - Every convex cone is the direct sum of a strictly convex cone and a - linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones - such that ``P`` is strictly convex, ``S`` is a subspace, and ``K`` - is the direct sum of ``P`` and ``S``. + - ``K2`` -- (default: ``K1``) the codomain cone; the image of this + cone under the returned operators is a subset of ``K2``. OUTPUT: - An ordered pair ``(P,S)`` of closed convex polyhedral cones where - ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the - direct sum of ``P`` and ``S``. + A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and + ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have + the property that ``P*x`` is an element of ``K2`` whenever ``x`` is + an element of ``K1``. Moreover, any nonnegative linear combination of + these matrices shares the same property. REFERENCES: - .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and - Optimization in Finite Dimensions I. Springer-Verlag, New - York, 1970. - - EXAMPLES: - - The nonnegative orthant is strictly convex, so it is its own - strictly convex component and its subspace component is trivial:: - - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: (P,S) = motzkin_decomposition(K) - sage: K.is_equivalent(P) - True - sage: S.is_trivial() - True - - Likewise, full spaces are their own subspace components:: - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: K.is_full_space() - True - sage: (P,S) = motzkin_decomposition(K) - sage: K.is_equivalent(S) - True - sage: P.is_trivial() - True - - TESTS: - - A random point in the cone should belong to either the strictly - convex component or the subspace component. If the point is nonzero, - it cannot be in both:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: (P,S) = motzkin_decomposition(K) - 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)) - True - - The strictly convex component should always be strictly convex, and - the subspace component should always be a subspace:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: (P,S) = motzkin_decomposition(K) - sage: P.is_strictly_convex() - True - sage: S.lineality() == S.dim() - True - - The generators of the components are obtained from orthogonal - projections of the original generators [Stoer-Witzgall]_:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: (P,S) = motzkin_decomposition(K) - sage: A = S.linear_subspace().complement().matrix() - sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A - sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice()) - sage: P.is_equivalent(expected_P) - True - sage: A = S.linear_subspace().matrix() - sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A - sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice()) - sage: S.is_equivalent(expected_S) - True - """ - # 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. - S_perp = Cone(S.dual(), K.lattice()) - P = K.intersection(S_perp) - - return (P,S) - - -def positive_operator_gens(K): - r""" - Compute generators of the cone of positive operators on this cone. - - OUTPUT: + .. [Orlitzky-Pi-Z] + M. Orlitzky. + Positive and Z-operators on closed convex cones. - A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. - Each matrix ``P`` in the list should have the property that ``P*x`` - is an element of ``K`` whenever ``x`` is an element of - ``K``. Moreover, any nonnegative linear combination of these - matrices shares the same property. + .. [Tam] + B.-S. Tam. + Some results of polyhedral cones and simplicial cones. + Linear and Multilinear Algebra, 4:4 (1977) 281--284. EXAMPLES: @@ -242,56 +168,70 @@ def positive_operator_gens(K): TESTS: - Each positive operator generator should send the generators of the - cone into the cone:: + Each positive operator generator should send the generators of one + cone into the other 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 ]) + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ]) True - Each positive operator generator should send a random element of the - cone into the cone:: + Each positive operator generator should send a random element of one + cone into the other 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 ]) + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ]) True A random element of the positive operator cone should send the - generators of the cone into the cone:: + generators of one cone into the other 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 ]) + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) + sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], + ....: lattice=L, + ....: check=False) + sage: P = matrix(K2.lattice_dim(), + ....: K1.lattice_dim(), + ....: pi_cone.random_element(QQ).list()) + sage: all([ K2.contains(P*x) for x in K1 ]) True A random element of the positive operator cone should send a random - element of the cone into the cone:: + element of one cone into the other 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)) + sage: K1 = random_cone(max_ambient_dim=4) + sage: K2 = random_cone(max_ambient_dim=4) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) + sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], + ....: lattice=L, + ....: check=False) + sage: P = matrix(K2.lattice_dim(), + ....: K1.lattice_dim(), + ....: pi_cone.random_element(QQ).list()) + sage: K2.contains(P*K1.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: K = random_cone(max_ambient_dim=4) 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: pi_cone = Cone([ g.list() for g in pi_of_K ], + ....: lattice=L, + ....: check=False) sage: actual = pi_cone.dual().linear_subspace() sage: U1 = [ vector((s.tensor_product(x)).list()) ....: for x in K.lines() @@ -307,13 +247,15 @@ def positive_operator_gens(K): is known from its lineality space:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) + sage: K = random_cone(max_ambient_dim=4) sage: n = K.lattice_dim() sage: m = K.dim() sage: l = K.lineality() sage: pi_of_K = positive_operator_gens(K) sage: L = ToricLattice(n**2) - sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) sage: actual = pi_cone.dual().lineality() sage: expected = l*(m - l) + m*(n - m) sage: actual == expected @@ -323,13 +265,16 @@ def positive_operator_gens(K): corollary in my paper:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) + sage: K = random_cone(max_ambient_dim=4) sage: n = K.lattice_dim() sage: m = K.dim() sage: l = K.lineality() sage: pi_of_K = positive_operator_gens(K) sage: L = ToricLattice(n**2) - sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim() + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.dim() sage: expected = n**2 - l*(m - l) - (n - m)*m sage: actual == expected True @@ -343,19 +288,25 @@ def positive_operator_gens(K): 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: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.