"""
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):
    r"""
    Return the Schur cone in ``n`` dimensions.

    INPUT:

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

    OUTPUT:

    A rational closed convex Schur cone of dimension ``n``.

    REFERENCES:

    .. [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

    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: set_random_seed()
        sage: def majorized_by(x,y):
        ....:     return (all(sum(x[0:i]) <= sum(y[0:i])
        ....:                 for i in xrange(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

    """

    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)

    # Likewise, when n == 0, we need to specify the lattice.
    return Cone(S.rows(), ToricLattice(n))