def __add__(self, other):
"""
Return sum of two modules.
"""
denominator = gcd.lcm(self.denominator, other.denominator)
row = self.dimension
assert row == other.dimension
column = self.rank + other.rank
mat = matrix.createMatrix(row, column)
adjust = denominator // self.denominator
for i, base in enumerate(self.basis):
mat.setColumn(i + 1, [c * adjust for c in base])
adjust = denominator // other.denominator
for i, base in enumerate(other.basis):
mat.setColumn(self.rank + i + 1, [c * adjust for c in base])
# Hermite normal form
hnf = mat.hermiteNormalForm()
# The hnf returned by the hermiteNormalForm method may have columns
# of zeros, and they are not needed.
zerovec = vector.Vector([0] * hnf.row)
while hnf.row < hnf.column or hnf[1] == zerovec:
hnf.deleteColumn(1)
omega = []
for j in range(1, hnf.column + 1):
omega.append(list(hnf[j]))
return ModuleWithDenominator(omega, denominator)
def intersect(self, other):
"""
return intersection of self and other
"""
# if self.number_field != other.number_field:
# raise ValueError("based on different number field")
if self.base != other.base:
new_self = self.change_base_module(other.base)
else:
new_self = self.copy()
new_denominator = gcd.lcm(new_self.denominator, other.denominator)
if new_self.denominator != new_denominator:
multiply_denom = ring.exact_division(
new_denominator, new_self.denominator)
new_self_mat_repr = multiply_denom * new_self.mat_repr
else:
new_self_mat_repr = new_self.mat_repr
if other.denominator != new_denominator:
multiply_denom = ring.exact_division(
new_denominator, other.denominator)
new_other_mat_repr = multiply_denom * other.mat_repr
else:
new_other_mat_repr = other.mat_repr
new_mat_repr = Submodule.fromMatrix(
new_self_mat_repr).intersectionOfSubmodules(new_other_mat_repr)
new_module = self.__class__(
[new_mat_repr, new_denominator], self.number_field, other.base)
new_module._simplify()
return new_module
def __add__(self, other):
"""
Return sum of two modules.
"""
denominator = gcd.lcm(self.denominator, other.denominator)
row = self.dimension
assert row == other.dimension
column = self.rank + other.rank
mat = matrix.createMatrix(row, column)
adjust = denominator // self.denominator
for i, base in enumerate(self.basis):
mat.setColumn(i + 1, [c * adjust for c in base])
adjust = denominator // other.denominator
for i, base in enumerate(other.basis):
mat.setColumn(self.rank + i + 1, [c * adjust for c in base])
# Hermite normal form
hnf = mat.hermiteNormalForm()
# The hnf returned by the hermiteNormalForm method may have columns
# of zeros, and they are not needed.
zerovec = vector.Vector([0] * hnf.row)
while hnf.row < hnf.column or hnf[1] == zerovec:
hnf.deleteColumn(1)
omega = []
for j in range(1, hnf.column + 1):
omega.append(list(hnf[j]))
return ModuleWithDenominator(omega, denominator)
def intersect(self, other):
"""
return intersection of self and other
"""
# if self.number_field != other.number_field:
# raise ValueError("based on different number field")
if self.base != other.base:
new_self = self.change_base_module(other.base)
else:
new_self = self.copy()
new_denominator = gcd.lcm(new_self.denominator, other.denominator)
if new_self.denominator != new_denominator:
multiply_denom = ring.exact_division(new_denominator,
new_self.denominator)
new_self_mat_repr = multiply_denom * new_self.mat_repr
else:
new_self_mat_repr = new_self.mat_repr
if other.denominator != new_denominator:
multiply_denom = ring.exact_division(new_denominator,
other.denominator)
new_other_mat_repr = multiply_denom * other.mat_repr
else:
new_other_mat_repr = other.mat_repr
new_mat_repr = Submodule.fromMatrix(
new_self_mat_repr).intersectionOfSubmodules(new_other_mat_repr)
new_module = self.__class__([new_mat_repr, new_denominator],
self.number_field, other.base)
new_module._simplify()
return new_module
def order(self):
"""
This method returns order for permutation element.
"""
data = self.simplify().data
sol = 1
for x in data:
sol = gcd.lcm(sol, len(x))
return sol
def order(self):
"""
This method returns order for permutation element.
