X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fschur.py;h=1fa1e52407fe2782b3852cf1482970724abb2003;hb=928b7d49fda98ff105c92293b5797bb7a2b9873a;hp=bea76b6f91f00d688690b283d54091c7f014e87e;hpb=4f05d94f49ebe477fb310cee9e62ab1052932a04;p=sage.d.git

diff --git a/mjo/cone/schur.py b/mjo/cone/schur.py
index bea76b6..1fa1e52 100644
--- a/mjo/cone/schur.py
+++ b/mjo/cone/schur.py
@@ -1,45 +1,131 @@
 """
 The Schur cone, as described in the "Critical angles..." papers by
-Seeger and Sossa.
+Iusem, Seeger, and Sossa. It defines the Schur ordering on `R^{n}`.
 """
 
 from sage.all import *
 
-def is_pointed(P,Q):
-    newP = Cone([ list(p) for p in P ])
-    newQ = Cone([ list(q) for q in Q ])
-    return newP.intersection(-newQ).is_trivial()
-
-def ext_angles(P,Q):
-    angles = []
-    for p in P:
-        for q in Q:
-            p = vector(SR, p)
-            q = vector(SR, q)
-            p = p/p.norm()
-            q = q/q.norm()
-            angles.append(arccos(p.inner_product(q)))
-
-    return angles
-
-def schur(n):
+def schur_cone(n, lattice=None):
     r"""
-    Return the Schur cone in ``n`` dimensions.
+    Return the Schur cone in ``n`` dimensions that induces the
+    majorization ordering.
 
     INPUT:
 
-    - ``n`` -- the ambient dimension of the Schur cone and its ambient space.
+      - ``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``.
+    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
 
-    hs = []
-    for i in range(1,n):
-        h_i = [0]*n
-        h_i[i-1] = QQ(1)
-        h_i[i] = -QQ(1)
-        hs.append(vector(QQ,n,h_i))
+    # The "max" below catches the trivial case where n == 0.
+    S = matrix(ZZ, max(0,n-1), n, _f)
 
-    return Cone(hs)
+    return Cone(S.rows(), lattice)