README: rewrite it, it was rather out-of-date

[sage.d.git] / mjo / cone / schur.py

diff --git a/mjo/cone/schur.py b/mjo/cone/schur.py
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-"""
-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 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
-
-    """
-
-    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))