"""
data = self.simplify().data
sol = 1
for x in data:
sol = gcd.lcm(sol, len(x))
return sol
def __add__(self, other):
if self.size == other.size:
m = self.size
zr_a = Zeta(m)
for i in range(m):
zr_a.z[i] = self.z[i] + other.z[i]
return zr_a
else:
m = gcd.lcm(self.size, other.size)
return self.promote(m) + other.promote(m)
def __add__(self, other):
if self.size == other.size:
m = self.size
zr_a = Zeta(m)
for i in range(m):
zr_a.z[i] = self.z[i] + other.z[i]
return zr_a
else:
m = gcd.lcm(self.size, other.size)
return self.promote(m) + other.promote(m)
def ch_basic(self):
denom = 1
for i in range(self.degree):
if not isinstance(self.coeff[i], int):
denom *= gcd.lcm(denom, (self.coeff[i]).denominator)
coeff = []
for i in range(self.degree):
if isinstance(self.coeff[i], int):
coeff.append(self.coeff[i] * denom)
else:
coeff.append(int((self.coeff[i]).numerator * denom / (self.coeff[i]).denominator))
return BasicAlgNumber([coeff, denom], self.polynomial)
def ch_basic(self):
denom = 1
for i in range(self.degree):
if not isinstance(self.coeff[i], int):
denom *= gcd.lcm(denom, (self.coeff[i]).denominator)
coeff = []
for i in range(self.degree):
if isinstance(self.coeff[i], int):
coeff.append(self.coeff[i] * denom)
else:
coeff.append(int((self.coeff[i]).numerator * denom / (self.coeff[i]).denominator))
return BasicAlgNumber([coeff, denom], self.polynomial)
def _toIntegerMatrix(mat, option=0):
"""
transform a (including integral) rational matrix to
some integral matrix as the following
[option]
0: return integral-matrix, denominator
(mat = 1/denominator * integral-matrix)
1: return integral-matrix, denominator, numerator
(mat = numerator/denominator * reduced-int-matrix)
2: return integral-matrix, numerator (assuming mat is integral)
(mat = numerator * numerator-reduced-rational-matrix)
"""
def getDenominator(ele):
"""
get the denominator of ele
"""
if rational.isIntegerObject(ele):
return 1
else: # rational
return ele.denominator
def getNumerator(ele):
"""
get the numerator of ele
"""
if rational.isIntegerObject(ele):
return ele
else: # rational
return ele.numerator
if isinstance(mat, vector.Vector):
mat = mat.toMatrix(True)
if option <= 1:
denom = mat.reduce(
lambda x, y: gcd.lcm(x, getDenominator(y)), 1)
new_mat = mat.map(
lambda x: getNumerator(x) * ring.exact_division(
denom, getDenominator(x)))
if option == 0:
return Submodule.fromMatrix(new_mat), denom
else:
new_mat = mat
numer = new_mat.reduce(
lambda x, y: gcd.gcd(x, getNumerator(y)))
if numer != 0:
new_mat2 = new_mat.map(
lambda x: ring.exact_division(getNumerator(x), numer))
else:
new_mat2 = new_mat
if option == 1:
return Submodule.fromMatrix(new_mat2), denom, numer
else:
return Submodule.fromMatrix(new_mat2), numer
def _toIntegerMatrix(mat, option=0):
"""
transform a (including integral) rational matrix to
some integral matrix as the following
[option]
0: return integral-matrix, denominator
(mat = 1/denominator * integral-matrix)
1: return integral-matrix, denominator, numerator
(mat = numerator/denominator * reduced-int-matrix)
2: return integral-matrix, numerator (assuming mat is integral)
(mat = numerator * numerator-reduced-rational-matrix)
"""
def getDenominator(ele):
"""
get the denominator of ele
"""
if rational.isIntegerObject(ele):
return 1
else: # rational
return ele.denominator
def getNumerator(ele):
"""
get the numerator of ele
"""
if rational.isIntegerObject(ele):
return ele
else: # rational
return ele.numerator
if isinstance(mat, vector.Vector):
mat = mat.toMatrix(True)
if option <= 1:
denom = mat.reduce(
lambda x, y: gcd.lcm(x, getDenominator(y)), 1)
new_mat = mat.map(
lambda x: getNumerator(x) * ring.exact_division(
denom, getDenominator(x)))
if option == 0:
return Submodule.fromMatrix(new_mat), denom
else:
new_mat = mat
numer = new_mat.reduce(
lambda x, y: gcd.gcd(x, getNumerator(y)))
if numer != 0:
new_mat2 = new_mat.map(
lambda x: ring.exact_division(getNumerator(x), numer))
else:
new_mat2 = new_mat
if option == 1:
return Submodule.fromMatrix(new_mat2), denom, numer
else:
return Submodule.fromMatrix(new_mat2), numer
def polyfactorizationNF(f):
"""
NFPolynomialFactorization(f) -> factors
Return a number field factorization of given number field
coefficient squarefree polynomial f. The result factors have
number field coefficients, its minimum absolute representation.
