def dup_prem(f, g, K):
"""Polynomial pseudo-remainder in `K[x]`. """
df = dup_degree(f)
dg = dup_degree(g)
r = f
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return r
N = df - dg + 1
lc_g = dup_LC(g, K)
while True:
dr = dup_degree(r)
if dr < dg:
break
lc_r = dup_LC(r, K)
j, N = dr-dg, N-1
R = dup_mul_ground(r, lc_g, K)
G = dup_mul_term(g, lc_r, j, K)
r = dup_sub(R, G, K)
return dup_mul_ground(r, lc_g**N, K)
def dup_ff_div(f, g, K):
"""Polynomial division with remainder over a field. """
df = dup_degree(f)
dg = dup_degree(g)
q, r = [], f
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
dr = dup_degree(r)
if dr < dg:
break
lc_r = dup_LC(r, K)
c = K.exquo(lc_r, lc_g)
j = dr - dg
q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)
r = dup_sub(r, h, K)
if not K.is_Exact:
r = dup_normal(r, K)
return q, r
def dup_rr_div(f, g, K):
"""Univariate division with remainder over a ring. """
df = dup_degree(f)
dg = dup_degree(g)
q, r = [], f
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
dr = dup_degree(r)
if dr < dg:
break
lc_r = dup_LC(r, K)
if lc_r % lc_g:
break
c = K.exquo(lc_r, lc_g)
j = dr - dg
q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)
r = dup_sub(r, h, K)
return q, r
def dup_sqf_list(f, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_list
>>> f = ZZ.map([2, 16, 50, 76, 56, 16])
>>> dup_sqf_list(f, ZZ)
(2, [([1, 1], 2), ([1, 2], 3)])
>>> dup_sqf_list(f, ZZ, all=True)
(2, [([1], 1), ([1, 1], 2), ([1, 2], 3)])
"""
if not K.has_CharacteristicZero:
return dup_gf_sqf_list(f, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dup_LC(f, K)
f = dup_monic(f, K)
else:
coeff, f = dup_primitive(f, K)
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
coeff = -coeff
if dup_degree(f) <= 0:
return coeff, []
result, i = [], 1
h = dup_diff(f, 1, K)
g, p, q = dup_inner_gcd(f, h, K)
while True:
d = dup_diff(p, 1, K)
h = dup_sub(q, d, K)
if not h:
result.append((p, i))
break
g, p, q = dup_inner_gcd(p, h, K)
if all or dup_degree(g) > 0:
result.append((g, i))
i += 1
return coeff, result
def _dup_ff_trivial_gcd(f, g, K):
"""Handle trivial cases in GCD algorithm over a field. """
if not (f or g):
return [], [], []
elif not f:
return dup_monic(g, K), [], [dup_LC(g, K)]
elif not g:
return dup_monic(f, K), [dup_LC(f, K)], []
else:
return None
def dup_sqf_list(f, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16
>>> R.dup_sqf_list(f)
(2, [(x + 1, 2), (x + 2, 3)])
>>> R.dup_sqf_list(f, all=True)
(2, [(1, 1), (x + 1, 2), (x + 2, 3)])
"""
if K.is_FiniteField:
return dup_gf_sqf_list(f, K, all=all)
if K.has_Field:
coeff = dup_LC(f, K)
f = dup_monic(f, K)
else:
coeff, f = dup_primitive(f, K)
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
coeff = -coeff
if dup_degree(f) <= 0:
return coeff, []
result, i = [], 1
h = dup_diff(f, 1, K)
g, p, q = dup_inner_gcd(f, h, K)
while True:
d = dup_diff(p, 1, K)
h = dup_sub(q, d, K)
if not h:
result.append((p, i))
break
g, p, q = dup_inner_gcd(p, h, K)
if all or dup_degree(g) > 0:
result.append((g, i))
i += 1
return coeff, result
def dup_prs_resultant(f, g, K):
"""
