X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=aeec0c90b5f582c8ba858d4b616fff191fb6d529;hb=c2d15c0884c8b92483f58826747887bd2bcdcdeb;hp=8790c30673a25af96c29bf22ff9e84458516da09;hpb=744bc420b47449500f0365e093467ffd268aeda2;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 8790c30..aeec0c9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,569 +1,455 @@ from sage.all import * +from sage.geometry.cone import is_Cone -def is_lyapunov_like(L,K): +def is_positive_on(L,K): r""" - Determine whether or not ``L`` is Lyapunov-like on ``K``. + Determine whether or not ``L`` is positive on ``K``. + + We say that ``L`` is positive on a closed convex cone ``K`` if + `L\left\lparen x \right\rparen` belongs to ``K`` for all `x` in + ``K``. This property need only be checked for generators of ``K``. - We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle - L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs - `\left\langle x,s \right\rangle` in the complementarity set of - ``K``. It is known [Orlitzky]_ that this property need only be - checked for generators of ``K`` and its dual. + To reliably check whether or not ``L`` is positive, its base ring + must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. INPUT: - - ``L`` -- A linear transformation or matrix. + - ``L`` -- A matrix over either an exact ring or ``SR``. - ``K`` -- A polyhedral closed convex cone. OUTPUT: - ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, - and ``False`` otherwise. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is positive on ``K``. - .. WARNING:: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - If this function returns ``True``, then ``L`` is Lyapunov-like - on ``K``. However, if ``False`` is returned, that could mean one - of two things. The first is that ``L`` is definitely not - Lyapunov-like on ``K``. The second is more of an "I don't know" - answer, returned (for example) if we cannot prove that an inner - product is zero. + - ``True`` will be returned if it can be proven that ``L`` + is positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - REFERENCES: + .. SEEALSO:: - M. Orlitzky. The Lyapunov rank of an improper cone. - http://www.optimization-online.org/DB_HTML/2015/10/5135.html + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` EXAMPLES: - The identity is always Lyapunov-like in a nontrivial space:: + Nonnegative matrices are positive operators on the nonnegative + orthant:: + + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = random_matrix(QQ,3).apply_map(abs) + sage: is_positive_on(L,K) + True + + TESTS: + + The identity operator is always positive:: sage: set_random_seed() - sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: L = identity_matrix(K.lattice_dim()) - sage: is_lyapunov_like(L,K) + sage: is_positive_on(L,K) True - As is the "zero" transformation:: + The "zero" operator is always positive:: - sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8) + sage: K = random_cone(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) + sage: is_positive_on(L,K) True - Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like - on ``K``:: + Everything in ``K.positive_operators_gens()`` should be + positive on ``K``:: - 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() ]) + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_positive_on(L,K) # long time + ....: for L in K.positive_operators_gens() ]) # long time True + sage: all([ is_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.positive_operators_gens() ]) # long time + True + + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: + + sage: K = [ vector([1,2,3]), vector([5,-1,7]) ] + sage: L = identity_matrix(3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. + + We can't give reliable answers over inexact rings:: + + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. """ - return all([(L*x).inner_product(s) == 0 - for (x,s) in K.discrete_complementarity_set()]) + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') -def motzkin_decomposition(K): - r""" - Return the pair of components in the Motzkin decomposition of this cone. + if L.base_ring().is_exact(): + # This should be way faster than computing the dual and + # checking a bunch of inequalities, but it doesn't work if + # ``L*x`` is symbolic. For example, ``e in Cone([(1,)])`` + # is true, but returns ``False``. + return all([ L*x in K for x in K ]) + else: + # Fall back to inequality-checking when the entries of ``L`` + # might be symbolic. + return all([ s*(L*x) >= 0 for x in K for s in K.dual() ]) - 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``. - OUTPUT: +def is_cross_positive_on(L,K): + r""" + Determine whether or not ``L`` is cross-positive on ``K``. - 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``. + We say that ``L`` is cross-positive on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \ge 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - REFERENCES: + To reliably check whether or not ``L`` is cross-positive, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and - Optimization in Finite Dimensions I. Springer-Verlag, New - York, 1970. + INPUT: - EXAMPLES: + - ``L`` -- A matrix over either an exact ring or ``SR``. - The nonnegative orthant is strictly convex, so it is its own - strictly convex component and its subspace component is trivial:: + - ``K`` -- A polyhedral closed convex cone. - 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 + OUTPUT: - Likewise, full spaces are their own subspace components:: + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is cross-positive on ``K``. - 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 + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - TESTS: + - ``True`` will be returned if it can be proven that ``L`` + is cross-positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not cross-positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - 