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[sage.d.git] / mjo / cone / nonnegative_orthant.py
1 r"""
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, lattice=None):
9 r"""
10 The nonnegative orthant in ``n`` dimensions.
11
12 INPUT:
13
14 - ``n`` -- the dimension of the ambient space.
15
16 - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
17 to be passed to the :func:`Cone` constructor.
18
19 OUTPUT:
20
21 The convex cone having ``n`` standard basis vectors as its
22 generators. Each generating ray will have the integer ring as its
23 base ring.
24
25 If a ``lattice`` was specified, then the resulting cone will live in
26 that lattice unless its rank is incompatible with the dimension
27 ``n`` (in which case a ``ValueError`` is raised).
28
29 SETUP::
30
31 sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
32
33 EXAMPLES::
34
35 sage: nonnegative_orthant(3).rays()
36 N(1, 0, 0),
37 N(0, 1, 0),
38 N(0, 0, 1)
39 in 3-d lattice N
40
41 TESTS:
42
43 We can construct the trivial cone as the nonnegative orthant in a
44 trivial vector space::
45
46 sage: nonnegative_orthant(0)
47 0-d cone in 0-d lattice N
48
49 The nonnegative orthant is a proper cone::
50
51 sage: set_random_seed()
52 sage: n = ZZ.random_element(10)
53 sage: K = nonnegative_orthant(n)
54 sage: K.is_proper()
55 True
56
57 If a ``lattice`` was given, it is actually used::
58
59 sage: L = ToricLattice(3, 'M')
60 sage: nonnegative_orthant(3, lattice=L)
61 3-d cone in 3-d lattice M
62
63 Unless the rank of the lattice disagrees with ``n``::
64
65 sage: L = ToricLattice(1, 'M')
66 sage: nonnegative_orthant(3, lattice=L)
67 Traceback (most recent call last):
68 ...
69 ValueError: lattice rank=1 and dimension n=3 are incompatible
70
71 """
72 if lattice is None:
73 lattice = ToricLattice(n)
74
75 if lattice.rank() != n:
76 raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
77 %
78 (lattice.rank(), n))
79
80 I = identity_matrix(ZZ,n)
81 return Cone(I.rows(), lattice)