from sage.all import *
-def is_full_space(K):
- r"""
- Return whether or not this cone is equal to its ambient vector space.
-
- OUTPUT:
-
- ``True`` if this cone is the entire vector space and ``False``
- otherwise.
-
- EXAMPLES:
-
- A ray in two dimensions is not equal to the entire space::
-
- sage: K = Cone([(1,0)])
- sage: is_full_space(K)
- False
-
- Neither is the nonnegative orthant::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: is_full_space(K)
- False
-
- The right half-space contains a vector subspace, but it is still not
- equal to the entire plane::
-
- sage: K = Cone([(1,0),(-1,0),(0,1)])
- sage: is_full_space(K)
- False
-
- But if we include nonnegative sums from both axes, then the resulting
- cone is the entire two-dimensional space::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: is_full_space(K)
- True
-
- """
- return K.linear_subspace() == K.lattice().vector_space()
-
-
-def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None):
- r"""
- Generate a random rational convex polyhedral cone.
-
- Lower and upper bounds may be provided for both the dimension of the
- ambient space and the number of generating rays of the cone. If a
- lower bound is left unspecified, it defaults to zero. Unspecified
- upper bounds will be chosen randomly.
-
- The lower bound on the number of rays is limited to twice the
- maximum dimension of the ambient vector space. To see why, consider
- the space $\mathbb{R}^{2}$, and suppose we have generated four rays,
- $\left\{ \pm e_{1}, \pm e_{2} \right\}$. Clearly any other ray in
- the space is a nonnegative linear combination of those four,
- so it is hopeless to generate more. It is therefore an error
- to request more in the form of ``min_rays``.
-
- .. NOTE:
-
- If you do not explicitly request more than ``2 * max_dim`` rays,
- a larger number may still be randomly generated. In that case,
- the returned cone will simply be equal to the entire space.
-
- INPUT:
-
- - ``min_dim`` (default: zero) -- A nonnegative integer representing the
- minimum dimension of the ambient lattice.
-
- - ``max_dim`` (default: random) -- A nonnegative integer representing
- the maximum dimension of the ambient
- lattice.
-
- - ``min_rays`` (default: zero) -- A nonnegative integer representing the
- minimum number of generating rays of the
- cone.
-
- - ``max_rays`` (default: random) -- A nonnegative integer representing the
- maximum number of generating rays of
- the cone.
-
- OUTPUT:
-
- A new, randomly generated cone.
-
- A ``ValueError` will be thrown under the following conditions:
-
- * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays``
- are negative.
-
- * ``max_dim`` is less than ``min_dim``.
-
- * ``max_rays`` is less than ``min_rays``.
-
- * ``min_rays`` is greater than twice ``max_dim``.
-
- EXAMPLES:
-
- If we set the lower/upper bounds to zero, then our result is
- predictable::
-
- sage: random_cone(0,0,0,0)
- 0-d cone in 0-d lattice N
-
- We can predict the dimension when ``min_dim == max_dim``::
-
- sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0)
- 0-d cone in 4-d lattice N
-
- Likewise for the number of rays when ``min_rays == max_rays``::
-
- sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10)
- 10-d cone in 10-d lattice N
-
- TESTS:
-
- It's hard to test the output of a random process, but we can at
- least make sure that we get a cone back::
-
- sage: from sage.geometry.cone import is_Cone # long time
- sage: K = random_cone() # long time
- sage: is_Cone(K) # long time
- True
-
- The upper/lower bounds are respected::
-
- sage: K = random_cone(min_dim=5, max_dim=10, min_rays=3, max_rays=4)
- sage: 5 <= K.lattice_dim() and K.lattice_dim() <= 10
- True
- sage: 3 <= K.nrays() and K.nrays() <= 4
- True
-
- Ensure that an exception is raised when either lower bound is greater
- than its respective upper bound::
-
- sage: random_cone(min_dim=5, max_dim=2)
- Traceback (most recent call last):
- ...
- ValueError: max_dim cannot be less than min_dim.
-
- sage: random_cone(min_rays=5, max_rays=2)
- Traceback (most recent call last):
- ...
- ValueError: max_rays cannot be less than min_rays.
-
- And if we request too many rays::
-
- sage: random_cone(min_rays=5, max_dim=1)
- Traceback (most recent call last):
- ...
- ValueError: min_rays cannot be larger than twice max_dim.
-
- """
-
- # Catch obvious mistakes so that we can generate clear error
- # messages.
-
- if min_dim < 0:
- raise ValueError('min_dim must be nonnegative.')
-
- if min_rays < 0:
- raise ValueError('min_rays must be nonnegative.')
-
- if max_dim is not None:
- if max_dim < 0:
- raise ValueError('max_dim must be nonnegative.')
- if (max_dim < min_dim):
- raise ValueError('max_dim cannot be less than min_dim.')
- if min_rays > 2*max_dim:
- raise ValueError('min_rays cannot be larger than twice max_dim.')
-
- if max_rays is not None:
- if max_rays < 0:
- raise ValueError('max_rays must be nonnegative.')
- if (max_rays < min_rays):
- raise ValueError('max_rays cannot be less than min_rays.')
-
-
- def random_min_max(l,u):
- r"""
- We need to handle two cases for the upper bounds, and we need to do
- the same thing for max_dim/max_rays. So we consolidate the logic here.
- """
- if u is None:
- # The upper bound is unspecified; return a random integer
- # in [l,infinity).
- return l + ZZ.random_element().abs()
- else:
- # We have an upper bound, and it's greater than or equal
- # to our lower bound. So we generate a random integer in
- # [0,u-l], and then add it to l to get something in
- # [l,u]. To understand the "+1", check the
- # ZZ.random_element() docs.
- return l + ZZ.random_element(u - l + 1)
-
- d = random_min_max(min_dim, max_dim)
- r = random_min_max(min_rays, max_rays)
-
- L = ToricLattice(d)
-
- # The rays are trickier to generate, since we could generate v and
- # 2*v as our "two rays." In that case, the resuting cone would
- # have one generating ray. To avoid such a situation, we start by
- # generating ``r`` rays where ``r`` is the number we want to end
- # up with...
- rays = [L.random_element() for i in range(0, r)]
-
- # (The lattice parameter is required when no rays are given, so we
- # pass it just in case ``r == 0``).
- K = Cone(rays, lattice=L)
-
- # Now if we generated two of the "same" rays, we'll have fewer
- # generating rays than ``r``. In that case, we keep making up new
- # rays and recreating the cone until we get the right number of
- # independent generators. We can obviously stop if ``K`` is the
- # entire ambient vector space.
- while r > K.nrays() and not is_full_space(K):
- rays.append(L.random_element())
- K = Cone(rays)
-
- return K
-
-
def discrete_complementarity_set(K):
r"""
Compute the discrete complementarity set of this cone.
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