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:: + .. SEEALSO:: - 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 + :func:`is_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` + + EXAMPLES: - The strictly convex component should always be strictly convex, and - the subspace component should always be a subspace:: + The identity operator is always cross-positive:: 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() + sage: L = identity_matrix(K.lattice_dim()) + sage: is_cross_positive_on(L,K) True - The generators of the components are obtained from orthogonal - projections of the original generators [Stoer-Witzgall]_:: + The "zero" operator is always cross-positive:: - 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) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_cross_positive_on(L,K) 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) + TESTS: - return (P,S) + Everything in ``K.cross_positive_operators_gens()`` should be + cross-positive on ``K``:: + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_cross_positive_on(L,K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time + True + sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time + True -def positive_operator_gens(K): - r""" - Compute generators of the cone of positive operators on this cone. + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - OUTPUT: + sage: L = identity_matrix(3) + sage: K = [ vector([8,2,-8]), vector([5,-5,7]) ] + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - 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. + We can't give reliable answers over inexact rings:: - EXAMPLES: + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - Positive operators on the nonnegative orthant are nonnegative matrices:: + """ + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - sage: K = Cone([(1,)]) - sage: positive_operator_gens(K) - [[1]] + return all([ s*(L*x) >= 0 + for (x,s) in K.discrete_complementarity_set() ]) - sage: K = Cone([(1,0),(0,1)]) - sage: positive_operator_gens(K) - [ - [1 0] [0 1] [0 0] [0 0] - [0 0], [0 0], [1 0], [0 1] - ] +def is_Z_operator_on(L,K): + r""" + Determine whether or not ``L`` is a Z-operator on ``K``. - The trivial cone in a trivial space has no positive operators:: + We say that ``L`` is a Z-operator on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \le 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. It is known that this property need only be checked + for generators of ``K`` and its dual. - sage: K = Cone([], ToricLattice(0)) - sage: positive_operator_gens(K) - [] + A matrix is a Z-operator on ``K`` if and only if its negation is a + cross-positive operator on ``K``. - Every operator is positive on the trivial cone:: + To reliably check whether or not ``L`` is a Z operator, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - sage: K = Cone([(0,)]) - sage: positive_operator_gens(K) - [[1], [-1]] + INPUT: - 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] - ] + - ``L`` -- A matrix over either an exact ring or ``SR``. - Every operator is positive on the ambient vector space:: + - ``K`` -- A polyhedral closed convex cone. - sage: K = Cone([(1,),(-1,)]) - sage: K.is_full_space() - True - sage: positive_operator_gens(K) - [[1], [-1]] + OUTPUT: - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: K.is_full_space() - 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] - ] - - 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] - ] + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is a Z-operator on ``K``. - TESTS: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - Each positive operator generator should send the generators of the - cone into the cone:: + - ``True`` will be returned if it can be proven that ``L`` + is a Z-operator on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not a Z-operator on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - 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 + .. SEEALSO:: - Each positive operator generator should send a random element of the - cone into the cone:: + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_lyapunov_like_on` - 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 + EXAMPLES: - A random element of the positive operator cone should send a random - element of the cone into the cone:: + The identity operator is always a Z-operator:: 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: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_Z_operator_on(L,K) 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:: + The "zero" operator is always a Z-operator:: - 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 + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_Z_operator_on(L,K) 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: 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 + TESTS: - The dimension of the cone of positive operators is given by the - corollary in my paper:: + Everything in ``K.Z_operators_gens()`` should be a Z-operator + on ``K``:: - 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: L = ToricLattice(n**2) - sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim() - sage: expected = n**2 - l*(m - l) - (n - m)*m - 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 + sage: all([ is_Z_operator_on(L,K) # long time + ....: for L in K.Z_operators_gens() ]) # long time 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 + sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time + ....: for L in K.Z_operators_gens() ]) # long time True - The lineality of the cone of positive operators follows from the - description of its generators:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) - 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: expected = n**2 - K.dim()*K.dual().dim() - sage: actual == expected - True + sage: L = identity_matrix(3) + sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ] + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - The trivial cone, full space, and half-plane all give rise to the - expected linealities:: - 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).