From 446018aef1279ff14866ae5e1d803a4a2a7c8024 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 5 Jun 2015 18:49:52 -0400 Subject: [PATCH 01/16] Commit a big fucking mess while I refactor the span restriction. --- mjo/cone/cone.py | 308 ++++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 280 insertions(+), 28 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index ff7d195..6fb15ae 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,55 +8,263 @@ addsitedir(abspath('../../')) from sage.all import * -def project_span(K): +def drop_dependent(vs): r""" - Project ``K`` into its own span. + 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 + + return result + + +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. + 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 unrestrict_span(K, K2=None): + if K2 is None: + K2 = K + + _,phi_inv = ips_iso(K2) + V_iso = iso_space(K2) + (W, W_perp) = V_iso.cartesian_factors() + + rays = [] + for r in K.rays(): + w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ]) + pair = V_iso( (w, W_perp.zero()) ) + rays.append( phi_inv(pair) ) + + L = ToricLattice(W.dimension() + W_perp.dimension()) + + return Cone(rays, lattice=L) + + + +def intersect_span(K1, K2): + r""" + Return a new cone obtained by intersecting ``K1`` with the span of ``K2``. + """ + L = K1.lattice() + + if L.rank() != K2.lattice().rank(): + raise ValueError('K1 and K2 must belong to lattices of the same rank.') + + SL_gens = list(K2.rays()) + span_K2_gens = SL_gens + [ -g for g in SL_gens ] + + # The lattices have the same rank (see above) so this should work. + span_K2 = Cone(span_K2_gens, L) + return K1.intersection(span_K2) + + + +def restrict_span(K, K2=None): + r""" + Restrict ``K`` into its own span, or the span of another cone. + + INPUT: + + - ``K2`` -- another cone whose lattice has the same rank as this cone. + + OUTPUT: + + A new cone in a sublattice. EXAMPLES:: sage: K = Cone([(1,)]) - sage: project_span(K) == K + sage: restrict_span(K) == K True sage: K2 = Cone([(1,0)]) - sage: project_span(K2).rays() + sage: restrict_span(K2).rays() N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: project_span(K3).rays() + sage: restrict_span(K3).rays() N(1) in 1-d lattice N - sage: project_span(K2) == project_span(K3) + sage: restrict_span(K2) == restrict_span(K3) True TESTS: The projected cone should always be solid:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) + sage: K_S = restrict_span(K) sage: K_S.is_solid() True + And the resulting cone should live in a space having the same + dimension as the space we restricted it to:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() ) + sage: K_S.lattice_dim() == K.dual().dim() + True + + This function has ``unrestrict_span()`` as its inverse:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10, solid=True) + sage: J = restrict_span(K) + sage: K == unrestrict_span(J,K) + True + + This function should not affect the dimension of a cone:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: K.dim() == restrict_span(K).dim() + True + + Nor should it affect the lineality of a cone:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: lineality(K) == lineality(restrict_span(K)) + True + + No matter which space we restrict to, the lineality should not + increase:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: J = intersect_span(K, K.dual()) + sage: lineality(K) >= lineality(restrict_span(J, 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 = 10) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() + sage: K_S = restrict_span(K) + sage: P = restrict_span(K_S.dual()).dual() sage: P.is_proper() True + If ``K`` is strictly convex, then both ``K_W`` and + ``K_star_W.dual()`` should equal ``K`` (after we unrestrict):: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10, strictly_convex=True) + sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) + sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: j1 = unrestrict_span(K_W, K.dual()) + sage: j2 = unrestrict_span(K_star_W_star, K.dual()) + sage: j1 == j2 + True + sage: j1 == K + True + sage: K; [ list(r) for r in K.rays() ] + + Test the proposition in our paper concerning the duals, where the + subspace `W` is the span of `K^{*}`:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10, solid=False, strictly_convex=False) + sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) + sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual() + sage: K_W.nrays() == K_star_W_star.nrays() + True + sage: K_W.dim() == K_star_W_star.dim() + True + sage: lineality(K_W) == lineality(K_star_W_star) + True + sage: K_W.is_solid() == K_star_W_star.is_solid() + True + sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex() + True + """ - L = K.lattice() - F = L.base_field() - Q = L.quotient(K.sublattice_complement()) - vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + if K2 is None: + K2 = K + + phi,_ = ips_iso(K2) + (W, W_perp) = iso_space(K2).cartesian_factors() + + ray_pairs = [ phi(r) for r in K.rays() ] + + if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]): + msg = 'Cone has nonzero components in W-perp!' + raise ValueError(msg) - newL = None - if len(vecs) == 0: - newL = ToricLattice(0) + # 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 ] - return Cone(vecs, lattice=newL) + L = ToricLattice(W.dimension()) + + return Cone(ws, lattice=L) @@ -112,6 +320,7 @@ def lineality(K): The lineality of a cone should be an integer between zero and the dimension of the ambient space, inclusive:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10) sage: l = lineality(K) sage: l in ZZ @@ -121,6 +330,7 @@ def lineality(K): A strictly convex cone should have lineality zero:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10, strictly_convex = True) sage: lineality(K) 0 @@ -191,6 +401,7 @@ def codim(K): The codimension of a cone should be an integer between zero and the dimension of the ambient space, inclusive:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10) sage: c = codim(K) sage: c in ZZ @@ -200,12 +411,14 @@ def codim(K): A solid cone should have codimension zero:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10, solid = True) sage: codim(K) 0 The codimension of a cone is equal to the lineality of its dual:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10, solid = True) sage: codim(K) == lineality(K.dual()) True @@ -268,11 +481,12 @@ def discrete_complementarity_set(K): The complementarity set of the dual can be obtained by switching the components of the complementarity set of the original cone:: - sage: K1 = random_cone(max_dim=10, max_rays=10) + sage: set_random_seed() + sage: K1 = random_cone(max_dim=6) sage: K2 = K1.dual() sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] sage: actual = discrete_complementarity_set(K1) - sage: actual == expected + sage: sorted(actual) == sorted(expected) True """ @@ -352,18 +566,33 @@ def LL(K): every pair `\left( x,s \right)` in the discrete complementarity set of the cone:: + sage: set_random_seed() sage: K = random_cone(max_dim=8, max_rays=10) sage: C_of_K = discrete_complementarity_set(K) sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] sage: sum(map(abs, l)) 0 + The Lyapunov-like transformations on a cone and its dual are related + by transposition, but we're not guaranteed to compute transposed + elements of `LL\left( K \right)` as our basis for `LL\left( K^{*} + \right)` + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, max_rays=10) + sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] + sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) + sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] + sage: