X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=4221566a52c1210fa1045b9df9305f04a45924a6;hb=3f2f1e2cc17d8796878a6971050f7aed7a37f737;hp=865a0700eac91a6af5550477eed6ebe2500cd304;hpb=4332dec9566bd956c4c4cd64f30510930edc95e6;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 865a070..4221566 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -35,6 +35,12 @@ def is_positive_on(L,K): - ``False`` will also be returned if we can't decide; specifically if we arrive at a symbolic inequality that cannot be resolved. + .. SEEALSO:: + + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` + EXAMPLES: Nonnegative matrices are positive operators on the nonnegative @@ -45,6 +51,34 @@ def is_positive_on(L,K): sage: is_positive_on(L,K) True + Your matrix can be over any exact ring, but you may get unexpected + answers with weirder rings. For example, any rational matrix is + positive on the plane, but if your matrix contains polynomial + variables, the answer will be negative:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: L = matrix(QQ[x], [[x,0],[0,1]]) + sage: is_positive_on(L,K) + False + + The previous example is "unexpected" because it depends on how we + check whether or not ``L`` is positive. For exact base rings, we + check whether or not ``L*z`` belongs to ``K`` for each ``z in K``. + If ``K`` is closed, then an equally-valid test would be to check + whether the inner product of ``L*z`` and ``s`` is nonnegative for + every ``z`` in ``K`` and ``s`` in ``K.dual()``. In fact, that is + what we do over inexact rings. In the previous example, that test + would return an affirmative answer:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: L = matrix(QQ[x], [[x,0],[0,1]]) + sage: all([ (L*z).inner_product(s) for z in K for s in K.dual() ]) + True + sage: is_positive_on(L.change_ring(SR), K) + True + TESTS: The identity operator is always positive:: @@ -77,14 +111,24 @@ def is_positive_on(L,K): Technically we could test this, but for now only closed convex cones are supported as our ``K`` argument:: - sage: L = identity_matrix(3) 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. + """ + 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: @@ -138,6 +182,12 @@ def is_cross_positive_on(L,K): - ``False`` will also be returned if we can't decide; specifically if we arrive at a symbolic inequality that cannot be resolved. + .. SEEALSO:: + + :func:`is_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` + EXAMPLES: The identity operator is always cross-positive:: @@ -179,6 +229,15 @@ def is_cross_positive_on(L,K): ... 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_cross_positive_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.') @@ -188,7 +247,7 @@ def is_cross_positive_on(L,K): return all([ s*(L*x) >= 0 for (x,s) in K.discrete_complementarity_set() ]) -def is_Z_on(L,K): +def is_Z_operator_on(L,K): r""" Determine whether or not ``L`` is a Z-operator on ``K``. @@ -227,6 +286,12 @@ def is_Z_on(L,K): - ``False`` will also be returned if we can't decide; specifically if we arrive at a symbolic inequality that cannot be resolved. + .. SEEALSO:: + + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_lyapunov_like_on` + EXAMPLES: The identity operator is always a Z-operator:: @@ -234,7 +299,7 @@ def is_Z_on(L,K): 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) + sage: is_Z_operator_on(L,K) True The "zero" operator is always a Z-operator:: @@ -242,7 +307,7 @@ def is_Z_on(L,K): 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) + sage: is_Z_operator_on(L,K) True TESTS: @@ -251,11 +316,11 @@ def is_Z_on(L,K): on ``K``:: sage: K = random_cone(max_ambient_dim=5) - sage: all([ is_Z_on(L,K) # long time + sage: all([ is_Z_operator_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 + sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time + ....: for L in K.Z_operators_gens() ]) # long time True Technically we could test this, but for now only closed convex cones @@ -263,11 +328,21 @@ def is_Z_on(L,K): sage: L = identity_matrix(3) sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ] - sage: is_Z_on(L,K) + sage: is_Z_operator_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_Z_operator_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. + """ return is_cross_positive_on(-L,K) @@ -282,6 +357,9 @@ def is_lyapunov_like_on(L,K): set of ``K``. This property need only be checked for generators of ``K`` and its dual. + An operator is Lyapunov-like on ``K`` if and only if both the + operator itself and its negation are cross-positive on ``K``. + 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 @@ -308,6 +386,12 @@ def is_lyapunov_like_on(L,K): - ``False`` will also be returned if we can't decide; specifically if we arrive at a symbolic inequality that cannot be resolved. + .. SEEALSO:: + + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on` + EXAMPLES: Diagonal matrices are Lyapunov-like operators on the nonnegative @@ -357,12 +441,37 @@ def is_lyapunov_like_on(L,K): ... 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_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. + + An operator is Lyapunov-like on a cone if and only if both the + operator and its negation are cross-positive on the cone:: + + sage: K = random_cone(max_ambient_dim=5) + sage: R = K.lattice().vector_space().base_ring() + sage: L = random_matrix(R, K.lattice_dim()) + sage: actual = is_lyapunov_like_on(L,K) # long time + sage: expected = (is_cross_positive_on(L,K) and # long time + ....: is_cross_positive_on(-L,K)) # long time + sage: actual == expected # 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('The base ring of L is neither SR nor exact.') + # Even though ``discrete_complementarity_set`` is a cached method + # of cones, this is faster than calling ``is_cross_positive_on`` + # twice: doing so checks twice as many inequalities as the number + # of equalities that we're about to check. return all([ s*(L*x) == 0 for (x,s) in K.discrete_complementarity_set() ])