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: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.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 = Cone([p.list() for p in pi_of_K], check=False).dim() sage: actual == 3 True @@ -363,11 +314,14 @@ def positive_operator_gens(K): description of its generators:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) + sage: K = random_cone(max_ambient_dim=4) sage: n = K.lattice_dim() sage: pi_of_K = positive_operator_gens(K) sage: L = ToricLattice(n**2) - sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality() + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.lineality() sage: expected = n**2 - K.dim()*K.dual().dim() sage: actual == expected True @@ -381,19 +335,23 @@ def positive_operator_gens(K): 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).lineality() + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: actual = pi_cone.lineality() 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).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) - sage: actual = Cone([p.list() for p in pi_of_K]).lineality() + sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False) + sage: actual = pi_cone.lineality() sage: actual == 2 True @@ -401,127 +359,363 @@ def positive_operator_gens(K): is proper:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) + sage: K = random_cone(max_ambient_dim=4) 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 = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: 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 + + A transformation is positive on a cone if and only if its adjoint is + positive on the dual of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: F = K.lattice().vector_space().base_field() + sage: n = K.lattice_dim() + sage: L = ToricLattice(n**2) + sage: W = VectorSpace(F, n**2) + sage: pi_of_K = positive_operator_gens(K) + sage: pi_of_K_star = positive_operator_gens(K.dual()) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: pi_star = Cone([p.list() for p in pi_of_K_star], + ....: lattice=L, + ....: check=False) + sage: M = MatrixSpace(F, n) + sage: L = M(pi_cone.random_element(ring=QQ).list()) + sage: pi_star.contains(W(L.transpose().list())) + True + + sage: L = W.random_element() + sage: L_star = W(M(L.list()).transpose().list()) + sage: pi_cone.contains(L) == pi_star.contains(L_star) + True + + The Lyapunov rank of the positive operator cone is the product of + the Lyapunov ranks of the associated cones if they're all proper:: + + sage: K1 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) + sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], + ....: lattice=L, + ....: check=False) + sage: beta1 = K1.lyapunov_rank() + sage: beta2 = K2.lyapunov_rank() + sage: pi_cone.lyapunov_rank() == beta1*beta2 + True + + The Lyapunov-like operators on a proper polyhedral positive operator + cone can be computed from the Lyapunov-like operators on the cones + with respect to which the operators are positive:: + + sage: K1 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=4, + ....: strictly_convex=True, + ....: solid=True) + sage: pi_K1_K2 = positive_operator_gens(K1,K2) + sage: F = K1.lattice().base_field() + sage: m = K1.lattice_dim() + sage: n = K2.lattice_dim() + sage: L = ToricLattice(m*n) + sage: M1 = MatrixSpace(F, m, m) + sage: M2 = MatrixSpace(F, n, n) + sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ] + sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ] + sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ] + sage: W = VectorSpace(F, (m**2)*(n**2)) + sage: expected = span(F, [ W(x.list()) for x in tps ]) + sage: pi_cone = Cone([p.list() for p in pi_K1_K2], + ....: lattice=L, + ....: check=False) + sage: LL_pi = pi_cone.lyapunov_like_basis() + sage: actual = span(F, [ W(x.list()) for x in LL_pi ]) + sage: actual == expected + True + """ + if K2 is None: + K2 = K1 + # 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() + F = K1.lattice().base_field() + n = K1.lattice_dim() + m = K2.lattice_dim() - tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] + tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ] # Convert those tensor products to long vectors. - W = VectorSpace(F, n**2) + W = VectorSpace(F, n*m) 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 K1.is_proper() and K2.is_proper(): + # All of the generators involved are extreme vectors and + # therefore minimal [Tam]_. If this cone is neither solid nor + # strictly convex, then the tensor product of ``s`` and ``x`` + # is the same as that of ``-s`` and ``-x``. However, as a + # /set/, ``tensor_products`` may still be minimal. + 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() ] + M = MatrixSpace(F, m, n) + return [ M(v.list()) for v in pi_cone ] -def Z_transformation_gens(K): +def Z_operator_gens(K): r""" - Compute generators of the cone of Z-transformations on this cone. + Compute generators of the cone of Z-operators on this cone. OUTPUT: A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. Each matrix ``L`` in the list should have the property that - ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the - discrete complementarity set of ``K``. Moreover, any nonnegative - linear combination of these matrices shares the same property. + ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of + this cone's :meth:`discrete_complementarity_set`. Moreover, any + conic (nonnegative linear) combination of these matrices shares the + same property. + + REFERENCES: + + M. Orlitzky. + Positive and Z-operators on closed convex cones. EXAMPLES: - Z-transformations on the nonnegative orthant are just