+++ /dev/null
-"""
-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.
-
- 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: set_random_seed()
- 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 xrange(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: 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
-
- 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)
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