-def random_element(K):
- r"""
- Return a random element of ``K`` from its ambient vector space.
-
- ALGORITHM:
-
- 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.
-
- 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.
-
- 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::
-
- 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) ])
- True
-
- TESTS:
-
- Any cone should contain a random element of itself::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: K.contains(random_element(K))
- 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)
- True
-
- The sum of random elements of a cone lies in the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: K.contains(sum([random_element(K) for i in range(10)]))
- 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 pointed_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::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=8)
- sage: (P,S) = pointed_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))
- True
- """
- linspace_gens = [ copy(b) for b in K.linear_subspace().basis() ]
- linspace_gens += [ -b for b in linspace_gens ]
-
- S = Cone(linspace_gens, K.lattice())
-
- # Since ``S`` is a subspace, its dual is its orthogonal complement
- # (albeit in the wrong lattice).
- S_perp = Cone(S.dual(), K.lattice())
- P = K.intersection(S_perp)
-
- return (P,S)
-