LL2_vecs = [ V(m.list()) for m in LL2 ] + sage: V.span(LL1_vecs) == V.span(LL2_vecs) + True + """ V = K.lattice().vector_space() C_of_K = discrete_complementarity_set(K) - tensor_products = [s.tensor_product(x) for (x,s) in C_of_K] + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] # Sage doesn't think matrices are vectors, so we have to convert # our matrices to vectors explicitly before we can figure out how @@ -465,8 +694,8 @@ def lyapunov_rank(K): [Orlitzky/Gowda]_:: sage: R5 = VectorSpace(QQ, 5) - sage: gens = R5.basis() + [ -r for r in R5.basis() ] - sage: K = Cone(gens) + sage: gs = R5.basis() + [ -r for r in R5.basis() ] + sage: K = Cone(gs) sage: lyapunov_rank(K) 25 @@ -499,8 +728,8 @@ def lyapunov_rank(K): sage: neg_e1 = (-1,0,0,0,0) sage: e2 = (0,1,0,0,0) sage: neg_e2 = (0,-1,0,0,0) - sage: zero = (0,0,0,0,0) - sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero]) + sage: z = (0,0,0,0,0) + sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z]) sage: lyapunov_rank(K) 19 sage: K.lattice_dim()**2 - K.dim()*codim(K) @@ -535,6 +764,7 @@ def lyapunov_rank(K): The Lyapunov rank should be additive on a product of proper cones [Rudolf et al.]_:: + sage: set_random_seed() sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: K = K1.cartesian_product(K2) @@ -544,16 +774,25 @@ def lyapunov_rank(K): The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` itself [Rudolf et al.]_:: + sage: set_random_seed() sage: K = random_cone(max_dim=10, max_rays=10) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True + Make sure we exercise the non-strictly-convex/non-solid case:: + + sage: set_random_seed() + sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True + The Lyapunov rank of a proper polyhedral cone in `n` dimensions can be any number between `1` and `n` inclusive, excluding `n-1` [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the trivial cone in a trivial space as well. However, in zero dimensions, the Lyapunov rank of the trivial cone will be zero:: + sage: set_random_seed() sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() @@ -565,6 +804,7 @@ def lyapunov_rank(K): In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: + sage: set_random_seed() sage: K = random_cone(max_dim=10) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() @@ -574,10 +814,11 @@ 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: set_random_seed() sage: K = random_cone(max_dim=10) sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() + sage: K_S = restrict_span(K) + sage: P = restrict_span(K_S.dual()).dual() sage: l = lineality(K) sage: c = codim(K) sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 @@ -586,11 +827,13 @@ def lyapunov_rank(K): The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: + sage: set_random_seed() sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: lyapunov_rank(K) == len(LL(K)) True """ + K_orig = K beta = 0 m = K.dim() @@ -599,14 +842,23 @@ def lyapunov_rank(K): if m < n: # K is not solid, project onto its span. - K = project_span(K) + K = restrict_span(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 if l > 0: # K is not pointed, project its dual onto its span. - K = project_span(K.dual()).dual() + # 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.dual()).dual() + + #Ks = [ list(r) for r in sorted(K.rays()) ] + #Js = [ list(r) for r in sorted(J.rays()) ] + + #if Ks != Js: + # print [ list(r) for r in K_orig.rays() ] # Lemma 3 beta += m * l -- 2.44.2 From 44802773ad9e5151890ed37e7bb2463ff9fc4135 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 5 Jun 2015 23:40:29 -0400 Subject: [PATCH 02/16] Use built-in Gram-Schmidt to make things a little better. --- mjo/cone/cone.py | 25 ++++--------------------- 1 file changed, 4 insertions(+), 21 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 6fb15ae..f2e8b2e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,26 +8,6 @@ addsitedir(abspath('../../')) from sage.all import * -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 - - return result - - def iso_space(K): r""" Construct the space `W \times W^{\perp}` isomorphic to the ambient space @@ -36,7 +16,10 @@ def iso_space(K): V = K.lattice().vector_space() # Create the space W \times W^{\perp} isomorphic to V. - W_basis = drop_dependent(K.rays()) + # First we get an orthogonal (but not normal) basis... + M = matrix(V.base_field(), K.rays()) + W_basis,_ = M.gram_schmidt() + W = V.subspace_with_basis(W_basis) W_perp = W.complement() -- 2.44.2 From 0bdf2bb8ca97eeb065e7dd3c36bdac6879a52116 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sat, 6 Jun 2015 21:16:27 -0400 Subject: [PATCH 03/16] Begin to clear up the mysteries of why the restriction doesn't work. Everything's still a mess, but at least the tests are passing. No more do we check for exact equality between K_W and K_star_W_star; instead we just check that many properties agree. In the paper we have an isomorphism, and equality holds. However, my isomorphism isn't an isomorphism when using a coordinate system -- and we have to use a non-normal one in Sage because we can't normalize vectors over QQ. Huh. --- mjo/cone/cone.py | 263 ++++++++++++++++++++++++++++++++++------------- 1 file changed, 190 insertions(+), 73 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index f2e8b2e..87cdf70 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,6 +8,44 @@ addsitedir(abspath('../../')) from sage.all import * +def basically_the_same(K1,K2): + r""" + ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` + otherwise. + """ + if K1.lattice_dim() != K2.lattice_dim(): + return False + + if K1.nrays() != K2.nrays(): + return False + + if K1.dim() != K2.dim(): + return False + + if lineality(K1) != lineality(K2): + return False + + if K1.is_solid() != K2.is_solid(): + return False + + if K1.is_strictly_convex() != K2.is_strictly_convex(): + return False + + if len(LL(K1)) != len(LL(K2)): + return False + + C_of_K1 = discrete_complementarity_set(K1) + C_of_K2 = discrete_complementarity_set(K2) + if len(C_of_K1) != len(C_of_K2): + return False + + if len(K1.facets()) != len(K2.facets()): + return False + + return True + + + def iso_space(K): r""" Construct the space `W \times W^{\perp}` isomorphic to the ambient space @@ -89,24 +127,6 @@ def unrestrict_span(K, K2=None): -def intersect_span(K1, K2): - r""" - Return a new cone obtained by intersecting ``K1`` with the span of ``K2``. - """ - L = K1.lattice() - - if L.rank() != K2.lattice().rank(): - raise ValueError('K1 and K2 must belong to lattices of the same rank.') - - SL_gens = list(K2.rays()) - span_K2_gens = SL_gens + [ -g for g in SL_gens ] - - # The lattices have the same rank (see above) so this should work. - span_K2 = Cone(span_K2_gens, L) - return K1.intersection(span_K2) - - - def restrict_span(K, K2=None): r""" Restrict ``K`` into its own span, or the span of another cone. @@ -141,7 +161,7 @@ def restrict_span(K, K2=None): The projected cone should always be solid:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) + sage: K = random_cone(max_dim = 8) sage: K_S = restrict_span(K) sage: K_S.is_solid() True @@ -150,15 +170,15 @@ def restrict_span(K, K2=None): dimension as the space we restricted it to:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() ) + sage: K = random_cone(max_dim = 8) + sage: K_S = restrict_span(K, K.dual() ) sage: K_S.lattice_dim() == K.dual().dim() True This function has ``unrestrict_span()`` as its inverse:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid=True) + sage: K = random_cone(max_dim = 8, solid=True) sage: J = restrict_span(K) sage: K == unrestrict_span(J,K) True @@ -166,14 +186,14 @@ def restrict_span(K, K2=None): This function should not affect the dimension of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) + sage: K = random_cone(max_dim = 8) sage: K.dim() == restrict_span(K).dim() True Nor should it affect the lineality of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) + sage: K = random_cone(max_dim = 8) sage: lineality(K) == lineality(restrict_span(K)) True @@ -181,51 +201,95 @@ def restrict_span(K, K2=None): increase:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: J = intersect_span(K, K.dual()) - sage: lineality(K) >= lineality(restrict_span(J, K.dual())) + sage: K = random_cone(max_dim = 8) + sage: lineality(K) >= lineality(restrict_span(K)) + True + sage: lineality(K) >= lineality(restrict_span(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 = 10) + 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: P.is_proper() True + sage: P = restrict_span(K_S, K_S.dual()) + sage: P.is_proper() + True - If ``K`` is strictly convex, then both ``K_W`` and - ``K_star_W.dual()`` should equal ``K`` (after we unrestrict):: + :: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, strictly_convex=True) - sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() - sage: j1 = unrestrict_span(K_W, K.dual()) - sage: j2 = unrestrict_span(K_star_W_star, K.dual()) - sage: j1 == j2 + 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: P.is_proper() True - sage: j1 == K + sage: P = restrict_span(K_S, K_S.dual()) + sage: P.is_proper() + True + + :: + + 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: P.is_proper() + True + sage: P = restrict_span(K_S, K_S.dual()) + sage: P.is_proper() + True + + :: + + 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: P.is_proper() + True + sage: P = restrict_span(K_S, K_S.dual()) + sage: P.is_proper() True - sage: K; [ list(r) for r in K.rays() ] Test the proposition in our paper concerning the duals, where the subspace `W` is the span of `K^{*}`:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid=False, strictly_convex=False) - sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) - sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual() - sage: K_W.nrays() == K_star_W_star.nrays() - True - sage: K_W.dim() == K_star_W_star.dim() + 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: basically_the_same(K_W, K_star_W_star) True - sage: lineality(K_W) == lineality(K_star_W_star) + + :: + + 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: basically_the_same(K_W, K_star_W_star) True - sage: K_W.is_solid() == K_star_W_star.is_solid() + + :: + + 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: basically_the_same(K_W, K_star_W_star) True - sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex() + + :: + + 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: basically_the_same(K_W, K_star_W_star) True """ @@ -237,9 +301,11 @@ def restrict_span(K, K2=None): ray_pairs = [ phi(r) for r in K.rays() ] - if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]): - msg = 'Cone has nonzero components in W-perp!' - raise ValueError(msg) + # 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) # Represent the cone in terms of a basis for W, i.e. with smaller # vectors. @@ -304,7 +370,7 @@ def lineality(K): dimension of the ambient space, inclusive:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) + sage: K = random_cone(max_dim = 8) sage: l = lineality(K) sage: l in ZZ True @@ -314,7 +380,7 @@ def lineality(K): A strictly convex cone should have lineality zero:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, strictly_convex = True) + sage: K = random_cone(max_dim = 8, strictly_convex = True) sage: lineality(K) 0 @@ -368,14 +434,20 @@ def codim(K): equal to the dimension of the ambient space:: sage: K = Cone([], lattice=ToricLattice(0)) + sage: K.lattice_dim() + 0 sage: codim(K) 0 sage: K = Cone([(0,)]) + sage: K.lattice_dim() + 1 sage: codim(K) 1 sage: K = Cone([(0,0)]) + sage: K.lattice_dim() + 2 sage: codim(K) 2 @@ -385,7 +457,7 @@ def codim(K): the dimension of the ambient space, inclusive:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) + sage: K = random_cone(max_dim = 8) sage: c = codim(K) sage: c in ZZ True @@ -395,14 +467,14 @@ def codim(K): A solid cone should have codimension zero:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid = True) + sage: K = random_cone(max_dim = 8, solid = True) sage: codim(K) 0 The codimension of a cone is equal to the lineality of its dual:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid = True) + sage: K = random_cone(max_dim = 8, solid = True) sage: codim(K) == lineality(K.dual()) True @@ -543,6 +615,14 @@ def LL(K): [0 0 1] ] + If our cone is the entire space, then every transformation on it is + Lyapunov-like:: + + sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) + sage: M = MatrixSpace(QQ,2) + sage: M.basis() == LL(K) + True + TESTS: The inner product `\left< L\left(x\right), s \right>` is zero for @@ -550,7 +630,7 @@ def LL(K): of the cone:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, max_rays=10) + sage: K = random_cone(max_dim=8) sage: C_of_K = discrete_complementarity_set(K) sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] sage: sum(map(abs, l)) @@ -562,7 +642,7 @@ def LL(K): \right)` sage: set_random_seed() - sage: K = random_cone(max_dim=8, max_rays=10) + sage: K = random_cone(max_dim=8) sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] @@ -748,8 +828,8 @@ def lyapunov_rank(K): [Rudolf et al.]_:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True) sage: K = K1.cartesian_product(K2) sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True @@ -758,14 +838,35 @@ def lyapunov_rank(K): itself [Rudolf et al.]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, max_rays=10) + sage: K = random_cone(max_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True Make sure we exercise the non-strictly-convex/non-solid case:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False) + sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True + + Let's check the other permutations as well, just to be sure:: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True + + :: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True + + :: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -776,7 +877,7 @@ def lyapunov_rank(K): the Lyapunov rank of the trivial cone will be zero:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: (n == 0 or 1 <= b) and b <= n @@ -788,7 +889,7 @@ def lyapunov_rank(K): Lyapunov rank `n-1` in `n` dimensions:: sage: set_random_seed() - sage: K = random_cone(max_dim=10) + sage: K = random_cone(max_dim=8) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: b == n-1 @@ -798,7 +899,7 @@ def lyapunov_rank(K): reduced to that of a proper cone [Orlitzky/Gowda]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=10) + 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() @@ -811,7 +912,29 @@ def lyapunov_rank(K): The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: lyapunov_rank(K) == len(LL(K)) + True + + In fact the same can be said of any cone. These additional tests + just increase our confidence that the reduction scheme works:: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: lyapunov_rank(K) == len(LL(K)) + True + + :: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: lyapunov_rank(K) == len(LL(K)) + True + + :: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) sage: lyapunov_rank(K) == len(LL(K)) True @@ -834,14 +957,8 @@ def lyapunov_rank(K): # 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.dual()).dual() - - #Ks = [ list(r) for r in sorted(K.rays()) ] - #Js = [ list(r) for r in sorted(J.rays()) ] - - #if Ks != Js: - # print [ list(r) for r in K_orig.rays() ] + #K = restrict_span(intersect_span(K,K.dual()), K.dual()) + K = restrict_span(K, K.dual()) # Lemma 3 beta += m * l -- 2.44.2 From c4fdc3d232c1c6c179f7919ebb4fd169017edafd Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 7 Jun 2015 00:42:39 -0400 Subject: [PATCH 04/16] Add back the drop_dependent() function. It turns out that it doesn't matter if our basis is orthogonal, so we don't need to do Gram-Schmidt. Since this will be relied upon in the paper, we go back to using (a subset of) the rays of the cone as our basis. --- mjo/cone/cone.py | 22 +++++++++++++++++++++- 1 file changed, 21 insertions(+), 1 deletion(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 87cdf70..8adc51c 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,6 +8,26 @@ addsitedir(abspath('../../')) from sage.all import * +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 + + return result + + def basically_the_same(K1,K2): r""" ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` @@ -56,7 +76,7 @@ def iso_space(K): # 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,_ = M.gram_schmidt() + W_basis = drop_dependent(K.rays()) W = V.subspace_with_basis(W_basis) W_perp = W.complement() -- 2.44.2 From 4418c497a443fb1f5cb068ced5a2ddd5a9a0ad05 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 18:25:00 -0400 Subject: [PATCH 05/16] Remove unused codim() function. --- mjo/cone/cone.py | 94 ------------------------------------------------ 1 file changed, 94 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 8adc51c..f5371d6 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -408,100 +408,6 @@ def lineality(K): return K.linear_subspace().dimension() -def codim(K): - r""" - Compute the codimension of this cone. - - The codimension of a cone is the dimension of the space of all - elements perpendicular to every element of the cone. In other words, - the codimension is the difference between the dimension of the - ambient space and the dimension of the cone itself. - - OUTPUT: - - A nonnegative integer representing the dimension of the space of all - elements perpendicular to this cone. - - .. seealso:: - - :meth:`dim`, :meth:`lattice_dim` - - EXAMPLES: - - The codimension of the nonnegative orthant is zero, since the span of - its generators equals the entire ambient space:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: codim(K) - 0 - - However, if we remove a ray so that the entire cone is contained - within the `x-y`-plane, then the resulting cone will have - codimension one, because the `z`-axis is perpendicular to every - element of the cone:: - - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: codim(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: codim(K) - 0 - - And if the cone is trivial in any space, then its codimension is - equal to the dimension of the ambient space:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: K.lattice_dim() - 0 - sage: codim(K) - 0 - - sage: K = Cone([(0,)]) - sage: K.lattice_dim() - 1 - sage: codim(K) - 1 - - sage: K = Cone([(0,0)]) - sage: K.lattice_dim() - 2 - sage: codim(K) - 2 - - TESTS: - - The codimension of a cone should be an integer between zero and - the dimension of the ambient space, inclusive:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: c = codim(K) - sage: c in ZZ - True - sage: (0 <= c) and (c <= K.lattice_dim()) - True - - A solid cone should have codimension zero:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid = True) - sage: codim(K) - 0 - - The codimension of a cone is equal to the lineality of its dual:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid = True) - sage: codim(K) == lineality(K.dual()) - True - - """ - return (K.lattice_dim() - K.dim()) - - def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. -- 2.44.2 From a96fd3b734f5819bfa8f408f0a682fec286a380c Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 18:27:07 -0400 Subject: [PATCH 06/16] Remove unused unrestrict_span() function. --- mjo/cone/cone.py | 23 +---------------------- 1 file changed, 1 insertion(+), 22 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index f5371d6..61914fa 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -126,28 +126,7 @@ def ips_iso(K): return (phi,phi_inv) - -def unrestrict_span(K, K2=None): - if K2 is None: - K2 = K - - _,phi_inv = ips_iso(K2) - V_iso = iso_space(K2) - (W, W_perp) = V_iso.cartesian_factors() - - rays = [] - for r in K.rays(): - w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ]) - pair = V_iso( (w, W_perp.zero()) ) - rays.append( phi_inv(pair) ) - - L = ToricLattice(W.dimension() + W_perp.dimension()) - - return Cone(rays, lattice=L) - - - -def restrict_span(K, K2=None): +def rho(K, K2=None): r""" Restrict ``K`` into its own span, or the span of another cone. -- 2.44.2 From a2ad3efc39da8dbcc497c0ac861e1df200c6de5e Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 18:27:33 -0400 Subject: [PATCH 07/16] Aaaand the test that was using that "unused" function. --- mjo/cone/cone.py | 8 -------- 1 file changed, 8 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 61914fa..2d84337 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -174,14 +174,6 @@ def rho(K, K2=None): sage: K_S.lattice_dim() == K.dual().dim() True - This function has ``unrestrict_span()`` as its inverse:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True) - sage: J = restrict_span(K) - sage: K == unrestrict_span(J,K) - True - This function should not affect the dimension of a cone:: sage: set_random_seed() -- 2.44.2 From 6bd30534d5aa984c73f511121efa8fda4386c51a Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 19:13:31 -0400 Subject: [PATCH 08/16] 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 -- 2.44.2 From e041595c10751828f196db2cda86bd0f15a81191 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 19:13:40 -0400 Subject: [PATCH 09/16] Revert "Remove unused codim() function." This reverts commit 4418c497a443fb1f5cb068ced5a2ddd5a9a0ad05. --- mjo/cone/cone.py | 94 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 94 insertions(+) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index b9e930e..ba5f51e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -313,6 +313,100 @@ def lineality(K): return K.linear_subspace().dimension() +def codim(K): + r""" + Compute the codimension of this cone. + + The codimension of a cone is the dimension of the space of all + elements perpendicular to every element of the cone. In other words, + the codimension is the difference between the dimension of the + ambient space and the dimension of the cone itself. + + OUTPUT: + + A nonnegative integer representing the dimension of the space of all + elements perpendicular to this cone. + + .. seealso:: + + :meth:`dim`, :meth:`lattice_dim` + + EXAMPLES: + + The codimension of the nonnegative orthant is zero, since the span of + its generators equals the entire ambient space:: + + sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) + sage: codim(K) + 0 + + However, if we remove a ray so that the entire cone is contained + within the `x-y`-plane, then the resulting cone will have + codimension one, because the `z`-axis is perpendicular to every + element of the cone:: + + sage: K = Cone([(1,0,0), (0,1,0)]) + sage: codim(K) + 1 + + If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: + + sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) + sage: codim(K) + 0 + + And if the cone is trivial in any space, then its codimension is + equal to the dimension of the