From 6bd30534d5aa984c73f511121efa8fda4386c51a Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 19:13:31 -0400 Subject: [PATCH] In the middle of mangling things. --- mjo/cone/cone.py | 175 +++++++++++++++-------------------------------- 1 file changed, 54 insertions(+), 121 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 2d84337..b9e930e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -12,18 +12,11 @@ def drop_dependent(vs): r""" Return the largest linearly-independent subset of ``vs``. """ - if len(vs) == 0: - # ...for lazy enough definitions of linearly-independent - return vs - result = [] - old_V = VectorSpace(vs[0].parent().base_field(), 0) - - for v in vs: - new_V = span(result + [v]) - if new_V.dimension() > old_V.dimension(): - result.append(v) - old_V = new_V + m = matrix(vs).echelon_form() + for idx in range(0, m.nrows()): + if not m[idx].is_zero(): + result.append(m[idx]) return result @@ -66,66 +59,6 @@ def basically_the_same(K1,K2): -def iso_space(K): - r""" - Construct the space `W \times W^{\perp}` isomorphic to the ambient space - of ``K`` where `W` is equal to the span of ``K``. - """ - V = K.lattice().vector_space() - - # Create the space W \times W^{\perp} isomorphic to V. - # First we get an orthogonal (but not normal) basis... - M = matrix(V.base_field(), K.rays()) - W_basis = drop_dependent(K.rays()) - - W = V.subspace_with_basis(W_basis) - W_perp = W.complement() - - return W.cartesian_product(W_perp) - - -def ips_iso(K): - r""" - Construct the IPS isomorphism and its inverse from our paper. - - Given a cone ``K``, the returned isomorphism will split its ambient - vector space `V` into a cartesian product `W \times W^{\perp}` where - `W` equals the span of ``K``. - """ - V = K.lattice().vector_space() - V_iso = iso_space(K) - (W, W_perp) = V_iso.cartesian_factors() - - # A space equivalent to V, but using our basis. - V_user = V.subspace_with_basis( W.basis() + W_perp.basis() ) - - def phi(v): - # Write v in terms of our custom basis, where the first dim(W) - # coordinates are for the W-part of the basis. - cs = V_user.coordinates(v) - - w1 = sum([ V_user.basis()[idx]*cs[idx] - for idx in range(0, W.dimension()) ]) - w2 = sum([ V_user.basis()[idx]*cs[idx] - for idx in range(W.dimension(), V.dimension()) ]) - - return V_iso( (w1, w2) ) - - - def phi_inv( pair ): - # Crash if the arguments are in the wrong spaces. - V_iso(pair) - - #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ]) - #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx] - # for idx in range(0,n-m) ]) - - return sum( pair.cartesian_factors() ) - - - return (phi,phi_inv) - - def rho(K, K2=None): r""" Restrict ``K`` into its own span, or the span of another cone. @@ -141,18 +74,18 @@ def rho(K, K2=None): EXAMPLES:: sage: K = Cone([(1,)]) - sage: restrict_span(K) == K + sage: rho(K) == K True sage: K2 = Cone([(1,0)]) - sage: restrict_span(K2).rays() + sage: rho(K2).rays() N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: restrict_span(K3).rays() + sage: rho(K3).rays() N(1) in 1-d lattice N - sage: restrict_span(K2) == restrict_span(K3) + sage: rho(K2) == rho(K3) True TESTS: @@ -161,7 +94,7 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K_S = restrict_span(K) + sage: K_S = rho(K) sage: K_S.is_solid() True @@ -170,7 +103,7 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K_S = restrict_span(K, K.dual() ) + sage: K_S = rho(K, K.dual() ) sage: K_S.lattice_dim() == K.dual().dim() True @@ -178,14 +111,14 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K.dim() == restrict_span(K).dim() + sage: K.dim() == rho(K).dim() True Nor should it affect the lineality of a cone:: sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: lineality(K) == lineality(restrict_span(K)) + sage: lineality(K) == lineality(rho(K)) True No matter which space we restrict to, the lineality should not @@ -193,20 +126,20 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: lineality(K) >= lineality(restrict_span(K)) + sage: lineality(K) >= lineality(rho(K)) True - sage: lineality(K) >= lineality(restrict_span(K, K.dual())) + sage: lineality(K) >= lineality(rho(K, K.dual())) True If we do this according to our paper, then the result is proper:: sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() + sage: K_S = rho(K) + sage: P = rho(K_S.dual()).dual() sage: P.