Resultant algorithm in `K[x]` using subresultant PRS.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_prs_resultant
>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([1, 0, -1])
>>> dup_prs_resultant(f, g, ZZ)
(4, [[1, 0, 1], [1, 0, -1], [-2]])
"""
if not f or not g:
return (K.zero, [])
R, B, D = dup_inner_subresultants(f, g, K)
if dup_degree(R[-1]) > 0:
return (K.zero, R)
if R[-2] == [K.one]:
return (dup_LC(R[-1], K), R)
s, i = 1, 1
p, q = K.one, K.one
for b, d in list(zip(B, D))[:-1]:
du = dup_degree(R[i - 1])
dv = dup_degree(R[i ])
dw = dup_degree(R[i + 1])
if du % 2 and dv % 2:
s = -s
lc, i = dup_LC(R[i], K), i + 1
p *= b**dv * lc**(du - dw)
q *= lc**(dv*(1 + d))
if s < 0:
p = -p
i = dup_degree(R[-2])
res = dup_LC(R[-1], K)**i
res = K.quo(res*p, q)
return res, R
def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K):
"""Wang/EEZ: Compute correct leading coefficients. """
C, J, v = [], [0]*len(E), u-1
for h in H:
c = dmp_one(v, K)
d = dup_LC(h, K)*cs
for i in reversed(xrange(len(E))):
k, e, (t, _) = 0, E[i], T[i]
while not (d % e):
d, k = d//e, k+1
if k != 0:
c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1
C.append(c)
if any([ not j for j in J ]):
raise ExtraneousFactors # pragma: no cover
CC, HH = [], []
for c, h in zip(C, H):
d = dmp_eval_tail(c, A, v, K)
lc = dup_LC(h, K)
if K.is_one(cs):
cc = lc//d
else:
g = K.gcd(lc, d)
d, cc = d//g, lc//g
h, cs = dup_mul_ground(h, d, K), cs//d
c = dmp_mul_ground(c, cc, v, K)
CC.append(c)
HH.append(h)
if K.is_one(cs):
return f, HH, CC
CCC, HHH = [], []
for c, h in zip(CC, HH):
CCC.append(dmp_mul_ground(c, cs, v, K))
HHH.append(dmp_mul_ground(h, cs, 0, K))
f = dmp_mul_ground(f, cs**(len(H)-1), u, K)
return f, HHH, CCC
def dup_pdiv(f, g, K):
"""
Polynomial pseudo-division in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_pdiv
>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])
>>> dup_pdiv(f, g, ZZ)
([2, 4], [20])
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r = [], f
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
N = df - dg + 1
lc_g = dup_LC(g, K)
while True:
dr = dup_degree(r)
if dr < dg:
break
lc_r = dup_LC(r, K)
j, N = dr-dg, N-1
Q = dup_mul_ground(q, lc_g, K)
q = dup_add_term(Q, lc_r, j, K)
R = dup_mul_ground(r, lc_g, K)
G = dup_mul_term(g, lc_r, j, K)
r = dup_sub(R, G, K)
c = lc_g**N
q = dup_mul_ground(q, c, K)
r = dup_mul_ground(r, c, K)
return q, r
def dup_pdiv(f, g, K):
"""
Polynomial pseudo-division in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_pdiv(x**2 + 1, 2*x - 4)
(2*x + 4, 20)
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r, dr = [], f, df
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
N = df - dg + 1
lc_g = dup_LC(g, K)
while True:
lc_r = dup_LC(r, K)
j, N = dr - dg, N - 1
Q = dup_mul_ground(q, lc_g, K)
q = dup_add_term(Q, lc_r, j, K)
R = dup_mul_ground(r, lc_g, K)
G = dup_mul_term(g, lc_r, j, K)
r = dup_sub(R, G, K)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
c = lc_g**N
q = dup_mul_ground(q, c, K)
r = dup_mul_ground(r, c, K)
return q, r
def dup_prs_resultant(f, g, K):
"""
