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, lattice=None):
   9     r"""
  10     Return the Schur cone in ``n`` dimensions.
  11
  12     INPUT:
  13
  14       - ``n`` -- the dimension 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     A rational closed convex Schur cone of dimension ``n``. Each
  22     generating ray will have the integer ring as its base ring.
  23
  24     If a ``lattice`` was specified, then the resulting cone will live in
  25     that lattice unless its rank is incompatible with the dimension
  26     ``n`` (in which case a ``ValueError`` is raised).
  27
  28     REFERENCES:
  29
  30     .. [GourionSeeger] Daniel Gourion and Alberto Seeger.
  31        Critical angles in polyhedral convex cones: numerical and
  32        statistical considerations. Mathematical Programming, 123:173-198,
  33        2010, doi:10.1007/s10107-009-0317-2.
  34
  35     .. [IusemSeegerOnPairs] Alfredo Iusem and Alberto Seeger.
  36        On pairs of vectors achieving the maximal angle of a convex cone.
  37        Mathematical Programming, 104(2-3):501-523, 2005,
  38        doi:10.1007/s10107-005-0626-z.
  39
  40     .. [SeegerSossaI] Alberto Seeger and David Sossa.
  41        Critical angles between two convex cones I. General theory.
  42        TOP, 24(1):44-65, 2016, doi:10.1007/s11750-015-0375-y.
  43
  44     SETUP::
  45
  46         sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
  47         sage: from mjo.cone.schur import schur_cone
  48
  49     EXAMPLES:
  50
  51     Verify the claim that the maximal angle between any two generators
  52     of the Schur cone and the nonnegative quintant is ``3*pi/4``::
  53
  54         sage: P = schur_cone(5)
  55         sage: Q = nonnegative_orthant(5)
  56         sage: G = ( g.change_ring(QQbar).normalized() for g in P )
  57         sage: H = ( h.change_ring(QQbar).normalized() for h in Q )
  58         sage: actual = max(arccos(u.inner_product(v)) for u in G for v in H)
  59         sage: expected = 3*pi/4
  60         sage: abs(actual - expected).n() < 1e-12
  61         True
  62
  63     The dual of the Schur cone is the "downward monotonic cone"
  64     [GourionSeeger]_, whose elements' entries are in non-increasing
  65     order::
  66
  67         sage: set_random_seed()
  68         sage: n = ZZ.random_element(10)
  69         sage: K = schur_cone(n).dual()
  70         sage: x = K.random_element()
  71         sage: all( x[i] >= x[i+1] for i in xrange(n-1) )
  72         True
  73
  74     TESTS:
  75
  76     We get the trivial cone when ``n`` is zero::
  77
  78         sage: schur_cone(0).is_trivial()
  79         True
  80
  81     The Schur cone induces the majorization ordering::
  82
  83         sage: set_random_seed()
  84         sage: def majorized_by(x,y):
  85         ....:     return (all(sum(x[0:i]) <= sum(y[0:i])
  86         ....:                 for i in xrange(x.degree()-1))
  87         ....:             and sum(x) == sum(y))
  88         sage: n = ZZ.random_element(10)
  89         sage: V = VectorSpace(QQ, n)
  90         sage: S = schur_cone(n)
  91         sage: majorized_by(V.zero(), S.random_element())
  92         True
  93         sage: x = V.random_element()
  94         sage: y = V.random_element()
  95         sage: majorized_by(x,y) == ( (y-x) in S )
  96         True
  97
  98     If a ``lattice`` was given, it is actually used::
  99
 100         sage: L = ToricLattice(3, 'M')
 101         sage: schur_cone(3, lattice=L)
 102         2-d cone in 3-d lattice M
 103
 104     Unless the rank of the lattice disagrees with ``n``::
 105
 106         sage: L = ToricLattice(1, 'M')
 107         sage: schur_cone(3, lattice=L)
 108         Traceback (most recent call last):
 109         ...
 110         ValueError: lattice rank=1 and dimension n=3 are incompatible
 111
 112     """
 113     if lattice is None:
 114         lattice = ToricLattice(n)
 115
 116     if lattice.rank() != n:
 117         raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
 118                          %
 119                          (lattice.rank(), n))
 120
 121     def _f(i,j):
 122         if i == j:
 123             return 1
 124         elif j - i == 1:
 125             return -1
 126         else:
 127             return 0
 128
 129     # The "max" below catches the trivial case where n == 0.
 130     S = matrix(ZZ, max(0,n-1), n, _f)
 131
 132     return Cone(S.rows(), lattice)