From: Michael Orlitzky Date: Mon, 4 Jan 2016 19:13:10 +0000 (-0500) Subject: Remove random_element() for sage and start cleanup on motzkin_decomposition(). X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=090b2c77aa4bd371d66885451f9df44c6b6d818f;p=sage.d.git Remove random_element() for sage and start cleanup on motzkin_decomposition(). --- diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index c9d6a17..a1ded52 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -65,108 +65,82 @@ def is_lyapunov_like(L,K): for (x,s) in K.discrete_complementarity_set()]) -def random_element(K): +def motzkin_decomposition(K): r""" - Return a random element of ``K`` from its ambient vector space. + Return the pair of components in the motzkin decomposition of this cone. - ALGORITHM: + Every convex cone is the direct sum of a strictly convex cone and a + linear subspace. Return a pair ``(P,S)`` of cones such that ``P`` is + strictly convex, ``S`` is a subspace, and ``K`` is the direct sum of + ``P`` and ``S``. - The cone ``K`` is specified in terms of its generators, so that - ``K`` is equal to the convex conic combination of those generators. - To choose a random element of ``K``, we assign random nonnegative - coefficients to each generator of ``K`` and construct a new vector - from the scaled rays. + OUTPUT: - A vector, rather than a ray, is returned so that the element may - have non-integer coordinates. Thus the element may have an - arbitrarily small norm. + An ordered pair ``(P,S)`` of closed convex polyhedral cones where + ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the + direct sum of ``P`` and ``S``. EXAMPLES: - A random element of the trivial cone is zero:: - - sage: set_random_seed() - sage: K = Cone([], ToricLattice(0)) - sage: random_element(K) - () - sage: K = Cone([(0,)]) - sage: random_element(K) - (0) - sage: K = Cone([(0,0)]) - sage: random_element(K) - (0, 0) - sage: K = Cone([(0,0,0)]) - sage: random_element(K) - (0, 0, 0) - - A random element of the nonnegative orthant should have all - components nonnegative:: + The nonnegative orthant is strictly convex, so it is its own + strictly convex component and its subspace component is trivial:: - sage: set_random_seed() sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: all([ x >= 0 for x in random_element(K) ]) + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(P) + True + sage: S.is_trivial() + True + + Likewise, full spaces are their own subspace components:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(S) + True + sage: P.is_trivial() True TESTS: - Any cone should contain a random element of itself:: + A random point in the cone should belong to either the strictly + convex component or the subspace component. If the point is nonzero, + it cannot be in both:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) - sage: K.contains(random_element(K)) + sage: (P,S) = motzkin_decomposition(K) + sage: x = K.random_element() + sage: P.contains(x) or S.contains(x) True - - A strictly convex cone contains no lines, and thus no negative - multiples of any of its elements besides zero:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, strictly_convex=True) - sage: x = random_element(K) - sage: x.is_zero() or not K.contains(-x) + sage: x.is_zero() or (P.contains(x) != S.contains(x)) True - The sum of random elements of a cone lies in the cone:: + The strictly convex component should always be strictly convex, and + the subspace component should always be a subspace:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) - sage: K.contains(sum([random_element(K) for i in range(10)])) + sage: (P,S) = motzkin_decomposition(K) + sage: P.is_strictly_convex() + True + sage: S.lineality() == S.dim() True - """ - V = K.lattice().vector_space() - scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ] - - # Make sure we return a vector. Without the coercion, we might - # return ``0`` when ``K`` has no rays. - return V(sum(scaled_gens)) - - -def motzkin_decomposition(K): - """ - Every convex cone is the direct sum of a pointed cone and a linear - subspace. Return a pair ``(P,S)`` of cones such that ``P`` is - pointed, ``S`` is a subspace, and ``K`` is the direct sum of ``P`` - and ``S``. - - OUTPUT: - - An ordered pair ``(P,S)`` of closed convex polyhedral cones where - ``P`` is pointed, ``S`` is a subspace, and ``K`` is the direct sum - of ``P`` and ``S``. - - TESTS: - - A random point in the cone should belong to either the pointed - subcone ``P`` or the subspace ``S``. If the point is nonzero, it - should lie in one but not both of them:: + The generators of the strictly convex component are obtained from + the orthogonal projections of the original generators onto the + orthogonal complement of the subspace component:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) sage: (P,S) = motzkin_decomposition(K) - sage: x = random_element(K) - sage: P.contains(x) or S.contains(x) - True - sage: x.is_zero() or (P.contains(x) != S.contains(x)) + sage: S_perp = S.linear_subspace().complement() + sage: A = S_perp.matrix().transpose() + sage: proj = A * (A.transpose()*A).inverse() * A.transpose() + sage: expected = Cone([ proj*g for g in K ], K.lattice()) + sage: P.is_equivalent(expected) True """ linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ] diff --git a/mjo/cone/rearrangement.py b/mjo/cone/rearrangement.py index 9dde4ec..0df1daf 100644 --- a/mjo/cone/rearrangement.py +++ b/mjo/cone/rearrangement.py @@ -6,7 +6,6 @@ from site import addsitedir addsitedir(abspath('../../')) from sage.all import * -from mjo.cone.cone import random_element def rearrangement_cone(p,n): r""" @@ -134,7 +133,7 @@ def has_rearrangement_property(v, p): sage: for n in range(2,10): ....: for p in range(1, n-1): ....: K = rearrangement_cone(p,n) - ....: v = random_element(K) + ....: v = K.random_element() ....: if not has_rearrangement_property(v,p): print v """