Resultant algorithm in `K[x]` using subresultant PRS.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_prs_resultant(x**2 + 1, x**2 - 1)
(4, [x**2 + 1, x**2 - 1, -2])
"""
if not f or not g:
return (K.zero, [])
R, B, D = dup_inner_subresultants(f, g, K)
if dup_degree(R[-1]) > 0:
return (K.zero, R)
if R[-2] == [K.one]:
return (dup_LC(R[-1], K), R)
s, i = 1, 1
p, q = K.one, K.one
for b, d in list(zip(B, D))[:-1]:
du = dup_degree(R[i - 1])
dv = dup_degree(R[i ])
dw = dup_degree(R[i + 1])
if du % 2 and dv % 2:
s = -s
lc, i = dup_LC(R[i], K), i + 1
p *= b**dv * lc**(du - dw)
q *= lc**(dv*(1 + d))
if s < 0:
p = -p
i = dup_degree(R[-2])
res = dup_LC(R[-1], K)**i
res = K.quo(res*p, q)
return res, R
def dup_ff_div(f, g, K):
"""
Polynomial division with remainder over a field.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> R.dup_ff_div(x**2 + 1, 2*x - 4)
(1/2*x + 1, 5)
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r, dr = [], f, df
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
lc_r = dup_LC(r, K)
c = K.exquo(lc_r, lc_g)
j = dr - dg
q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)
r = dup_sub(r, h, K)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif dr == _dr and not K.is_Exact:
# remove leading term created by rounding error
r = dup_strip(r[1:])
dr = dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
return q, r
def dup_ff_div(f, g, K):
"""
Polynomial division with remainder over a field.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densearith import dup_ff_div
>>> f = QQ.map([1, 0, 1])
>>> g = QQ.map([2, -4])
>>> dup_ff_div(f, g, QQ)
([1/2, 1/1], [5/1])
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r = [], f
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
dr = dup_degree(r)
if dr < dg:
break
lc_r = dup_LC(r, K)
c = K.exquo(lc_r, lc_g)
j = dr - dg
q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)
r = dup_sub(r, h, K)
if not K.is_Exact:
r = dup_normal(r, K)
return q, r
def dup_rr_div(f, g, K):
"""
Univariate division with remainder over a ring.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_rr_div
>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])
>>> dup_rr_div(f, g, ZZ)
([], [1, 0, 1])
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r = [], f
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
dr = dup_degree(r)
if dr < dg:
break
lc_r = dup_LC(r, K)
if lc_r % lc_g:
break
c = K.exquo(lc_r, lc_g)
j = dr - dg
q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)
r = dup_sub(r, h, K)
return q, r
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_sqf_part(f, K):
"""
Returns square-free part of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_part
>>> dup_sqf_part([ZZ(1), ZZ(0), -ZZ(3), -ZZ(2)], ZZ)
[1, -1, -2]
"""
if not K.has_CharacteristicZero:
return dup_gf_sqf_part(f, K)
if not f:
return f
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
gcd = dup_gcd(f, dup_diff(f, 1, K), K)
sqf = dup_quo(f, gcd, K)
if K.has_Field or not K.is_Exact:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def _dup_right_decompose(f, s, K):
"""Helper function for :func:`_dup_decompose`."""
