cone/nonnegative_orthant.py: add a word to the docs.

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

diff --git a/mjo/cone/schur.py b/mjo/cone/schur.py
index fd806d93eb6380574c69cabe5471dc938b7aebe6..53607a948ef26a3cb9f5fae2d86c8d895a4fab71 100644 (file)
--- a/mjo/cone/schur.py
+++ b/mjo/cone/schur.py
@@ -19,6 +19,16 @@ def schur_cone(n):
 
     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.
@@ -35,13 +45,24 @@ def schur_cone(n):
 
         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: 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::
@@ -49,6 +70,23 @@ def schur_cone(n):
         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):