X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=3f5a4fed4e1c49853f00eafcf6084744223ca296;hb=3e6f51aa1f2d6f300cb22281701901add3631904;hp=81698e467c603df3e57cb764c256a2e082df5ec6;hpb=b966ec383942ef1ff7837786c29d1f3edc33b84e;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 81698e4..3f5a4fe 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -7,57 +7,91 @@ addsitedir(abspath('../../')) from sage.all import * +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 project_span(K, K2 = None): +def span_iso(K): r""" - Return a "copy" of ``K`` embeded in a lower-dimensional space. + 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:: - 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. + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi(K).dim() == phi(K).lattice_dim() + True - EXAMPLES:: + The isomorphism should be an inner product space isomorphism, and + thus it should preserve dual cones (and commute with the "dual" + operation). But beware the automatic renaming of the dual lattice. + OH AND YOU HAVE TO SORT THE CONES:: - 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 = random_cone(max_dim=10, strictly_convex=False, solid=True) + sage: L = K.lattice() + sage: rename_lattice(L, 'L') + sage: (phi, phi_inv) = span_iso(K) + sage: sorted(phi_inv( phi(K).dual() )) == sorted(K.dual()) + True - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: K2 = Cone([(0,1)]) - sage: project_span(K, K2).rays() - N(1) - in 1-d lattice N + We may need to isomorph twice to make sure we stop moving down to + smaller spaces. (Once you've done this on a cone and its dual, the + result should be proper.) OH AND YOU HAVE TO SORT THE CONES:: + + sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False) + sage: L = K.lattice() + sage: rename_lattice(L, 'L') + sage: (phi, phi_inv) = span_iso(K) + sage: K_S = phi(K) + sage: (phi_dual, phi_dual_inv) = span_iso(K_S.dual()) + sage: J_T = phi_dual(K_S.dual()).dual() + sage: phi_inv(phi_dual_inv(J_T)) == K + True """ - # Allow us to use a second cone to generate the subspace into - # which we're "projecting." - if K2 is None: - K2 = K + 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 +293,7 @@ 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 +413,7 @@ 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 @@ -388,15 +422,14 @@ 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, phi1_inv) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, phi2_inv) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: phi1_inv(phi2_inv(J_T)) == K True - sage: J_T = J_T1 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