cone/symmetric_psd.py: use two more generator expressions.

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

diff --git a/mjo/cone/schur.py b/mjo/cone/schur.py
index fd806d93eb6380574c69cabe5471dc938b7aebe6..dca7292807096997a18bf5a57ae7dc400f3be187 100644 (file)
--- a/mjo/cone/schur.py
+++ b/mjo/cone/schur.py
@@ -19,6 +19,11 @@ def schur_cone(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.
@@ -35,9 +40,9 @@ 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
@@ -49,6 +54,23 @@ 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):