"""
The Schur cone, as described in the "Critical angles..." papers by
Iusem, Seeger, and Sossa. It defines the Schur ordering on `R^{n}`.
"""

from sage.all import *

def schur_cone(n, lattice=None):
    r"""
    Return the Schur cone in ``n`` dimensions that induces the
    majorization ordering.

    INPUT:

      - ``n`` -- the dimension the ambient space.

      - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
                       to be passed to the :func:`Cone` constructor.

    OUTPUT:

    A rational closed convex Schur cone of dimension ``n``. Each
    generating ray will have the integer ring as its base ring.

    If a ``lattice`` was specified, then the resulting cone will live in
    that lattice unless its rank is incompatible with the dimension
    ``n`` (in which case a ``ValueError`` is raised).

    REFERENCES:

    .. [GourionSeeger] Daniel Gourion and Alberto Seeger.
       Critical angles in polyhedral convex cones: numerical and
       statistical considerations. Mathematical Programming, 123:173-198,
       2010, doi:10.1007/s10107-009-0317-2.

    .. [IusemSeegerOnPairs] Alfredo Iusem and Alberto Seeger.
       On pairs of vectors achieving the maximal angle of a convex cone.
       Mathematical Programming, 104(2-3):501-523, 2005,
       doi:10.1007/s10107-005-0626-z.

    .. [SeegerSossaI] Alberto Seeger and David Sossa.
       Critical angles between two convex cones I. General theory.
       TOP, 24(1):44-65, 2016, doi:10.1007/s11750-015-0375-y.

    SETUP::

        sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
        sage: from mjo.cone.schur import schur_cone

    EXAMPLES:

    Verify the claim that the maximal angle between any two generators
    of the Schur cone and the nonnegative quintant is ``3*pi/4``::

        sage: P = schur_cone(5)
        sage: Q = nonnegative_orthant(5)
        sage: G = ( g.change_ring(QQbar).normalized() for g in P )
        sage: H = ( h.change_ring(QQbar).normalized() for h in Q )
        sage: actual = max(arccos(u.inner_product(v)) for u in G for v in H)
        sage: expected = 3*pi/4
        sage: abs(actual - expected).n() < 1e-12
        True

    The dual of the Schur cone is the "downward monotonic cone"
    [GourionSeeger]_, whose elements' entries are in non-increasing
    order::

        sage: n = ZZ.random_element(10)
        sage: K = schur_cone(n).dual()
        sage: x = K.random_element()
        sage: all( x[i] >= x[i+1] for i in range(n-1) )
        True

    TESTS:

    We get the trivial cone when ``n`` is zero::

        sage: schur_cone(0).is_trivial()
        True

    The Schur cone induces the majorization ordering::

        sage: def majorized_by(x,y):
        ....:     return (all(sum(x[0:i]) <= sum(y[0:i])
        ....:                 for i in range(x.degree()-1))
        ....:             and sum(x) == sum(y))
        sage: n = ZZ.random_element(10)
        sage: V = VectorSpace(QQ, n)
        sage: S = schur_cone(n)
        sage: majorized_by(V.zero(), S.random_element())
        True
        sage: x = V.random_element()
        sage: y = V.random_element()
        sage: majorized_by(x,y) == ( (y-x) in S )
        True

    If a ``lattice`` was given, it is actually used::

        sage: L = ToricLattice(3, 'M')
        sage: schur_cone(3, lattice=L)
        2-d cone in 3-d lattice M

    Unless the rank of the lattice disagrees with ``n``::

        sage: L = ToricLattice(1, 'M')
        sage: schur_cone(3, lattice=L)
        Traceback (most recent call last):
        ...
        ValueError: lattice rank=1 and dimension n=3 are incompatible

    """
    if lattice is None:
        lattice = ToricLattice(n)

    if lattice.rank() != n:
        raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
                         %
                         (lattice.rank(), n))

    def _f(i,j):
        if i == j:
            return 1
        elif j - i == 1:
            return -1
        else:
            return 0

    # The "max" below catches the trivial case where n == 0.
    S = matrix(ZZ, max(0,n-1), n, _f)

    return Cone(S.rows(), lattice)