def rs_sin(p, x, prec):
"""
Sine of a series
Returns the series expansion of the sin of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_sin
>>> R, x, y = ring('x, y', QQ)
>>> rs_sin(x + x*y, x, 4)
-1/6*x**3*y**3 - 1/2*x**3*y**2 - 1/2*x**3*y - 1/6*x**3 + x*y + x
See Also
========
sin
"""
R = x.ring
if not p:
return R(0)
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
c_expr = c.as_expr()
if R.domain is EX:
t1, t2 = sin(c_expr), cos(c_expr)
elif isinstance(c, PolyElement):
try:
t1, t2 = R(sin(c_expr)), R(cos(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in this "
"domain.")
else:
raise DomainError("The given series can't be expanded in this "
"domain")
raise NotImplementedError
p1 = p - c
# Makes use of sympy cos, sin fuctions to evaluate the values of the cos/sin
# of the constant term.
return rs_sin(p1, x, prec) * t2 + rs_cos(p1, x, prec) * t1
# Series is calculated in terms of tan as its evaluation is fast.
if len(p) > 20 and p.ngens == 1:
t = rs_tan(p / 2, x, prec)
t2 = rs_square(t, x, prec)
p1 = rs_series_inversion(1 + t2, x, prec)
return rs_mul(p1, 2 * t, x, prec)
one = R(1)
n = 1
c = [0]
for k in range(2, prec + 2, 2):
c.append(one / n)
c.append(0)
n *= -k * (k + 1)
return rs_series_from_list(p, c, x, prec)
def dup_factor_list(f, K0, **args):
"""Factor polynomials into irreducibles in `K[x]`. """
if not K0.has_CharacteristicZero: # pragma: no cover
raise DomainError('only characteristic zero allowed')
if K0.is_Algebraic:
coeff, factors = dup_ext_factor(f, K0)
else:
if not K0.is_Exact:
K0_inexact, K0 = K0, K0.get_exact()
f = dup_convert(f, K0_inexact, K0)
else:
K0_inexact = None
if K0.has_Field:
K = K0.get_ring()
denom, f = dup_ground_to_ring(f, K0, K)
f = dup_convert(f, K0, K)
else:
K = K0
if K.is_ZZ:
coeff, factors = dup_zz_factor(f, K, **args)
elif K.is_Poly:
f, u = dmp_inject(f, 0, K)
coeff, factors = dmp_factor_list(f, u, K.dom, **args)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_eject(f, u, K), k)
coeff = K.convert(coeff, K.dom)
else: # pragma: no cover
raise DomainError('factorization not supported over %s' % K0)
if K0.has_Field:
for i, (f, k) in enumerate(factors):
factors[i] = (dup_convert(f, K, K0), k)
coeff = K0.convert(coeff, K)
denom = K0.convert(denom, K)
coeff = K0.quo(coeff, denom)
if K0_inexact is not None:
for i, (f, k) in enumerate(factors):
factors[i] = (dup_convert(f, K0, K0_inexact), k)
coeff = K0_inexact.convert(coeff, K0)
if not args.get('include', False):
return coeff, factors
else:
if not factors:
return [(dup_strip([coeff]), 1)]
else:
g = dup_mul_ground(factors[0][0], coeff, K)
return [(g, factors[0][1])] + factors[1:]
def rs_tan(p, x, prec):
"""
Tangent of a series
Returns the series expansion of the tan of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_tan
>>> R, x, y = ring('x, y', QQ)
>>> rs_tan(x + x*y, x, 4)
1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x
See Also
========
tan
"""
if rs_is_puiseux(p, x):
r = rs_puiseux(rs_tan, p, x, prec)
return r
R = p.ring
const = 0
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
if R.domain is EX:
c_expr = c.as_expr()
const = tan(c_expr)
elif isinstance(c, PolyElement):
try:
c_expr = c.as_expr()
const = R(tan(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
else:
try:
const = R(tan(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
p1 = p - c
# Makes use of sympy fuctions to evaluate the values of the cos/sin
# of the constant term.
t2 = rs_tan(p1, x, prec)
t = rs_series_inversion(1 - const * t2, x, prec)
return rs_mul(const + t2, t, x, prec)
if R.ngens == 1:
return _tan1(p, x, prec)
else:
return rs_fun(p, rs_tan, x, prec)
def rs_exp(p, x, prec):
"""
Exponentiation of a series modulo ``O(x**prec)``
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_exp
>>> R, x = ring('x', QQ)
>>> rs_exp(x**2, x, 7)
1/6*x**6 + 1/2*x**4 + x**2 + 1
"""
if rs_is_puiseux(p, x):
return rs_puiseux(rs_exp, p, x, prec)
R = p.ring
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
if R.domain is EX:
c_expr = c.as_expr()
const = exp(c_expr)
elif isinstance(c, PolyElement):
try:
c_expr = c.as_expr()
const = R(exp(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
else:
try:
const = R(exp(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
p1 = p - c
# Makes use of sympy fuctions to evaluate the values of the cos/sin
# of the constant term.
return const * rs_exp(p1, x, prec)
if len(p) > 20:
return _exp1(p, x, prec)
one = R(1)
n = 1
k = 1
c = []
for k in range(prec):
c.append(one / n)
k += 1
n *= k
r = rs_series_from_list(p, c, x, prec)
return r
def rs_log(p, x, prec):
"""
The Logarithm of ``p`` modulo ``O(x**prec)``
Notes
=====
truncation of ``integral dx p**-1*d p/dx`` is used.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_log
>>> R, x = ring('x', QQ)
>>> rs_log(1 + x, x, 8)
1/7*x**7 - 1/6*x**6 + 1/5*x**5 - 1/4*x**4 + 1/3*x**3 - 1/2*x**2 + x
>>> rs_log(x**QQ(3, 2) + 1, x, 5)
1/3*x**(9/2) - 1/2*x**3 + x**(3/2)
"""
if rs_is_puiseux(p, x):
return rs_puiseux(rs_log, p, x, prec)
if p == 1:
return 0
R = p.ring
if _has_constant_term(p, x):
const = 0
zm = R.zero_monom
c = p[zm]
if c == 1:
pass
else:
c_expr = c.as_expr()
if R.domain is EX:
const = log(c_expr)
elif isinstance(c, PolyElement):
try:
const = R(log(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
else:
try:
const = R(log(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
dlog = p.diff(x)
dlog = rs_mul(dlog, _series_inversion1(p, x, prec), x, prec - 1)
return rs_integrate(dlog, x) + const
else:
raise NotImplementedError
def rs_tanh(p, x, prec):
"""
Hyperbolic tangent of a series
Returns the series expansion of the tanh of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_tanh
>>> R, x, y = ring('x, y', QQ)
>>> rs_tanh(x + x*y, x, 4)
-1/3*x**3*y**3 - x**3*y**2 - x**3*y - 1/3*x**3 + x*y + x
See Also
========
tanh
"""
if rs_is_puiseux(p, x):
return rs_puiseux(rs_tanh, p, x, prec)
R = p.ring
const = 0
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
if R.domain is EX:
c_expr = c.as_expr()
const = tanh(c_expr)
elif isinstance(c, PolyElement):
try:
c_expr = c.as_expr()
const = R(tanh(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
else:
try:
const = R(tanh(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
p1 = p - c
t1 = rs_tanh(p1, x, prec)
t = rs_series_inversion(1 + const * t1, x, prec)
return rs_mul(const + t1, t, x, prec)
if R.ngens == 1:
return _tanh(p, x, prec)
else:
return rs_fun(p, _tanh, x, prec)
def rs_atanh(p, x, prec):
"""
Hyperbolic arctangent of a series
Returns the series expansion of the atanh of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_atanh
>>> R, x, y = ring('x, y', QQ)
>>> rs_atanh(x + x*y, x, 4)
1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x
See Also
========
atanh
"""
if rs_is_puiseux(p, x):
return rs_puiseux(rs_atanh, p, x, prec)
R = p.ring
const = 0
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
if R.domain is EX:
c_expr = c.as_expr()
const = atanh(c_expr)
elif isinstance(c, PolyElement):
try:
c_expr = c.as_expr()
const = R(atanh(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
else:
try:
const = R(atanh(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
# Instead of using a closed form formula, we differentiate atanh(p) to get
# `1/(1-p**2) * dp`, whose series expansion is much easier to calculate.
# Finally we integrate to get back atanh
dp = rs_diff(p, x)
p1 = -rs_square(p, x, prec) + 1
p1 = rs_series_inversion(p1, x, prec - 1)
p1 = rs_mul(dp, p1, x, prec - 1)
return rs_integrate(p1, x) + const
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in `F[x]`.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_half_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_half_gcdex(f, g, QQ)
([-1/5, 3/5], [1/1, 1/1])
"""
if not (K.has_Field or not K.is_Exact):
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_quo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dup_factor_list(f, K0):
"""Factor univariate polynomials into irreducibles in `K[x]`. """
j, f = dup_terms_gcd(f, K0)
cont, f = dup_primitive(f, K0)
if K0.is_FiniteField:
coeff, factors = dup_gf_factor(f, K0)
elif K0.is_Algebraic:
coeff, factors = dup_ext_factor(f, K0)
else:
if not K0.is_Exact:
K0_inexact, K0 = K0, K0.get_exact()
f = dup_convert(f, K0_inexact, K0)
else:
K0_inexact = None
if K0.is_Field:
K = K0.get_ring()
denom, f = dup_clear_denoms(f, K0, K)
f = dup_convert(f, K0, K)
else:
K = K0
if K.is_ZZ:
coeff, factors = dup_zz_factor(f, K)
elif K.is_Poly:
f, u = dmp_inject(f, 0, K)
coeff, factors = dmp_factor_list(f, u, K.dom)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_eject(f, u, K), k)
coeff = K.convert(coeff, K.dom)
else: # pragma: no cover
raise DomainError('factorization not supported over %s' % K0)
if K0.is_Field:
for i, (f, k) in enumerate(factors):
factors[i] = (dup_convert(f, K, K0), k)
coeff = K0.convert(coeff, K)
coeff = K0.quo(coeff, denom)
if K0_inexact:
for i, (f, k) in enumerate(factors):
max_norm = dup_max_norm(f, K0)
f = dup_quo_ground(f, max_norm, K0)
f = dup_convert(f, K0, K0_inexact)
factors[i] = (f, k)
coeff = K0.mul(coeff, K0.pow(max_norm, k))
coeff = K0_inexact.convert(coeff, K0)
K0 = K0_inexact
if j:
factors.insert(0, ([K0.one, K0.zero], j))
return coeff*cont, _sort_factors(factors)
def iter_groebner(seq, n, ring, method=None):
"""
Computes Groebner basis for a set of polynomials in `K[X]`.
Wrapper around the (default) improved Buchberger and the other algorithms
for computing Groebner bases. The choice of algorithm can be changed via
``method`` argument or :func:`setup` from :mod:`sympy.polys.polyconfig`,
where ``method`` can be either ``buchberger`` or ``f5b``.
"""
domain, orig = ring.domain, None
if not domain.is_Field or not domain.has_assoc_Field:
try:
orig, ring = ring, ring.clone(domain=domain.get_field())
except DomainError:
raise DomainError("can't compute a Groebner basis over %s" %
domain)
else:
seq = [s.set_ring(ring) for s in seq]
G = _incr_buch(seq, n, ring)
if orig is not None:
G = [g.clear_denoms()[1].set_ring(orig) for g in G]
return G
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in `F[x]`.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_half_gcdex(f, g)
(-1/5*x + 3/5, x + 1)
"""
if not K.has_Field:
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_quo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dup_factor_list(f, K0):
"""Factor polynomials into irreducibles in `K[x]`. """
j, f = dup_terms_gcd(f, K0)
if not K0.has_CharacteristicZero:
coeff, factors = dup_gf_factor(f, K0)
elif K0.is_Algebraic:
coeff, factors = dup_ext_factor(f, K0)
else:
if not K0.is_Exact:
K0_inexact, K0 = K0, K0.get_exact()
f = dup_convert(f, K0_inexact, K0)
else:
K0_inexact = None
if K0.has_Field:
K = K0.get_ring()
denom, f = dup_clear_denoms(f, K0, K)
f = dup_convert(f, K0, K)
else:
K = K0
if K.is_ZZ:
coeff, factors = dup_zz_factor(f, K)
elif K.is_Poly:
f, u = dmp_inject(f, 0, K)
coeff, factors = dmp_factor_list(f, u, K.dom)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_eject(f, u, K), k)
coeff = K.convert(coeff, K.dom)
else: # pragma: no cover
raise DomainError('factorization not supported over %s' % K0)
if K0.has_Field:
for i, (f, k) in enumerate(factors):
factors[i] = (dup_convert(f, K, K0), k)
coeff = K0.convert(coeff, K)
denom = K0.convert(denom, K)
coeff = K0.quo(coeff, denom)
if K0_inexact is not None:
for i, (f, k) in enumerate(factors):
factors[i] = (dup_convert(f, K0, K0_inexact), k)
coeff = K0_inexact.convert(coeff, K0)
if j:
factors.insert(0, ([K0.one, K0.zero], j))
return coeff, _sort_factors(factors)
def dmp_norm(f, u, K):
"""
Norm of ``f`` in ``K[X1, ..., Xn]``, often not square-free.
