X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b7456e21abc170d6479f9010a2b7ccdd5ab9438d;hb=1bbade9f41ffbfe366b15d0db657f666bc1f025d;hp=cbbfe9b692f3bdf36f68f202deedd467c684cef0;hpb=115ecffcebcd0b7f86358aa38ddb52f1115e966c;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index cbbfe9b..b7456e2 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,350 +1,23 @@ from sage.all import * -from sage.geometry.cone import is_Cone - -def is_positive_on(L,K): - r""" - 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``. - - 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 matrix over either an exact ring or ``SR``. - - - ``K`` -- A polyhedral closed convex cone. - - OUTPUT: - - If the base ring of ``L`` is exact, then ``True`` will be returned if - and only if ``L`` is positive on ``K``. - - If the base ring of ``L`` is ``SR``, then the situation is more - complicated: - - - ``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. - - EXAMPLES: - - 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(max_ambient_dim=8) - sage: L = identity_matrix(K.lattice_dim()) - sage: is_positive_on(L,K) - True - - The "zero" operator is always positive:: - - 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_positive_on(L,K) - True - - Everything in ``K.positive_operators_gens()`` should be - positive on ``K``:: - - 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 - - """ - 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('base ring of operator L is neither SR nor exact') - - 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() ]) - - -def is_cross_positive_on(L,K): - r""" - Determine whether or not ``L`` is cross-positive on ``K``. - - 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. - - 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. - - INPUT: - - - ``L`` -- A matrix over either an exact ring or ``SR``. - - - ``K`` -- A polyhedral closed convex cone. - - OUTPUT: - - If the base ring of ``L`` is exact, then ``True`` will be returned if - and only if ``L`` is cross-positive on ``K``. - - If the base ring of ``L`` is ``SR``, then the situation is more - complicated: - - - ``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. - - EXAMPLES: - - The identity operator is always cross-positive:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: L = identity_matrix(K.lattice_dim()) - sage: is_cross_positive_on(L,K) - True - - The "zero" operator is always cross-positive:: - - 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_cross_positive_on(L,K) - True - - TESTS: - - 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 - - """ - 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('base ring of operator L is neither SR nor exact') - - return all([ s*(L*x) >= 0 - for (x,s) in K.discrete_complementarity_set() ]) - -def is_Z_on(L,K): - r""" - Determine whether or not ``L`` is a Z-operator on ``K``. - - 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. - - A matrix is a Z-operator on ``K`` if and only if its negation is a - cross-positive operator on ``K``. - - 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. - - INPUT: - - - ``L`` -- A matrix over either an exact ring or ``SR``. - - - ``K`` -- A polyhedral closed convex cone. - - OUTPUT: - - If the base ring of ``L`` is exact, then ``True`` will be returned if - and only if ``L`` is a Z-operator on ``K``. - - If the base ring of ``L`` is ``SR``, then the situation is more - complicated: - - - ``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. - - EXAMPLES: - - The identity operator is always a Z-operator:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: L = identity_matrix(K.lattice_dim()) - sage: is_Z_on(L,K) - True - - The "zero" operator is always a Z-operator:: - - 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_on(L,K) - True - - TESTS: - - Everything in ``K.Z_operators_gens()`` should be a Z-operator - on ``K``:: - - sage: K = random_cone(max_ambient_dim=5) - sage: all([ is_Z_on(L,K) # long time - ....: for L in K.Z_operators_gens() ]) # long time - True - sage: all([ is_Z_on(L.change_ring(SR),K) # long time - ....: for L in K.Z_operators_gens() ]) # long time - True - - """ - return is_cross_positive_on(-L,K) - - -def is_lyapunov_like_on(L,K): - r""" - Determine whether or not ``L`` is Lyapunov-like on ``K``. - - 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. - - 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. - - INPUT: - - - ``L`` -- A matrix over either an exact ring or ``SR``. - - - ``K`` -- A polyhedral closed convex cone. - - OUTPUT: - - If the base ring of ``L`` is exact, then ``True`` will be returned if - and only if ``L`` is Lyapunov-like on ``K``. - - If the base ring of ``L`` is ``SR``, then the situation is more - complicated: - - - ``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. - - EXAMPLES: - - Diagonal matrices are Lyapunov-like operators on the nonnegative - orthant:: - - 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 identity operator is always Lyapunov-like:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: L = identity_matrix(K.lattice_dim()) - sage: is_lyapunov_like_on(L,K) - True - - The "zero" operator is always Lyapunov-like:: - - 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 - - Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like - on ``K``:: - - 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 - - """ - 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('base ring of operator L is neither SR nor exact') - - return all([ s*(L*x) == 0 - for (x,s) in K.discrete_complementarity_set() ]) - 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) + 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) + return Cone(( g.list() for g in gens ), lattice=L, check=False) def Z_cone(K): gens = K.Z_operators_gens() L = ToricLattice(K.lattice_dim()**2) - return Cone([ g.list() for g in gens ], lattice=L, check=False) + 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) + return Cone(( g.list() for g in gens ), lattice=L, check=False)