X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=60f9c34ec8bc271d65812859f51ca77636c8cbbc;hb=874e3ce831e0b1901b3c280a32ffe18e36f54959;hp=81698e467c603df3e57cb764c256a2e082df5ec6;hpb=b966ec383942ef1ff7837786c29d1f3edc33b84e;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 81698e4..60f9c34 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -7,57 +7,115 @@ addsitedir(abspath('../../')) from sage.all import * - -def project_span(K, K2 = None): +def project_span(K): r""" - Return a "copy" of ``K`` embeded in a lower-dimensional space. - - By default, we will project ``K`` into the subspace spanned by its - rays. However, if ``K2`` is not ``None``, we will project into the - space spanned by the rays of ``K2`` instead. + Project ``K`` into its own span. EXAMPLES:: - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: project_span(K) - 2-d cone in 2-d lattice N - sage: project_span(K).rays() - N(1, 0), - N(0, 1) - in 2-d lattice N - - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: K2 = Cone([(0,1)]) - sage: project_span(K, K2).rays() + sage: K = Cone([(1,)]) + sage: project_span(K) == K + True + + sage: K2 = Cone([(1,0)]) + sage: project_span(K2).rays() N(1) in 1-d lattice N + sage: K3 = Cone([(1,0,0)]) + sage: project_span(K3).rays() + N(1) + in 1-d lattice N + sage: project_span(K2) == project_span(K3) + True + + TESTS: + + The projected cone should always be solid:: + + sage: K = random_cone() + sage: K_S = project_span(K) + sage: K_S.is_solid() + True + + If we do this according to our paper, then the result is proper:: + + sage: K = random_cone() + sage: K_S = project_span(K) + sage: P = project_span(K_S.dual()).dual() + sage: P.is_proper() + True """ - # Allow us to use a second cone to generate the subspace into - # which we're "projecting." - if K2 is None: - K2 = K + F = K.lattice().base_field() + Q = K.lattice().quotient(K.sublattice_complement()) + vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + + L = None + if len(vecs) == 0: + L = ToricLattice(0) + + return Cone(vecs, lattice=L) + + +def rename_lattice(L,s): + r""" + Change all names of the given lattice to ``s``. + """ + L._name = s + L._dual_name = s + L._latex_name = s + L._latex_dual_name = s + +def span_iso(K): + r""" + Return an isomorphism (and its inverse) that will send ``K`` into a + lower-dimensional space isomorphic to its span (and back). + + EXAMPLES: + + The inverse composed with the isomorphism should be the identity:: + + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi_inv(phi(K)) == K + True + + The image of ``K`` under the isomorphism should have full dimension:: + + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi(K).dim() == phi(K).lattice_dim() + True + + """ + phi_domain = K.sublattice().vector_space() + phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension()) + + # S goes from the new space to the cone space. + S = linear_transformation(phi_codo, phi_domain, phi_domain.basis()) - # Use these to generate the new cone. - cs1 = K.rays().matrix().columns() + # phi goes from the cone space to the new space. + def phi(J_orig): + r""" + Takes a cone ``J`` and sends it into the new space. + """ + newrays = map(S.inverse(), J_orig.rays()) + L = None + if len(newrays) == 0: + L = ToricLattice(0) - # And use these to figure out which indices to drop. - cs2 = K2.rays().matrix().columns() + return Cone(newrays, lattice=L) - perp_idxs = [] + def phi_inverse(J_sub): + r""" + The inverse to phi which goes from the new space to the cone space. + """ + newrays = map(S, J_sub.rays()) + return Cone(newrays, lattice=K.lattice()) - for idx in range(0, len(cs2)): - if cs2[idx].is_zero(): - perp_idxs.append(idx) - solid_cols = [ cs1[idx] for idx in range(0,len(cs1)) - if not idx in perp_idxs - and not idx >= len(cs2) ] + return (phi, phi_inverse) - m = matrix(solid_cols) - L = ToricLattice(len(m.rows())) - J = Cone(m.transpose(), lattice=L) - return J def discrete_complementarity_set(K): @@ -259,7 +317,7 @@ def lyapunov_rank(K): An integer representing the Lyapunov rank of the cone. If the dimension of the ambient vector space is `n`, then the Lyapunov rank will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible for any cone. + not possible (see the first reference). .. note:: @@ -379,7 +437,7 @@ def lyapunov_rank(K): In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: - sage: K = random_cone(max_dim=10, max_rays=16) + sage: K = random_cone(max_dim=10) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: b == n-1 @@ -388,20 +446,79 @@ def lyapunov_rank(K): The calculation of the Lyapunov rank of an improper cone can be reduced to that of a proper cone [Orlitzky/Gowda]_:: - sage: K = random_cone(max_dim=15, max_rays=25) + sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False) sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: J_T1 = project_span(K_S.dual()).dual() - sage: J_T2 = project_span(K, K_S.dual()) - sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice()) - sage: J_T1 == J_T2 + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected True - sage: J_T = J_T1 + + Repeat the previous test with different ``random_cone()`` params:: + + sage: K = random_cone(max_dim=15, solid=False, strictly_convex=True) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() sage: l = K.linear_subspace().dimension() sage: codim = K.lattice_dim() - K.dim() sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 sage: actual == expected True + sage: K = random_cone(max_dim=15, solid=True, strictly_convex=False) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + sage: K = random_cone(max_dim=15, solid=True, strictly_convex=True) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + sage: K = random_cone(max_dim=15) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + And test with the project_span function:: + + sage: K = random_cone(max_dim=15) + sage: actual = lyapunov_rank(K) + sage: K_S = project_span(K) + sage: P = project_span(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + """ return len(LL(K))