mjo/cone/schur.py

   1 """
   2 The Schur cone, as described in the "Critical angles..." papers by
   3 Iusem, Seeger, and Sossa. It defines the Schur ordering on `R^{n}`.
   4 """
   5
   6 from sage.all import *
   7
   8 def schur_cone(n):
   9     r"""
  10     Return the Schur cone in ``n`` dimensions.
  11
  12     INPUT:
  13
  14     - ``n`` -- the dimension the ambient space.
  15
  16     OUTPUT:
  17
  18     A rational closed convex Schur cone of dimension ``n``.
  19
  20     REFERENCES:
  21
  22     .. [IusemSeegerOnPairs] Alfredo Iusem and Alberto Seeger.
  23        On pairs of vectors achieving the maximal angle of a convex cone.
  24        Mathematical Programming, 104(2-3):501-523, 2005,
  25        doi:10.1007/s10107-005-0626-z.
  26
  27     .. [SeegerSossaI] Alberto Seeger and David Sossa.
  28        Critical angles between two convex cones I. General theory.
  29        TOP, 24(1):44-65, 2016, doi:10.1007/s11750-015-0375-y.
  30
  31     SETUP::
  32
  33         sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
  34         sage: from mjo.cone.schur import schur_cone
  35
  36     EXAMPLES:
  37
  38     Verify the claim that the maximal angle between any two generators
  39     of the Schur cone and the nonnegative quintant is ``3*pi/4``::
  40
  41         sage: P = schur_cone(5)
  42         sage: Q = nonnegative_orthant(5)
  43         sage: G = [ g.change_ring(QQbar).normalized() for g in P ]
  44         sage: H = [ h.change_ring(QQbar).normalized() for h in Q ]
  45         sage: actual = max([arccos(u.inner_product(v)) for u in G for v in H])
  46         sage: expected = 3*pi/4
  47         sage: abs(actual - expected).n() < 1e-12
  48         True
  49
  50     TESTS:
  51
  52     We get the trivial cone when ``n`` is zero::
  53
  54         sage: schur_cone(0).is_trivial()
  55         True
  56
  57     The Schur cone induces the majorization ordering::
  58
  59         sage: set_random_seed()
  60         sage: def majorized_by(x,y):
  61         ....:     return (all(sum(x[0:i]) <= sum(y[0:i])
  62         ....:                 for i in range(x.degree()-1))
  63         ....:             and sum(x) == sum(y))
  64         sage: n = ZZ.random_element(10)
  65         sage: V = VectorSpace(QQ, n)
  66         sage: S = schur_cone(n)
  67         sage: majorized_by(V.zero(), S.random_element())
  68         True
  69         sage: x = V.random_element()
  70         sage: y = V.random_element()
  71         sage: majorized_by(x,y) == ( (y-x) in S )
  72         True
  73
  74     """
  75
  76     def _f(i,j):
  77         if i == j:
  78             return 1
  79         elif j - i == 1:
  80             return -1
  81         else:
  82             return 0
  83
  84     # The "max" below catches the trivial case where n == 0.
  85     S = matrix(ZZ, max(0,n-1), n, _f)
  86
  87     # Likewise, when n == 0, we need to specify the lattice.
  88     return Cone(S.rows(), ToricLattice(n))