"""
# initialize
Q = rational.RationalField() # Q as RationalField
K = f.getCoefficientRing() # K as NumberField
# to squarefree
u = f / f.gcd(f.differentiate())
# partition each degree.
degree = g.degree()
Gcoeffs = {} # Gcoeffs as coefficient as G (TBA)
for i in range(degree + 1):
U[i] = u[i]
if not isinstance(U[i], algfield.BasicAlgNumber):
raise NotImplementError("FIXME: support only BasicAlgNumber .")
g[i] = uniutil.OneVariableDensePolynomial(U[i].coeff, "Y",
Q) / U[i].denom
for j in range(g[i].degree() + 1):
Gcoeffs[(i, j)] = g[i][j]
G = multiutil.polynomial(Gcoeffs, Q) # G as partition form of u
# search for squarefree polynomial norm
k = 0
# FIXME: symbolic computation, but also slower.
x = multiutil.polynomial((1, 0), Q)
y = multiutil.polynomial((0, 1), Q)
while True:
N = T(y).resultant(G(x - k * y, y), 2)
if (N == N / N.gcd(N.differentiate())):
break
k = k + 1
# Now, here N is squarefree.
# assume all the coefficients of N is rational.Rational .
denom = 1
for coeff in N.coefficients():
denom = gcd.lcm(denom, coeff.denominator)
Nz = N * denom # NZ be a integer, but probably not monic.
for index, coeff in Nz.terms():
V[index] = coeff.numerator
NZ = uniutil.polynomial(V, rational.IntegerRing())
F = integerpolynomialfactorization(NZ)
def double_embeddings(f_q1, f_q2):
"""
Return embedding homomorphism functions from f_q1 and f_q2
to the composite field.
"""
identity = lambda x: x
if f_q1 is f_q2:
return (identity, identity)
p = f_q2.getCharacteristic()
k1, k2 = f_q1.degree, f_q2.degree
if not k2 % k1:
return (embedding(f_q1, f_q2), identity)
if not k1 % k2:
return (identity, embedding(f_q2, f_q1))
composite = FiniteExtendedField(p, gcd.lcm(k1, k2))
return (embedding(f_q1, composite), embedding(f_q2, composite))
def double_embeddings(f_q1, f_q2):
"""
Return embedding homomorphism functions from f_q1 and f_q2
to the composite field.
"""
identity = lambda x: x
if f_q1 is f_q2:
return (identity, identity)
p = f_q2.getCharacteristic()
k1, k2 = f_q1.degree, f_q2.degree
if not k2 % k1:
return (embedding(f_q1, f_q2), identity)
if not k1 % k2:
return (identity, embedding(f_q2, f_q1))
composite = FiniteExtendedField(p, gcd.lcm(k1, k2))
return (embedding(f_q1, composite), embedding(f_q2, composite))
def inverse(self):
f = qpoly(self.polynomial)
g = qpoly(self.coeff)
quot = f.extgcd(g)
icoeff = [self.denom * quot[1][i] for i in range(self.degree)]
#print icoeff
denom_list = []
for i in range(self.degree):
if icoeff[i] == 0:
denom_list.append(1)
else:
denom_list.append(icoeff[i].denominator)
new_denom = 1
for i in range(self.degree):
new_denom = gcd.lcm(new_denom, denom_list[i])
icoeff = [icoeff[i].numerator * new_denom // icoeff[i].denominator for i in range(self.degree)]
return BasicAlgNumber([icoeff, new_denom], self.polynomial)
def inverse(self):
f = qpoly(self.polynomial)
g = qpoly(self.coeff)
quot = f.extgcd(g)
icoeff = [self.denom * quot[1][i] for i in range(self.degree)]
#print icoeff
denom_list = []
for i in range(self.degree):
if icoeff[i] == 0:
denom_list.append(1)
else:
denom_list.append(icoeff[i].denominator)
new_denom = 1
for i in range(self.degree):
new_denom = gcd.lcm(new_denom, denom_list[i])
icoeff = [icoeff[i].numerator * new_denom // icoeff[i].denominator for i in range(self.degree)]
return BasicAlgNumber([icoeff, new_denom], self.polynomial)
def __mul__(self, other):
if not isinstance(other, Zeta):
zr_m = Zeta(self.size)
zr_m.z = [x * other for x in self.z]
return zr_m
elif self.size == other.size:
zr_m = Zeta(self.size)
other = +other
for k in range(other.size):
if not other.z[k]:
continue
elif other.z[k] == 1:
zr_m = zr_m + (self << k)
else:
zr_m = zr_m + (self << k) * other.z[k]
return zr_m
else:
m = gcd.lcm(self.size, other.size)
return self.promote(m) * other.promote(m)
def __mul__(self, other):
if not isinstance(other, Zeta):
zr_m = Zeta(self.size)
zr_m.z = [x*other for x in self.z]
return zr_m
elif self.size == other.size:
zr_m = Zeta(self.size)
other = +other
for k in range(other.size):
if not other.z[k]:
continue
elif other.z[k] == 1:
zr_m = zr_m + (self<<k)
else:
zr_m = zr_m + (self<<k)*other.z[k]
return zr_m
else:
m = gcd.lcm(self.size,other.size)
return self.promote(m)*other.promote(m)
def testLcm(self):
self.assertEqual(2, gcd.lcm(1, 2))
self.assertEqual(4, gcd.lcm(2, 4))
self.assertEqual(0, gcd.lcm(0, 10))
self.assertEqual(0, gcd.lcm(10, 0))
self.assertEqual(273, gcd.lcm(13, 21))