ambient space:: + + sage: K = Cone([], lattice=ToricLattice(0)) + sage: K.lattice_dim() + 0 + sage: codim(K) + 0 + + sage: K = Cone([(0,)]) + sage: K.lattice_dim() + 1 + sage: codim(K) + 1 + + sage: K = Cone([(0,0)]) + sage: K.lattice_dim() + 2 + sage: codim(K) + 2 + + TESTS: + + The codimension of a cone should be an integer between zero and + the dimension of the ambient space, inclusive:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 8) + sage: c = codim(K) + sage: c in ZZ + True + sage: (0 <= c) and (c <= K.lattice_dim()) + True + + A solid cone should have codimension zero:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 8, solid = True) + sage: codim(K) + 0 + + The codimension of a cone is equal to the lineality of its dual:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 8, solid = True) + sage: codim(K) == lineality(K.dual()) + True + + """ + return (K.lattice_dim() - K.dim()) + + def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. -- 2.44.2 From 2e8613b1e875a5a2eee6688cdaa9a39923a0eb07 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 21:39:12 -0400 Subject: [PATCH 10/16] Finish test cleanup, notation updates, and dead code removal. --- mjo/cone/cone.py | 116 ++++++++++++++++++++++------------------------- 1 file changed, 53 insertions(+), 63 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index ba5f51e..c6d2682 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,23 +8,12 @@ addsitedir(abspath('../../')) from sage.all import * -def drop_dependent(vs): - r""" - Return the largest linearly-independent subset of ``vs``. - """ - result = [] - m = matrix(vs).echelon_form() - for idx in range(0, m.nrows()): - if not m[idx].is_zero(): - result.append(m[idx]) - - return result - - def basically_the_same(K1,K2): r""" ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. + otherwise. This is intended as a lazy way to check whether or not + ``K1`` and ``K2`` are linearly isomorphic (i.e. ``A(K1) == K2`` for + some invertible linear transformation ``A``). """ if K1.lattice_dim() != K2.lattice_dim(): return False @@ -65,7 +54,8 @@ def rho(K, K2=None): INPUT: - - ``K2`` -- another cone whose lattice has the same rank as this cone. + - ``K2`` -- another cone whose lattice has the same rank as this + cone. OUTPUT: @@ -136,11 +126,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False) sage: K_S = rho(K) - sage: P = rho(K_S.dual()).dual() - sage: P.is_proper() + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = rho(K_S, K_S.dual()) - sage: P.is_proper() + sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP.is_proper() True :: @@ -148,11 +138,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) sage: K_S = rho(K) - sage: P = rho(K_S.dual()).dual() - sage: P.is_proper() + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = rho(K_S, K_S.dual()) - sage: P.is_proper() + sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP.is_proper() True :: @@ -160,11 +150,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) sage: K_S = rho(K) - sage: P = rho(K_S.dual()).dual() - sage: P.is_proper() + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = rho(K_S, K_S.dual()) - sage: P.is_proper() + sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP.is_proper() True :: @@ -172,47 +162,52 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) sage: K_S = rho(K) - sage: P = rho(K_S.dual()).dual() - sage: P.is_proper() + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = rho(K_S, K_S.dual()) - sage: P.is_proper() + sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP.is_proper() True - Test the proposition in our paper concerning the duals, where the - subspace `W` is the span of `K^{*}`:: + Test the proposition in our paper concerning the duals and + restrictions. Generate a random cone, then create a subcone of + it. The operation of dual-taking should then commute with rho:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() + sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W = rho(K, J) + sage: K_star_W_star = rho(K.dual(), J).dual() sage: basically_the_same(K_W, K_star_W_star) True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() + sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W = rho(K, J) + sage: K_star_W_star = rho(K.dual(), J).dual() sage: basically_the_same(K_W, K_star_W_star) True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() + sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W = rho(K, J) + sage: K_star_W_star = rho(K.dual(), J).dual() sage: basically_the_same(K_W, K_star_W_star) True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() + sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W = rho(K, J) + sage: K_star_W_star = rho(K.dual(), J).dual() sage: basically_the_same(K_W, K_star_W_star) True @@ -220,18 +215,16 @@ def rho(K, K2=None): if K2 is None: K2 = K - # 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. + # First we project K onto the span of K2. This will explode if the + # rank of ``K2.lattice()`` doesn't match ours. span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice()) K = K.intersection(span_K2) - V = K.lattice().vector_space() - - # 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) + # Cheat a little to get the subspace span(K2). The paper uses the + # rays of K2 as a basis, but everything is invariant under linear + # isomorphism (i.e. a change of basis), and this is a little + # faster. + W = span_K2.linear_subspace() # We've already intersected K with the span of K2, so every # generator of K should belong to W now. @@ -420,9 +413,7 @@ def discrete_complementarity_set(K): A list of pairs `(x,s)` such that, - * `x` is in this cone. * `x` is a generator of this cone. - * `s` is in this cone's dual. * `s` is a generator of this cone's dual. * `x` and `s` are orthogonal. @@ -827,10 +818,10 @@ def lyapunov_rank(K): sage: K = random_cone(max_dim=8) sage: actual = lyapunov_rank(K) sage: K_S = rho(K) - sage: P = rho(K_S.dual()).dual() + sage: K_SP = rho(K_S.dual()).dual() sage: l = lineality(K) sage: c = codim(K) - sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 + sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 sage: actual == expected True @@ -864,7 +855,6 @@ def lyapunov_rank(K): True """ - K_orig = K beta = 0 m = K.dim() @@ -872,16 +862,16 @@ def lyapunov_rank(K): l = lineality(K) if m < n: - # K is not solid, project onto its span. + # K is not solid, restrict to its span. K = rho(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 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 = rho(K.dual()).dual() + # K is not pointed, restrict to the span of its dual. Uses a + # proposition from our paper, i.e. this is equivalent to K = + # rho(K.dual()).dual(). K = rho(K, K.dual()) # Lemma 3 -- 2.44.2 From 7c71dbc3454b5211269b879462f3530d76ad6991 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 14 Jun 2015 12:25:55 -0400 Subject: [PATCH 11/16] Remove the lineality() and codim() functions (now depends on branch u/mjo/ticket/18701). --- mjo/cone/cone.py | 181 +++-------------------------------------------- 1 file changed, 8 insertions(+), 173 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index c6d2682..f4b2244 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -24,7 +24,7 @@ def basically_the_same(K1,K2): if K1.dim() != K2.dim(): return False - if lineality(K1) != lineality(K2): + if K1.lineality() != K2.lineality(): return False if K1.is_solid() != K2.is_solid(): @@ -108,7 +108,7 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: lineality(K) == lineality(rho(K)) + sage: K.lineality() == rho(K).lineality() True No matter which space we restrict to, the lineality should not @@ -116,9 +116,9 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: lineality(K) >= lineality(rho(K)) + sage: K.lineality() >= rho(K).lineality() True - sage: lineality(K) >= lineality(rho(K, K.dual())) + sage: K.lineality() >= rho(K, K.dual()).lineality() True If we do this according to our paper, then the result is proper:: @@ -235,171 +235,6 @@ def rho(K, K2=None): -def lineality(K): - r""" - Compute the lineality of this cone. - - The lineality of a cone is the dimension of the largest linear - subspace contained in that cone. - - OUTPUT: - - A nonnegative integer; the dimension of the largest subspace - contained within this cone. - - REFERENCES: - - .