n = len(f) - 1
lc = dup_LC(f, K)
f = dup_to_raw_dict(f)
g = { s: K.one }
r = n // s
for i in range(1, s):
coeff = K.zero
for j in range(0, i):
if not n + j - i in f:
continue
if not s - j in g:
continue
fc, gc = f[n + j - i], g[s - j]
coeff += (i - r*j)*fc*gc
g[s - i] = K.quo(coeff, i*r*lc)
return dup_from_raw_dict(g, K)
def dup_ext_factor(f, K):
"""Factor univariate 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_include(r, K.dom)
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_shift(h, H, K)
factors[i] = h
factors = dup_trial_division(F, factors, K)
return lc, factors
def dup_monic(f, K):
"""
Divide all coefficients by ``LC(f)`` in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ, QQ
>>> R, x = ring("x", ZZ)
>>> R.dup_monic(3*x**2 + 6*x + 9)
x**2 + 2*x + 3
>>> R, x = ring("x", QQ)
>>> R.dup_monic(3*x**2 + 4*x + 2)
x**2 + 4/3*x + 2/3
"""
if not f:
return f
lc = dup_LC(f, K)
if K.is_one(lc):
return f
else:
return dup_exquo_ground(f, lc, K)
def dup_zz_mignotte_bound(f, K):
"""Mignotte bound for univariate polynomials in `K[x]`. """
a = dup_max_norm(f, K)
b = abs(dup_LC(f, K))
n = dup_degree(f)
return K.sqrt(K(n+1))*2**n*a*b
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_exquo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dup_sqf_part(f, K):
"""
Returns square-free part of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_sqf_part(x**3 - 3*x - 2)
x**2 - x - 2
"""
if K.is_FiniteField:
return dup_gf_sqf_part(f, K)
if not f:
return f
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
gcd = dup_gcd(f, dup_diff(f, 1, K), K)
sqf = dup_quo(f, gcd, K)
if K.has_Field:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def dup_zz_factor_sqf(f, K):
"""Factor square-free (non-primitive) polyomials in `Z[x]`. """
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, []
elif n == 1:
return cont, [(g, 1)]
if query('USE_IRREDUCIBLE_IN_FACTOR'):
if dup_zz_irreducible_p(g, K):
return cont, [(g, 1)]
factors = None
if query('USE_CYCLOTOMIC_FACTOR'):
factors = dup_zz_cyclotomic_factor(g, K)
if factors is None:
factors = dup_zz_zassenhaus(g, K)
return cont, _sort_factors(factors, multiple=False)
def dup_compose(f, g, K):
"""
Evaluate functional composition ``f(g)`` in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_compose(x**2 + x, x - 1)
x**2 - x
"""
if len(g) <= 1:
return dup_strip([dup_eval(f, dup_LC(g, K), K)])
if not f:
return []
h = [f[0]]
for c in f[1:]:
h = dup_mul(h, g, K)
h = dup_add_term(h, c, 0, K)
return h
def _dup_rr_trivial_gcd(f, g, K):
"""Handle trivial cases in GCD algorithm over a ring. """
if not (f or g):
return [], [], []
elif not f:
if K.is_nonnegative(dup_LC(g, K)):
return g, [], [K.one]
else:
return dup_neg(g, K), [], [-K.one]
elif not g:
if K.is_nonnegative(dup_LC(f, K)):
return f, [K.one], []
else:
return dup_neg(f, K), [-K.one], []
return None
def dup_compose(f, g, K):
"""
Evaluate functional composition ``f(g)`` in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_compose
>>> f = ZZ.map([1, 1, 0])
>>> g = ZZ.map([1, -1])
>>> dup_compose(f, g, ZZ)
[1, -1, 0]
"""
if len(g) <= 1:
return dup_strip([dup_eval(f, dup_LC(g, K), K)])
if not f:
return []
h = [f[0]]
for c in f[1:]:
h = dup_mul(h, g, K)
h = dup_add_term(h, c, 0, K)
return h
def dup_monic(f, K):
"""
Divides all coefficients by ``LC(f)`` in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densetools import dup_monic
>>> dup_monic([ZZ(3), ZZ(6), ZZ(9)], ZZ)
[1, 2, 3]
>>> dup_monic([QQ(3), QQ(4), QQ(2)], QQ)
[1/1, 4/3, 2/3]
"""
if not f:
return f
lc = dup_LC(f, K)
if K.is_one(lc):
return f
else:
return dup_exquo_ground(f, lc, K)