Z-matrices. + Z-operators on the nonnegative orthant are just Z-matrices. That is, matrices whose off-diagonal elements are nonnegative:: sage: K = Cone([(1,0),(0,1)]) - sage: Z_transformation_gens(K) + sage: Z_operator_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_transformation_gens(K) + sage: all([ z[i][j] <= 0 for z in Z_operator_gens(K) ....: for i in range(z.nrows()) ....: for j in range(z.ncols()) ....: if i != j ]) True - The trivial cone in a trivial space has no Z-transformations:: + The trivial cone in a trivial space has no Z-operators:: sage: K = Cone([], ToricLattice(0)) - sage: Z_transformation_gens(K) + sage: Z_operator_gens(K) [] - Z-transformations on a subspace are Lyapunov-like and vice-versa:: + Every operator is a Z-operator on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: Z_operator_gens(K) + [[-1], [1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: Z_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] + ] + + A non-obvious application is to find the Z-operators on the + right half-plane:: + + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_operator_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-operators on a subspace are Lyapunov-like and vice-versa:: sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) 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_transformation_gens(K) ]) + sage: zs = span([ vector(z.list()) for z in Z_operator_gens(K) ]) sage: zs == lls True TESTS: - The Z-property is possessed by every Z-transformation:: + The Z-property is possessed by every Z-operator:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=6) - sage: Z_of_K = Z_transformation_gens(K) + sage: K = random_cone(max_ambient_dim=4) + sage: Z_of_K = Z_operator_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]) True - The lineality space of Z is LL:: + The lineality space of the cone of Z-operators is the space of + Lyapunov-like operators:: 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_transformation_gens(K) ]) - sage: z_cone.linear_subspace() == lls + 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_operator_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 True - And thus, the lineality of Z is the Lyapunov rank:: + The lineality of the Z-operators on a cone is the Lyapunov + rank of that cone:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=6) - sage: Z_of_K = Z_transformation_gens(K) + sage: K = random_cone(max_ambient_dim=4) + sage: Z_of_K = Z_operator_gens(K) sage: L = ToricLattice(K.lattice_dim()**2) - sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L) - sage: z_cone.lineality() == K.lyapunov_rank() + sage: Z_cone = Cone([ z.list() for z in Z_of_K ], + ....: lattice=L, + ....: check=False) + 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 and Z-operator + cones are equal. From this it follows that the dimensions of the + Z-operator cone and positive operator cone are equal:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) + sage: K = random_cone(max_ambient_dim=4) sage: pi_of_K = positive_operator_gens(K) - sage: Z_of_K = Z_transformation_gens(K) + sage: Z_of_K = Z_operator_gens(K) sage: L = ToricLattice(K.lattice_dim()**2) - sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual() - sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual() + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: Z_cone = Cone([ z.list() for z in Z_of_K], + ....: lattice=L, + ....: check=False) + 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_operator_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_operator_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_operator_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False) + sage: Z_cone.dim() == 3 + True + + The Z-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: Z_of_pK = Z_operator_gens(pK) + sage: actual = Cone([t.list() for t in Z_of_pK], + ....: lattice=L, + ....: check=False) + sage: Z_of_K = Z_operator_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 + + An operator is a Z-operator on a cone if and only if its + adjoint is a Z-operator on the dual of that cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: F = K.lattice().vector_space().base_field() + sage: n = K.lattice_dim() + sage: L = ToricLattice(n**2) + sage: W = VectorSpace(F, n**2) + sage: Z_of_K = Z_operator_gens(K) + sage: Z_of_K_star = Z_operator_gens(K.dual()) + sage: Z_cone = Cone([p.list() for p in Z_of_K], + ....: lattice=L, + ....: check=False) + sage: Z_star = Cone([p.list() for p in Z_of_K_star], + ....: lattice=L, + ....: check=False) + sage: M = MatrixSpace(F, n) + sage: L = M(Z_cone.random_element(ring=QQ).list()) + sage: Z_star.contains(W(L.transpose().list())) + True + + sage: L = W.random_element() + sage: L_star = W(M(L.list()).transpose().list()) + sage: Z_cone.contains(L) == Z_star.contains(L_star) + 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 @@ -530,7 +724,7 @@ def Z_transformation_gens(K): n = K.lattice_dim() # These tensor products contain generators for the dual cone of - # the cross-positive transformations. + # the cross-positive operators. tensor_products = [ s.tensor_product(x) for (x,s) in K.discrete_complementarity_set() ] @@ -538,32 +732,40 @@ 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_proper(): + # All of the generators involved are extreme vectors and + # therefore minimal. If this cone is neither solid nor + # strictly convex, then the tensor product of ``s`` and ``x`` + # is the same as that of ``-s`` and ``-x``. However, as a + # /set/, ``tensor_products`` may still be minimal. + 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 + # But first, make them negative, so we get Z-operators 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 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) def Z_cone(K): - gens = Z_transformation_gens(K) - L = None - if len(gens) == 0: - L = ToricLattice(0) - return Cone([ g.list() for g in gens ], lattice=L) + gens = Z_operator_gens(K) + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) def pi_cone(K): gens = positive_operator_gens(K) - L = None - if len(gens) == 0: - L = ToricLattice(0) - return Cone([ g.list() for g in gens ], lattice=L) + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False)