"""
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
h, _ = dmp_inject(f, u, K, front=True)
return dmp_resultant(g, h, u + 1, K.dom)
def __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
if len(ext) == 1 and isinstance(ext[0], tuple):
orig_ext = ext[0][1:]
else:
orig_ext = ext
self.orig_ext = orig_ext
"""
Original elements given to generate the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
>>> K.orig_ext
(sqrt(2), sqrt(3))
"""
self.ext = to_number_field(ext)
"""
Primitive element used for the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2), sqrt(3))
>>> K.ext
sqrt(2) + sqrt(3)
"""
self.mod = self.ext.minpoly.rep
"""
Minimal polynomial for the primitive element of the extension.
>>> from sympy import QQ, sqrt
>>> K = QQ.algebraic_field(sqrt(2))
>>> K.mod
DMP([1, 0, -2], QQ, None)
"""
self.domain = self.dom = dom
self.ngens = 1
self.symbols = self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
self._maximal_order = None
self._discriminant = None
self._nilradicals_mod_p = {}
def __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
self.ext = to_number_field(ext)
self.mod = self.ext.minpoly.rep
self.dom = dom
self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
def dmp_sqf_norm(f, u, K):
"""
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> from sympy import I
>>> K = QQ.algebraic_field(I)
>>> R, x, y = ring("x,y", K)
>>> _, X, Y = ring("x,y", QQ)
>>> s, f, r = R.dmp_sqf_norm(x*y + y**2)
>>> s == 1
True
>>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
True
>>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
True
"""
if not u:
return dup_sqf_norm(f, K)
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
F = dmp_raise([K.one, -K.unit], u, 0, K)
s = 0
while True:
h, _ = dmp_inject(f, u, K, front=True)
r = dmp_resultant(g, h, u + 1, K.dom)
if dmp_sqf_p(r, u, K.dom):
break
else:
f, s = dmp_compose(f, F, u, K), s + 1
return s, f, r
def dmp_sqf_norm(f, u, K):
"""
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy import I
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.sqfreetools import dmp_sqf_norm
>>> K = QQ.algebraic_field(I)
>>> s, f, r = dmp_sqf_norm([[K(1), K(0)], [K(1), K(0), K(0)]], 1, K)
>>> s == 1
True
>>> f == [[K(1), K(0)], [K(1), K([QQ(-1), QQ(0)]), K(0)]]
True
>>> r == [[1, 0, 0], [2, 0, 0, 0], [1, 0, 1, 0, 0]]
True
"""
if not u:
return dup_sqf_norm(f, K)
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
F = dmp_raise([K.one, -K.unit], u, 0, K)
s = 0
while True:
h, _ = dmp_inject(f, u, K, front=True)
r = dmp_resultant(g, h, u + 1, K.dom)
if dmp_sqf_p(r, u, K.dom):
break
else:
f, s = dmp_compose(f, F, u, K), s + 1
return s, f, r
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_real_imag
>>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.iteritems():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dup_real_imag(x**3 + x**2 + x + 1)
(x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y)
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.items():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
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 K.is_GaussianField:
K1 = K.as_AlgebraicField()
f = dmp_convert(f, u, K, K1)
K = K1
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 __init__(self, dom, *ext):
if not dom.is_QQ:
raise DomainError("ground domain must be a rational field")
from sympy.polys.numberfields import to_number_field
if len(ext) == 1 and isinstance(ext[0], tuple):
self.orig_ext = ext[0][1:]
else:
self.orig_ext = ext
self.ext = to_number_field(ext)
self.mod = self.ext.minpoly.rep
self.domain = self.dom = dom
self.ngens = 1
self.symbols = self.gens = (self.ext, )
self.unit = self([dom(1), dom(0)])
self.zero = self.dtype.zero(self.mod.rep, dom)
self.one = self.dtype.one(self.mod.rep, dom)
def dup_sqf_norm(f, K):
"""
Square-free norm of ``f`` in ``K[x]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> from sympy import sqrt
>>> K = QQ.algebraic_field(sqrt(3))
>>> R, x = ring("x", K)
>>> _, X = ring("x", QQ)
>>> s, f, r = R.dup_sqf_norm(x**2 - 2)
>>> s == 1
True
>>> f == x**2 + K([QQ(-2), QQ(0)])*x + 1
True
>>> r == X**4 - 10*X**2 + 1
True
"""
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
s, g = 0, dmp_raise(K.mod.rep, 1, 0, K.dom)
while True:
h, _ = dmp_inject(f, 0, K, front=True)
r = dmp_resultant(g, h, 1, K.dom)
if dup_sqf_p(r, K.dom):
break
else:
f, s = dup_shift(f, -K.unit, K), s + 1
return s, f, r
def groebner(seq, ring, method=None):
"""
Computes Groebner basis for a set of polynomials in `K[X]`.
