def _triangulate_cone_internal(C, all_cones=False):
rays = list(C.rays())
if any(a < 0 for r in rays for a in r):
raise ValueError('Internal triangulation only implemented for non-negative cones.')
normalised_rays = [ZZ.one() / v.norm(1) * vector(QQ, v) for v in rays]
with TemporaryList() as maximal_faces:
for indices in PointConfiguration(normalised_rays).triangulate() if len(normalised_rays) > 1 else [[0]]:
subrays = [rays[i] for i in indices]
if all_cones:
maximal_faces.append(indices)
else:
multiplicity = prod(a for a in matrix(ZZ, subrays).elementary_divisors() if a != 0)
yield SCONE(rays=subrays, multiplicity=multiplicity)
if all_cones:
X = SimplicialComplex(maximal_faces=maximal_faces, maximality_check=False)
for f in X.face_iterator():
yield SCONE(rays=[rays[i] for i in f], multiplicity=-1)
def _mixed_volume_gfan(gen):
"""
Use gfan to compute the mixed volume of a collection of polytopes.
Just like Khovanskii, we use the normalised mixed volume which satisfies
mixed_volume([P,...,P]) = volume(P).
"""
P = list(gen)
n = len(P)
if any(Q.ambient_dim() != n for Q in P):
raise TypeError(
'Number of polytopes and ambient dimension do not match')
if n == 0:
return 0
elif n == 1:
return P[0].volume()
R = PolynomialRing(QQ, 'x', n)
I = R.ideal([sum(monomial_exp(R, e) for e in Q.vertices()) for Q in P])
return ZZ.one() / Integer(factorial(n)) * I.groebner_fan().mixed_volume()
def make_cond_key(D):
D1 = ZZ(D)
if D1 < 1:
D1 = ZZ.one()
D1 = int(D1.log(10))
return '%04d%s' % (D1, str(D))