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton - University Press, Princeton, 1970. - - EXAMPLES: - - The lineality of the nonnegative orthant is zero, since it clearly - contains no lines:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 0 - - However, if we add another ray so that the entire `x`-axis belongs - to the cone, then the resulting cone will have lineality one:: - - sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal - to the dimension of the ambient space (i.e. two):: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: lineality(K) - 2 - - Per the definition, the lineality of the trivial cone in a trivial - space is zero:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: lineality(K) - 0 - - TESTS: - - The lineality of a cone should be an integer between zero and the - dimension of the ambient space, inclusive:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: l = lineality(K) - sage: l in ZZ - True - sage: (0 <= l) and (l <= K.lattice_dim()) - True - - A strictly convex cone should have lineality zero:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex = True) - sage: lineality(K) - 0 - - """ - return K.linear_subspace().dimension() - - -def codim(K): - r""" - Compute the codimension of this cone. - - The codimension of a cone is the dimension of the space of all - elements perpendicular to every element of the cone. In other words, - the codimension is the difference between the dimension of the - ambient space and the dimension of the cone itself. - - OUTPUT: - - A nonnegative integer representing the dimension of the space of all - elements perpendicular to this cone. - - .. seealso:: - - :meth:`dim`, :meth:`lattice_dim` - - EXAMPLES: - - The codimension of the nonnegative orthant is zero, since the span of - its generators equals the entire ambient space:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: codim(K) - 0 - - However, if we remove a ray so that the entire cone is contained - within the `x-y`-plane, then the resulting cone will have - codimension one, because the `z`-axis is perpendicular to every - element of the cone:: - - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: codim(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: codim(K) - 0 - - And if the cone is trivial in any space, then its codimension is - equal to the dimension of the ambient space:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: K.lattice_dim() - 0 - sage: codim(K) - 0 - - sage: K = Cone([(0,)]) - sage: K.lattice_dim() - 1 - sage: codim(K) - 1 - - sage: K = Cone([(0,0)]) - sage: K.lattice_dim() - 2 - sage: codim(K) - 2 - - TESTS: - - The codimension of a cone should be an integer between zero and - the dimension of the ambient space, inclusive:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: c = codim(K) - sage: c in ZZ - True - sage: (0 <= c) and (c <= K.lattice_dim()) - True - - A solid cone should have codimension zero:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid = True) - sage: codim(K) - 0 - - The codimension of a cone is equal to the lineality of its dual:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid = True) - sage: codim(K) == lineality(K.dual()) - True - - """ - return (K.lattice_dim() - K.dim()) - - def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. @@ -711,7 +546,7 @@ def lyapunov_rank(K): sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z]) sage: lyapunov_rank(K) 19 - sage: K.lattice_dim()**2 - K.dim()*codim(K) + sage: K.lattice_dim()**2 - K.dim()*K.codim() 19 The Lyapunov rank should be additive on a product of proper cones @@ -819,8 +654,8 @@ def lyapunov_rank(K): sage: actual = lyapunov_rank(K) sage: K_S = rho(K) sage: K_SP = rho(K_S.dual()).dual() - sage: l = lineality(K) - sage: c = codim(K) + sage: l = K.lineality() + sage: c = K.codim() sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 sage: actual == expected True @@ -859,7 +694,7 @@ def lyapunov_rank(K): m = K.dim() n = K.lattice_dim() - l = lineality(K) + l = K.lineality() if m < n: # K is not solid, restrict to its span. -- 2.44.2 From d27a3f72ac6b898534615de9111ace4082a4c55e Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 15 Jun 2015 13:29:00 -0400 Subject: [PATCH 12/16] Add a new test for a theorem in the improper paper. --- mjo/cone/cone.py | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index f4b2244..e9d0f1e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -689,6 +689,15 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == len(LL(K)) True + Test Theorem 3 in [Orlitzky/Gowda]_:: + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: L = ToricLattice(K.lattice_dim() + 1) + sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) + sage: lyapunov_rank(K) >= K.lattice_dim() + True + """ beta = 0 -- 2.44.2 From c687f64bed580a9f3be4b98b5188a42caaf27eba Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 15 Jun 2015 14:54:10 -0400 Subject: [PATCH 13/16] Make basically_the_same() and rho() functions private. Add examples/tests for basically_the_same(). Add linear isomorphism test for lyapunov_rank(). --- mjo/cone/cone.py | 174 +++++++++++++++++++++++++++++++++++------------ 1 file changed, 129 insertions(+), 45 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index e9d0f1e..e2c43d8 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,12 +8,55 @@ addsitedir(abspath('../../')) from sage.all import * -def basically_the_same(K1,K2): +def _basically_the_same(K1, K2): r""" + Test whether or not ``K1`` and ``K2`` are "basically the same." + + This is a hack to get around the fact that it's difficult to tell + when two cones are linearly isomorphic. We have a proposition that + equates two cones, but represented over `\mathbb{Q}`, they are + merely linearly isomorphic (not equal). So rather than test for + equality, we test a list of properties that should be preserved + under an invertible linear transformation. + + OUTPUT: + ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. This is intended as a lazy way to check whether or not - ``K1`` and ``K2`` are linearly isomorphic (i.e. ``A(K1) == K2`` for - some invertible linear transformation ``A``). + otherwise. + + EXAMPLES: + + Any proper cone with three generators in `\mathbb{R}^{3}` is + basically the same as the nonnegative orthant:: + + sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)]) + sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)]) + sage: _basically_the_same(K1, K2) + True + + Negating a cone gives you another cone that is basically the same:: + + sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)]) + sage: _basically_the_same(K, -K) + True + + TESTS: + + Any cone is basically the same as itself:: + + sage: K = random_cone(max_dim = 8) + sage: _basically_the_same(K, K) + True + + After applying an invertible matrix to the rows of a cone, the + result should be basically the same as the cone we started with:: + + sage: K1 = random_cone(max_dim = 8) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: _basically_the_same(K1, K2) + True + """ if K1.lattice_dim() != K2.lattice_dim(): return False @@ -48,7 +91,7 @@ def basically_the_same(K1,K2): -def rho(K, K2=None): +def _rho(K, K2=None): r""" Restrict ``K`` into its own span, or the span of another cone. @@ -64,18 +107,18 @@ def rho(K, K2=None): EXAMPLES:: sage: K = Cone([(1,)]) - sage: rho(K) == K + sage: _rho(K) == K True sage: K2 = Cone([(1,0)]) - sage: rho(K2).rays() + sage: _rho(K2).rays() N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: rho(K3).rays() + sage: _rho(K3).rays() N(1) in 1-d lattice N - sage: rho(K2) == rho(K3) + sage: _rho(K2) == _rho(K3) True TESTS: @@ -84,7 +127,7 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K_S = rho(K) + sage: K_S = _rho(K) sage: K_S.is_solid() True @@ -93,7 +136,7 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K_S = rho(K, K.dual() ) + sage: K_S = _rho(K, K.dual() ) sage: K_S.lattice_dim() == K.dual().dim() True @@ -101,14 +144,14 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K.dim() == rho(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: K.lineality() == rho(K).lineality() + sage: K.lineality() == _rho(K).lineality() True No matter which space we restrict to, the lineality should not @@ -116,20 +159,20 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K.lineality() >= rho(K).lineality() + sage: K.lineality() >= _rho(K).lineality() True - sage: K.lineality() >= rho(K, K.dual()).lineality() + sage: K.lineality() >= _rho(K, K.dual()).lineality() 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 = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True @@ -137,11 +180,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True @@ -149,11 +192,11 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True @@ -161,24 +204,24 @@ def rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True - Test the proposition in our paper concerning the duals and + Test Proposition 7 in our paper concerning the duals and restrictions. Generate a random cone, then create a subcone of it. The operation of dual-taking should then commute with rho:: sage: set_random_seed() sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True :: @@ -186,9 +229,9 @@ def rho(K, K2=None): sage: set_random_seed() sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True :: @@ -196,9 +239,9 @@ def rho(K, K2=None): sage: set_random_seed() sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True :: @@ -206,9 +249,9 @@ def rho(K, K2=None): sage: set_random_seed() sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True """ @@ -585,6 +628,47 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True + The Lyapunov rank is invariant under a linear isomorphism + [Orlitzky/Gowda]_:: + + sage: K1 = random_cone(max_dim = 8) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + Just to be sure, test a few more:: + + sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + :: + + sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + :: + + sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + :: + + sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` itself [Rudolf et al.]_:: @@ -652,8 +736,8 @@ def lyapunov_rank(K): sage: set_random_seed() sage: K = random_cone(max_dim=8) sage: actual = lyapunov_rank(K) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: l = K.lineality() sage: c = K.codim() sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 @@ -707,7 +791,7 @@ def lyapunov_rank(K): if m < n: # K is not solid, restrict to its span. - K = rho(K) + K = _rho(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 @@ -715,8 +799,8 @@ def lyapunov_rank(K): if l > 0: # K is not pointed, restrict to the span of its dual. Uses a # proposition from our paper, i.e. this is equivalent to K = - # rho(K.dual()).dual(). - K = rho(K, K.dual()) + # _rho(K.dual()).dual(). + K = _rho(K, K.dual()) # Lemma 3 beta += m * l -- 2.44.2 From f463f748186b5d9808cc93a37a86dc41f901bea1 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 27 Jul 2015 10:33:55 -0400 Subject: [PATCH 14/16] mjo/interpolation.py: fix typo. --- mjo/interpolation.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/mjo/interpolation.py b/mjo/interpolation.py index 5c8fa28..e32ed6d 100644 --- a/mjo/interpolation.py +++ b/mjo/interpolation.py @@ -60,7 +60,7 @@ def lagrange_coefficient(k, x, xs): def lagrange_polynomial(x, xs, ys): """ - Return the Lagrange form of the interpolation polynomial in `x` of + Return the Lagrange form of the interpolating polynomial in `x` at the points (xs[k], ys[k]). INPUT: -- 2.44.2 From cbd0e9a0ed9133de08665a55f146d04cb9536ec6 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 27 Jul 2015 10:36:07 -0400 Subject: [PATCH 15/16] Update docs for cone.discrete_complementarity_set(). --- mjo/cone/cone.py | 40 ++++++++++++++++++++++++++++++++++------ 1 file changed, 34 insertions(+), 6 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index e2c43d8..baff1a7 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -282,19 +282,29 @@ def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. - The complementarity set of this cone is the set of all orthogonal - pairs `(x,s)` such that `x` is in this cone, and `s` is in its - dual. The discrete complementarity set restricts `x` and `s` to be - generators of their respective cones. + The complementarity set of a cone is the set of all orthogonal pairs + `(x,s)` such that `x` is in the cone, and `s` is in its dual. The + discrete complementarity set is a subset of the complementarity set + where `x` and `s` are required to be generators of their respective + cones. + + For polyhedral cones, the discrete complementarity set is always + finite. OUTPUT: A list of pairs `(x,s)` such that, + * Both `x` and `s` are vectors (not rays). * `x` is a generator of this cone. * `s` is a generator of this cone's dual. * `x` and `s` are orthogonal. + REFERENCES: + + .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an + Improper Cone. Work in-progress. + EXAMPLES: The discrete complementarity set of the nonnegative orthant consists @@ -325,6 +335,13 @@ def discrete_complementarity_set(K): sage: discrete_complementarity_set(K) [] + Likewise when this cone is trivial (its dual is the entire space):: + + sage: L = ToricLattice(0) + sage: K = Cone([], ToricLattice(0)) + sage: discrete_complementarity_set(K) + [] + TESTS: The complementarity set of the dual can be obtained by switching the @@ -338,12 +355,23 @@ def discrete_complementarity_set(K): sage: sorted(actual) == sorted(expected) True + The pairs in the discrete complementarity set are in fact + complementary:: + + sage: set_random_seed() + sage: K = random_cone(max_dim=6) + sage: dcs = discrete_complementarity_set(K) + sage: sum([x.inner_product(s).abs() for (x,s) in dcs]) + 0 + """ V = K.lattice().vector_space() - # Convert the rays to vectors so that we can compute inner - # products. + # Convert rays to vectors so that we can compute inner products. xs = [V(x) for x in K.rays()] + + # We also convert the generators of the dual cone so that we + # return pairs of vectors and not (vector, ray) pairs. ss = [V(s) for s in K.dual().rays()] return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] -- 2.44.2 From 98637c981445d35a061878923baf3ae4651ecb0b Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Tue, 4 Aug 2015 11:56:25 -0400 Subject: [PATCH 16/16] Update max_ambient_dim parameter name for random_cone(). --- mjo/cone/cone.py | 126 ++++++++++++++++++++++++++++++++--------------- 1 file changed, 87 insertions(+), 39 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index baff1a7..e40579f 