From 2e8613b1e875a5a2eee6688cdaa9a39923a0eb07 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 21:39:12 -0400 Subject: [PATCH] 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