def dup_ext_factor(f, K):
"""Factor polynomials over algebraic number fields. """
n, lc = dup_degree(f), dup_LC(f, K)
f = dup_monic(f, K)
if n <= 0:
return lc, []
if n == 1:
return lc, [(f, 1)]
f, F = dup_sqf_part(f, K), f
s, g, r = dup_sqf_norm(f, K)
factors = dup_factor_list(r, K.dom, include=True)
if len(factors) == 1:
return lc, [(f, n // dup_degree(f))]
H = s * K.unit
for i, (factor, _) in enumerate(factors):
h = dup_convert(factor, K.dom, K)
h, _, g = dup_inner_gcd(h, g, K)
h = dup_taylor(h, H, K)
factors[i] = h
factors = dup_trial_division(F, factors, K)
return lc, factors
def dup_ext_factor(f, K):
"""Factor polynomials over algebraic number fields. """
n, lc = dup_degree(f), dup_LC(f, K)
f = dup_monic(f, K)
if n <= 0:
return lc, []
if n == 1:
return lc, [(f, 1)]
f, F = dup_sqf_part(f, K), f
s, g, r = dup_sqf_norm(f, K)
factors = dup_factor_list(r, K.dom, include=True)
if len(factors) == 1:
return lc, [(f, n//dup_degree(f))]
H = s*K.unit
for i, (factor, _) in enumerate(factors):
h = dup_convert(factor, K.dom, K)
h, _, g = dup_inner_gcd(h, g, K)
h = dup_taylor(h, H, K)
factors[i] = h
factors = dup_trial_division(F, factors, K)
return lc, factors
def dup_zz_factor(f, K, **args):
"""Factor (non square-free) polynomials in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Zassenhaus algorithm. Trial division is used to recover the
multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*x**4 - 2`::
>>> from sympy.polys.factortools import dup_zz_factor
>>> from sympy.polys.algebratools import ZZ
>>> dup_zz_factor([2, 0, 0, 0, -2], ZZ)
(2, [([1, -1], 1), ([1, 1], 1), ([1, 0, 1], 1)])
In result we got the following factorization::
f = 2 (x - 1) (x + 1) (x**2 + 1)
Note that this is a complete factorization over integers,
however over Gaussian integers we can factor the last term.
By default, polynomials `x**n - 1` and `x**n + 1` are factored
using cyclotomic decomposition to speedup computations. To
disable this behaviour set cyclotomic=False.
References
==========
.. [Gathen99] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
First Edition, Cambridge University Press, 1999, pp. 427
"""
cont, g = dup_primitive(f, K)
n = dup_degree(g)
if dup_LC(g, K) < 0:
cont, g = -cont, dup_neg(g, K)
if n <= 0:
return cont, []
if n == 1 or dup_zz_irreducible_p(g, K):
return cont, [(g, 1)]
g = dup_sqf_part(g, K)
H, factors = None, []
if args.get('cyclotomic', True):
H = dup_zz_cyclotomic_factor(g, K)
if H is None:
H = dup_zz_zassenhaus(g, K)
for h in H:
k = 0
while True:
q, r = dup_div(f, h, K)
if not r:
f, k = q, k + 1
else:
break
factors.append((h, k))
return cont, _sort_factors(factors)
def dup_zz_factor(f, K, **args):
"""Factor (non square-free) polynomials in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Zassenhaus algorithm. Trial division is used to recover the
multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*x**4 - 2`::
>>> from sympy.polys.factortools import dup_zz_factor
>>> from sympy.polys.algebratools import ZZ
>>> dup_zz_factor([2, 0, 0, 0, -2], ZZ)
(2, [([1, -1], 1), ([1, 1], 1), ([1, 0, 1], 1)])
In result we got the following factorization::
f = 2 (x - 1) (x + 1) (x**2 + 1)
Note that this is a complete factorization over integers,
however over Gaussian integers we can factor the last term.
By default, polynomials `x**n - 1` and `x**n + 1` are factored
using cyclotomic decomposition to speedup computations. To
disable this behaviour set cyclotomic=False.
References
==========
.. [Gathen99] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
First Edition, Cambridge University Press, 1999, pp. 427
"""
cont, g = dup_primitive(f, K)
n = dup_degree(g)
if dup_LC(g, K) < 0:
cont, g = -cont, dup_neg(g, K)
if n <= 0:
return cont, []
if n == 1 or dup_zz_irreducible_p(g, K):
return cont, [(g, 1)]
g = dup_sqf_part(g, K)
H, factors = None, []
if args.get('cyclotomic', True):
H = dup_zz_cyclotomic_factor(g, K)
if H is None:
H = dup_zz_zassenhaus(g, K)
for h in H:
k = 0
while True:
q, r = dup_div(f, h, K)
if not r:
f, k = q, k+1
else:
break
factors.append((h, k))
return cont, _sort_factors(factors)
