mjo/cone/nonnegative_orthant.py

   1 """
   2 The nonnegative orthant in `\mathbb{Z}^{n}`. I'm sick and tired of
   3 typing it.
   4 """
   5
   6 from sage.all import *
   7
   8 def nonnegative_orthant(n):
   9     r"""
  10     The nonnegative orthant in ``n`` dimensions.
  11
  12     INPUT:
  13
  14       - ``n`` -- the dimension of the ambient space.
  15
  16     OUTPUT:
  17
  18     The convex cone having ``n`` standard basis vectors as its
  19     generators. Each generating ray will have the integer ring as its
  20     base ring.
  21
  22     SETUP::
  23
  24         sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
  25
  26     EXAMPLES::
  27
  28         sage: nonnegative_orthant(3).rays()
  29         N(1, 0, 0),
  30         N(0, 1, 0),
  31         N(0, 0, 1)
  32         in 3-d lattice N
  33
  34     TESTS:
  35
  36     We can construct the trivial cone as the nonnegative orthant in a
  37     trivial vector space::
  38
  39         sage: nonnegative_orthant(0)
  40         0-d cone in 0-d lattice N
  41
  42     The nonnegative orthant is a proper cone::
  43
  44         sage: set_random_seed()
  45         sage: n = ZZ.random_element(10)
  46         sage: K = nonnegative_orthant(n)
  47         sage: K.is_proper()
  48         True
  49
  50     """
  51     I = identity_matrix(ZZ,n)
  52     lattice = ToricLattice(n)
  53     return Cone(I.rows(), lattice)