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 - 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: actual == 2 - True + We can't give reliable answers over inexact rings:: - A cone is proper if and only if its cone of positive operators - is proper:: + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - 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([p.list() for p in pi_of_K], lattice=L) - sage: K.is_proper() == pi_cone.is_proper() - 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 - # two values to construct the appropriate "long vector" space. - F = K.lattice().base_field() - n = K.lattice_dim() + return is_cross_positive_on(-L,K) + - tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] +def is_lyapunov_like_on(L,K): + r""" + Determine whether or not ``L`` is Lyapunov-like on ``K``. - # Convert those tensor products to long vectors. - W = VectorSpace(F, n**2) - vectors = [ W(tp.list()) for tp in tensor_products ] + We say that ``L`` is Lyapunov-like on a closed convex cone ``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle = 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - # 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())) + An operator is Lyapunov-like on ``K`` if and only if both the + operator itself and its negation are cross-positive on ``K``. - # Now compute the desired cone from its dual... - pi_cone = pi_dual.dual() + To reliably check whether or not ``L`` is Lyapunov-like, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - # And finally convert its rays back to matrix representations. - M = MatrixSpace(F, n) - return [ M(v.list()) for v in pi_cone.rays() ] + INPUT: + - ``L`` -- A matrix over either an exact ring or ``SR``. -def Z_transformation_gens(K): - r""" - Compute generators of the cone of Z-transformations on this cone. + - ``K`` -- A polyhedral closed convex 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. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is Lyapunov-like on ``K``. - EXAMPLES: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - Z-transformations 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) - [ - [ 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) - ....: for i in range(z.nrows()) - ....: for j in range(z.ncols()) - ....: if i != j ]) - True + - ``True`` will be returned if it can be proven that ``L`` + is Lyapunov-like on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not Lyapunov-like on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. + + .. SEEALSO:: - The trivial cone in a trivial space has no Z-transformations:: + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on` - sage: K = Cone([], ToricLattice(0)) - sage: Z_transformation_gens(K) - [] + EXAMPLES: - Z-transformations on a subspace are Lyapunov-like and vice-versa:: + Diagonal matrices are Lyapunov-like operators on the nonnegative + orthant:: - 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 == lls + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = diagonal_matrix(random_vector(QQ,3)) + sage: is_lyapunov_like_on(L,K) True TESTS: - The Z-property is possessed by every Z-transformation:: + The identity operator is always Lyapunov-like:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=6) - 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]) + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - The lineality space of Z is LL:: + The "zero" operator is always Lyapunov-like:: - 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=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - And thus, the lineality of Z is the Lyapunov rank:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=6) - 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 ], lattice=L) - sage: z_cone.lineality() == K.lyapunov_rank() + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_lyapunov_like_on(L,K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time + True + sage: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time True - The lineality spaces of pi-star and Z-star are equal: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=5) - sage: pi_of_K = positive_operator_gens(K) - sage: Z_of_K = Z_transformation_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_star.linear_subspace() == z_star.linear_subspace() - 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 - # two values to construct the appropriate "long vector" space. - F = K.lattice().base_field() - n = K.lattice_dim() + sage: L = identity_matrix(3) + sage: K = [ vector([2,2,-1]), vector([5,4,-3]) ] + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - # 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() ] + We can't give reliable answers over inexact rings:: - # Turn our matrices into long vectors... - W = VectorSpace(F, n**2) - vectors = [ W(m.list()) for m in tensor_products ] + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. + + """ + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - # 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())) + return all([ s*(L*x) == 0 + for (x,s) in K.discrete_complementarity_set() ]) - # 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 - # not cross-positive ones. - M = MatrixSpace(F, n) - return [ -M(v.list()) for v in Sigma_cone.rays() ] +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 Sigma_cone(K): + gens = K.cross_positive_operators_gens() + 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) - -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) + gens = K.Z_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) + +def pi_cone(K1, K2=None): + if K2 is None: + K2 = K1 + gens = K1.positive_operators_gens(K2) + L = ToricLattice(K1.lattice_dim()*K2.lattice_dim()) + return Cone([ g.list() for g in gens ], lattice=L, check=False)