def test_smith_normal():
m = Matrix([[12, 6, 4, 8], [3, 9, 6, 12], [2, 16, 14, 28],
[20, 10, 10, 20]])
setattr(m, 'ring', ZZ)
smf = Matrix([[1, 0, 0, 0], [0, 10, 0, 0], [0, 0, -30, 0], [0, 0, 0, 0]])
assert smith_normal_form(m) == smf
x = Symbol('x')
m = Matrix([[Poly(x - 1), Poly(1, x), Poly(-1, x)],
[0, Poly(x), Poly(-1, x)], [Poly(0, x),
Poly(-1, x),
Poly(x)]])
setattr(m, 'ring', QQ[x])
invs = (Poly(1, x), Poly(x - 1), Poly(x**2 - 1))
assert invariant_factors(m) == invs
def _is_infinite(self):
"""
Test if the group is infinite. Return `True` if the test succeeds
and `None` otherwise
"""
used_gens = set()
for r in self.relators:
used_gens.update(r.contains_generators())
if any([g not in used_gens for g in self.generators]):
return True
# Abelianisation test: check is the abelianisation is infinite
abelian_rels = []
from sympy.polys.solvers import RawMatrix as Matrix
from sympy.polys.domains import ZZ
from sympy.matrices.normalforms import invariant_factors
for rel in self.relators:
abelian_rels.append([rel.exponent_sum(g) for g in self.generators])
m = Matrix(abelian_rels)
setattr(m, "ring", ZZ)
if 0 in invariant_factors(m):
return True
else:
return None