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -44,14 +44,14 @@ def _basically_the_same(K1, K2): Any cone is basically the same as itself:: - sage: K = random_cone(max_dim = 8) + sage: K = random_cone(max_ambient_dim = 8) sage: _basically_the_same(K, K) True After applying an invertible matrix to the rows of a cone, the result should be basically the same as the cone we started with:: - sage: K1 = random_cone(max_dim = 8) + sage: K1 = random_cone(max_ambient_dim = 8) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: _basically_the_same(K1, K2) @@ -126,7 +126,7 @@ def _rho(K, K2=None): The projected cone should always be solid:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) + sage: K = random_cone(max_ambient_dim = 8) sage: K_S = _rho(K) sage: K_S.is_solid() True @@ -135,7 +135,7 @@ def _rho(K, K2=None): dimension as the space we restricted it to:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) + sage: K = random_cone(max_ambient_dim = 8) sage: K_S = _rho(K, K.dual() ) sage: K_S.lattice_dim() == K.dual().dim() True @@ -143,14 +143,14 @@ def _rho(K, K2=None): This function should not affect the dimension of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) + sage: K = random_cone(max_ambient_dim = 8) 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: K = random_cone(max_ambient_dim = 8) sage: K.lineality() == _rho(K).lineality() True @@ -158,7 +158,7 @@ def _rho(K, K2=None): increase:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) + sage: K = random_cone(max_ambient_dim = 8) sage: K.lineality() >= _rho(K).lineality() True sage: K.lineality() >= _rho(K, K.dual()).lineality() @@ -167,7 +167,9 @@ def _rho(K, K2=None): 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 = random_cone(max_ambient_dim = 8, + ....: strictly_convex=False, + ....: solid=False) sage: K_S = _rho(K) sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() @@ -179,7 +181,9 @@ def _rho(K, K2=None): :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=True, + ....: solid=False) sage: K_S = _rho(K) sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() @@ -191,7 +195,9 @@ def _rho(K, K2=None): :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=False, + ....: solid=True) sage: K_S = _rho(K) sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() @@ -203,7 +209,9 @@ def _rho(K, K2=None): :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=True, + ....: solid=True) sage: K_S = _rho(K) sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() @@ -217,7 +225,9 @@ def _rho(K, K2=None): it. The operation of dual-taking should then commute with rho:: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=False, + ....: strictly_convex=False) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) sage: K_W_star = _rho(K, J).dual() sage: K_star_W = _rho(K.dual(), J) @@ -227,7 +237,9 @@ def _rho(K, K2=None): :: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=True, + ....: strictly_convex=False) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) sage: K_W_star = _rho(K, J).dual() sage: K_star_W = _rho(K.dual(), J) @@ -237,7 +249,9 @@ def _rho(K, K2=None): :: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=False, + ....: strictly_convex=True) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) sage: K_W_star = _rho(K, J).dual() sage: K_star_W = _rho(K.dual(), J) @@ -247,7 +261,9 @@ def _rho(K, K2=None): :: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=True, + ....: strictly_convex=True) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) sage: K_W_star = _rho(K, J).dual() sage: K_star_W = _rho(K.dual(), J) @@ -348,7 +364,7 @@ def discrete_complementarity_set(K): components of the complementarity set of the original cone:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=6) + sage: K1 = random_cone(max_ambient_dim=6) sage: K2 = K1.dual() sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] sage: actual = discrete_complementarity_set(K1) @@ -359,7 +375,7 @@ def discrete_complementarity_set(K): complementary:: sage: set_random_seed() - sage: K = random_cone(max_dim=6) + sage: K = random_cone(max_ambient_dim=6) sage: dcs = discrete_complementarity_set(K) sage: sum([x.inner_product(s).abs() for (x,s) in dcs]) 0 @@ -452,7 +468,7 @@ def LL(K): of the cone:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: C_of_K = discrete_complementarity_set(K) sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] sage: sum(map(abs, l)) @@ -464,7 +480,7 @@ def LL(K): \right)` sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] @@ -650,8 +666,12 @@ def lyapunov_rank(K): [Rudolf et al.]_:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: K = K1.cartesian_product(K2) sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True @@ -659,7 +679,7 @@ def lyapunov_rank(K): The Lyapunov rank is invariant under a linear isomorphism [Orlitzky/Gowda]_:: - sage: K1 = random_cone(max_dim = 8) + sage: K1 = random_cone(max_ambient_dim = 8) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: lyapunov_rank(K1) == lyapunov_rank(K2) @@ -667,7 +687,9 @@ def lyapunov_rank(K): Just to be sure, test a few more:: - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: lyapunov_rank(K1) == lyapunov_rank(K2) @@ -675,7 +697,9 @@ def lyapunov_rank(K): :: - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=False) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: lyapunov_rank(K1) == lyapunov_rank(K2) @@ -683,7 +707,9 @@ def lyapunov_rank(K): :: - sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=True) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: lyapunov_rank(K1) == lyapunov_rank(K2) @@ -691,7 +717,9 @@ def lyapunov_rank(K): :: - sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=False) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: lyapunov_rank(K1) == lyapunov_rank(K2) @@ -701,35 +729,43 @@ def lyapunov_rank(K): itself [Rudolf et al.]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True Make sure we exercise the non-strictly-convex/non-solid case:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=False) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True Let's check the other permutations as well, just to be sure:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=True) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=False) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -740,7 +776,9 @@ def lyapunov_rank(K): the Lyapunov rank of the trivial cone will be zero:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: (n == 0 or 1 <= b) and b <= n @@ -752,7 +790,7 @@ def lyapunov_rank(K): Lyapunov rank `n-1` in `n` dimensions:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: b == n-1 @@ -762,7 +800,7 @@ def lyapunov_rank(K): reduced to that of a proper cone [Orlitzky/Gowda]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: actual = lyapunov_rank(K) sage: K_S = _rho(K) sage: K_SP = _rho(K_S.dual()).dual() @@ -775,7 +813,9 @@ def lyapunov_rank(K): The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: lyapunov_rank(K) == len(LL(K)) True @@ -783,28 +823,36 @@ def lyapunov_rank(K): just increase our confidence that the reduction scheme works:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=False) sage: lyapunov_rank(K) == len(LL(K)) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=True) sage: lyapunov_rank(K) == len(LL(K)) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=False) sage: lyapunov_rank(K) == len(LL(K)) True Test Theorem 3 in [Orlitzky/Gowda]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: L = ToricLattice(K.lattice_dim() + 1) sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) sage: lyapunov_rank(K) >= K.lattice_dim() -- 2.44.2