Wrapper around the (default) improved Buchberger and the other algorithms
for computing Groebner bases. The choice of algorithm can be changed via
``method`` argument or :func:`setup` from :mod:`sympy.polys.polyconfig`,
where ``method`` can be either ``buchberger`` or ``f5b``.
"""
if method is None:
method = query('groebner')
_groebner_methods = {
'buchberger': _buchberger,
'f5b': _f5b,
}
try:
_groebner = _groebner_methods[method]
except KeyError:
raise ValueError(
"'%s' is not a valid Groebner bases algorithm (valid are 'buchberger' and 'f5b')"
% method)
domain, orig = ring.domain, None
if not domain.is_Field or not domain.has_assoc_Field:
try:
orig, ring = ring, ring.clone(domain=domain.get_field())
except DomainError:
raise DomainError("can't compute a Groebner basis over %s" %
domain)
else:
seq = [s.set_ring(ring) for s in seq]
G = _groebner(seq, ring)
if orig is not None:
G = [g.clear_denoms()[1].set_ring(orig) for g in G]
return G
def dup_sqf_norm(f, K):
"""
Square-free norm of ``f`` in ``K[x]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
**Examples**
>>> from sympy import sqrt
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.sqfreetools import dup_sqf_norm
>>> K = QQ.algebraic_field(sqrt(3))
>>> s, f, r = dup_sqf_norm([K(1), K(0), K(-2)], K)
>>> s == 1
True
>>> f == [K(1), K([QQ(-2), QQ(0)]), K(1)]
True
>>> r == [1, 0, -10, 0, 1]
True
"""
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
s, g = 0, dmp_raise(K.mod.rep, 1, 0, K.dom)
while True:
h, _ = dmp_inject(f, 0, K, front=True)
r = dmp_resultant(g, h, 1, K.dom)
if dup_sqf_p(r, K.dom):
break
else:
f, s = dup_shift(f, -K.unit, K), s+1
return s, f, r
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 dmp_factor_list(f, u, K0):
"""Factor polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list(f, K0)
J, f = dmp_terms_gcd(f, u, K0)
if not K0.has_CharacteristicZero: # pragma: no cover
coeff, factors = dmp_gf_factor(f, u, K0)
elif K0.is_Algebraic:
coeff, factors = dmp_ext_factor(f, u, K0)
else:
if not K0.is_Exact:
K0_inexact, K0 = K0, K0.get_exact()
f = dmp_convert(f, u, K0_inexact, K0)
else:
K0_inexact = None
if K0.has_Field:
K = K0.get_ring()
denom, f = dmp_clear_denoms(f, u, K0, K)
f = dmp_convert(f, u, K0, K)
else:
K = K0
if K.is_ZZ:
levels, f, v = dmp_exclude(f, u, K)
coeff, factors = dmp_zz_factor(f, v, K)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_include(f, levels, v, K), k)
elif K.is_Poly:
f, v = dmp_inject(f, u, K)
coeff, factors = dmp_factor_list(f, v, K.dom)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_eject(f, v, K), k)
coeff = K.convert(coeff, K.dom)
else: # pragma: no cover
raise DomainError('factorization not supported over %s' % K0)
if K0.has_Field:
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_convert(f, u, K, K0), k)
coeff = K0.convert(coeff, K)
denom = K0.convert(denom, K)
coeff = K0.quo(coeff, denom)
if K0_inexact is not None:
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_convert(f, u, K0, K0_inexact), k)
coeff = K0_inexact.convert(coeff, K0)
for i, j in enumerate(reversed(J)):
if not j:
continue
term = {(0, ) * (u - i) + (1, ) + (0, ) * i: K0.one}
factors.insert(0, (dmp_from_dict(term, u, K0), j))
return coeff, _sort_factors(factors)
def dmp_gf_factor(f, u, K):
"""Factor multivariate polynomials over finite fields. """
raise DomainError('multivariate polynomials over %s' % K)
def get_ring(self):
"""Returns a ring associated with ``self``. """
raise DomainError('there is no ring associated with %s' % self)
def algebraic_field(self, *extension):
"""Returns an algebraic field, i.e. `K(\\alpha, \dots)`. """
raise DomainError("can't create algebraic field over %s" % self)
def get_field(self):
"""Returns a field associated with ``self``. """
raise DomainError('there is no field associated with %s' % self)