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) + sage: P = rho(K_S, K_S.dual()) sage: P.is_proper() True @@ -214,11 +147,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() + sage: K_S = rho(K) + sage: P = rho(K_S.dual()).dual() sage: P.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) + sage: P = rho(K_S, K_S.dual()) sage: P.is_proper() True @@ -226,11 +159,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() + sage: K_S = rho(K) + sage: P = rho(K_S.dual()).dual() sage: P.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) + sage: P = rho(K_S, K_S.dual()) sage: P.is_proper() True @@ -238,11 +171,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() + sage: K_S = rho(K) + sage: P = rho(K_S.dual()).dual() sage: P.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) + sage: P = rho(K_S, K_S.dual()) sage: P.is_proper() True @@ -251,8 +184,8 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: K_W = rho(K, K.dual()) + sage: K_star_W_star = rho(K.dual()).dual() sage: basically_the_same(K_W, K_star_W_star) True @@ -260,8 +193,8 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: K_W = rho(K, K.dual()) + sage: K_star_W_star = rho(K.dual()).dual() sage: basically_the_same(K_W, K_star_W_star) True @@ -269,8 +202,8 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: K_W = rho(K, K.dual()) + sage: K_star_W_star = rho(K.dual()).dual() sage: basically_the_same(K_W, K_star_W_star) True @@ -278,8 +211,8 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: K_W = rho(K, K.dual()) + sage: K_star_W_star = rho(K.dual()).dual() sage: basically_the_same(K_W, K_star_W_star) True @@ -287,24 +220,25 @@ def rho(K, K2=None): if K2 is None: K2 = K - phi,_ = ips_iso(K2) - (W, W_perp) = iso_space(K2).cartesian_factors() + # First we project K onto the span of K2. This can be done with + # cones (i.e. without converting to vector spaces), but it's + # annoying to deal with lattice mismatches. + span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice()) + K = K.intersection(span_K2) - ray_pairs = [ phi(r) for r in K.rays() ] - - # Shouldn't matter? - # - #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]): - # msg = 'Cone has nonzero components in W-perp!' - # raise ValueError(msg) + V = K.lattice().vector_space() - # Represent the cone in terms of a basis for W, i.e. with smaller - # vectors. - ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ] + # Create the space W \times W^{\perp} isomorphic to V. + # First we get an orthogonal (but not normal) basis... + W_basis = drop_dependent(K2.rays()) + W = V.subspace_with_basis(W_basis) - L = ToricLattice(W.dimension()) + # We've already intersected K with the span of K2, so every + # generator of K should belong to W now. + W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - return Cone(ws, lattice=L) + L = ToricLattice(K2.dim()) + return Cone(W_rays, lattice=L) @@ -798,8 +732,8 @@ def lyapunov_rank(K): sage: set_random_seed() sage: K = random_cone(max_dim=8) sage: actual = lyapunov_rank(K) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() + sage: K_S = rho(K) + sage: P = rho(K_S.dual()).dual() sage: l = lineality(K) sage: c = codim(K) sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 @@ -845,7 +779,7 @@ def lyapunov_rank(K): if m < n: # K is not solid, project onto its span. - K = restrict_span(K) + K = rho(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 @@ -853,9 +787,8 @@ def lyapunov_rank(K): if l > 0: # K is not pointed, project its dual onto its span. # Uses a proposition from our paper, i.e. this is - # equivalent to K = restrict_span(K.dual()).dual() - #K = restrict_span(intersect_span(K,K.dual()), K.dual()) - K = restrict_span(K, K.dual()) + # equivalent to K = rho(K.dual()).dual() + K = rho(K, K.dual()) # Lemma 3 beta += m * l -- 2.43.2