def _coeffs_generator(n):
"""Generate coefficients for `primitive_element()`. """
for coeffs in variations([1, -1, 2, -2, 3, -3], n, repetition=True):
# Two linear combinations with coeffs of opposite signs are
# opposites of each other. Hence it suffices to test only one.
if coeffs[0] > 0:
yield list(coeffs)
def dmp_lift(f, u, K):
"""
Convert algebraic coefficients to integers in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> from sympy import I
>>> K = QQ.algebraic_field(I)
>>> R, x = ring("x", K)
>>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)])
>>> R.dmp_lift(f)
x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16
"""
if not K.is_Algebraic:
raise DomainError(
'computation can be done only in an algebraic domain')
F, monoms, polys = dmp_to_dict(f, u), [], []
for monom, coeff in F.items():
if not coeff.is_ground:
monoms.append(monom)
perms = variations([-1, 1], len(monoms), repetition=True)
for perm in perms:
G = dict(F)
for sign, monom in zip(perm, monoms):
if sign == -1:
G[monom] = -G[monom]
polys.append(dmp_from_dict(G, u, K))
return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom)
def dmp_lift(f, u, K):
"""
Convert algebraic coefficients to integers in ``K[X]``.
Examples
========
>>> from sympy import I
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densetools import dmp_lift
>>> K = QQ.algebraic_field(I)
>>> f = [K(1), K([QQ(1), QQ(0)]), K([QQ(2), QQ(0)])]
>>> dmp_lift(f, 0, K)
[1/1, 0/1, 2/1, 0/1, 9/1, 0/1, -8/1, 0/1, 16/1]
"""
if not K.is_Algebraic:
raise DomainError(
'computation can be done only in an algebraic domain')
F, monoms, polys = dmp_to_dict(f, u), [], []
for monom, coeff in F.iteritems():
if not coeff.is_ground:
monoms.append(monom)
perms = variations([-1, 1], len(monoms), repetition=True)
for perm in perms:
G = dict(F)
for sign, monom in zip(perm, monoms):
if sign == -1:
G[monom] = -G[monom]
polys.append(dmp_from_dict(G, u, K))
return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom)
def _coeffs_generator(n):
"""Generate coefficients for `primitive_element()`. """
for coeffs in variations([1, -1], n, repetition=True):
yield list(coeffs)
def _coeffs_generator(n):
"""Generate coefficients for `primitive_element()`. """
for coeffs in variations([1, -1], n, repetition=True):
yield list(coeffs)
