def __init__(self, L, O, V):
super(BorderBasis, self).__init__(L, L.as_polys(V))
self.O = O
self.dO = L.border(O)
self.V = V
self._quotient_basis = [L.monom(o) for o in O]
self.Q = BorderBasedUniverse(L.ring, O)
class BorderBasis(Basis):
"""
A border basis.
"""
def __init__(self, L, O, V):
super(BorderBasis, self).__init__(L, L.as_polys(V))
self.O = O
self.dO = L.border(O)
self.V = V
self._quotient_basis = [L.monom(o) for o in O]
self.Q = BorderBasedUniverse(L.ring, O)
def find_basis_nearest_lt(self, t):
"""
Find the basis element with the nearest leading term
"""
# Find the index closest to t
closest, dist = None, maxint
for t_ in self.dO:
d = tuple_diff(t, t_)
if any(idx < 0 for idx in d): continue
dist_ = sum(d)
if dist_ < dist:
closest, dist = t_, dist_
assert closest is not None
return self._generators[self.dO.index(closest)]
def find_basis_with_lt(self, t):
"""
Find the basis element with leading term
"""
idx = self.dO.index(t)
return self._generators[idx]
@property
def quotient_basis(self):
return self._quotient_basis
def mod(self, f):
"""
Compute the mod
"""
while True:
if f.LM in self.O:
return f
else:
# Find the nearest element on the border, and eliminate
b = self.find_basis_nearest_lt(f.LM)
f -= f.LC/b.LC * self.L.monom(tuple_diff(f.LM, b.LM)) * b
def formal_multiplication_matrices(self):
"""
With border bases, we know that the formal multiplication
matrices essentially only invoke terms on the border, so we can
compute this super efficiently.
"""
d = len(self.Q)
Vs = [np.zeros((d,d)) for _ in xrange(self.L.nsymbols)]
for i, V in enumerate(Vs):
for j, b in enumerate(self.O):
t = tuple_incr(b, i)
if t in self.O:
V[j, self.Q.index(t)] = 1
else:
b_ = self.find_basis_with_lt(t)
for t, c in b_.terms()[1:]:
V[j, self.Q.index(t)] = -c
return Vs