def dup_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_discriminant(x**2 + 2*x + 3)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d*(d - 1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c*K(s))
def dup_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_discriminant
>>> dup_discriminant([ZZ(1), ZZ(2), ZZ(3)], ZZ)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d*(d-1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c*K(s))
def dup_rr_div(f, g, K):
"""
Univariate division with remainder over a ring.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_div(x**2 + 1, 2*x - 4)
(0, x**2 + 1)
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r, dr = [], f, df
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
lc_r = dup_LC(r, K)
if lc_r % lc_g:
break
c = K.exquo(lc_r, lc_g)
j = dr - dg
q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)
r = dup_sub(r, h, K)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
return q, r
def test_dup_subresultants():
assert dup_resultant([], [], ZZ) == ZZ(0)
assert dup_resultant([ZZ(1)], [], ZZ) == ZZ(0)
assert dup_resultant([], [ZZ(1)], ZZ) == ZZ(0)
f = dup_normal([1,0,1,0,-3,-3,8,2,-5], ZZ)
g = dup_normal([3,0,5,0,-4,-9,21], ZZ)
a = dup_normal([15,0,-3,0,9], ZZ)
b = dup_normal([65,125,-245], ZZ)
c = dup_normal([9326,-12300], ZZ)
d = dup_normal([260708], ZZ)
assert dup_subresultants(f, g, ZZ) == [f, g, a, b, c, d]
assert dup_resultant(f, g, ZZ) == dup_LC(d, ZZ)
f = dup_normal([1,-2,1], ZZ)
g = dup_normal([1,0,-1], ZZ)
a = dup_normal([2,-2], ZZ)
assert dup_subresultants(f, g, ZZ) == [f, g, a]
assert dup_resultant(f, g, ZZ) == 0
f = dup_normal([1,0, 1], ZZ)
g = dup_normal([1,0,-1], ZZ)
a = dup_normal([-2], ZZ)
assert dup_subresultants(f, g, ZZ) == [f, g, a]
assert dup_resultant(f, g, ZZ) == 4
f = dup_normal([1,0,-1], ZZ)
g = dup_normal([1,-1,0,2], ZZ)
assert dup_resultant(f, g, ZZ) == 0
f = dup_normal([3,0,-1,0], ZZ)
g = dup_normal([5,0,1], ZZ)
assert dup_resultant(f, g, ZZ) == 64
f = dup_normal([1,-2,7], ZZ)
g = dup_normal([1,0,-1,5], ZZ)
assert dup_resultant(f, g, ZZ) == 265
f = dup_normal([1,-6,11,-6], ZZ)
g = dup_normal([1,-15,74,-120], ZZ)
assert dup_resultant(f, g, ZZ) == -8640
f = dup_normal([1,-6,11,-6], ZZ)
g = dup_normal([1,-10,29,-20], ZZ)
assert dup_resultant(f, g, ZZ) == 0
f = dup_normal([1,0,0,-1], ZZ)
g = dup_normal([1,2,2,-1], ZZ)
assert dup_resultant(f, g, ZZ) == 16
f = dup_normal([1,0,0,0,0,0,0,0,-2], ZZ)
g = dup_normal([1,-1], ZZ)
assert dup_resultant(f, g, ZZ) == -1
def dup_zz_heu_gcd(f, g, K):
"""
Heuristic polynomial GCD in `Z[x]`.
Given univariate polynomials `f` and `g` in `Z[x]`, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The final step is to verify if the result is the
correct GCD. This gives cofactors as a side effect.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_zz_heu_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
References
==========
1. [Liao95]_
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
df = dup_degree(f)
dg = dup_degree(g)
gcd, f, g = dup_extract(f, g, K)
if df == 0 or dg == 0:
return [gcd], f, g
f_norm = dup_max_norm(f, K)
g_norm = dup_max_norm(g, K)
B = K(2*min(f_norm, g_norm) + 29)
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dup_LC(f, K)),
g_norm // abs(dup_LC(g, K))) + 2)
for i in range(0, HEU_GCD_MAX):
ff = dup_eval(f, x, K)
gg = dup_eval(g, x, K)
if ff and gg:
h = K.gcd(ff, gg)
cff = ff // h
cfg = gg // h
h = _dup_zz_gcd_interpolate(h, x, K)
h = dup_primitive(h, K)[1]
cff_, r = dup_div(f, h, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg_
cff = _dup_zz_gcd_interpolate(cff, x, K)
h, r = dup_div(f, cff, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff, cfg_
cfg = _dup_zz_gcd_interpolate(cfg, x, K)
h, r = dup_div(g, cfg, K)
if not r:
cff_, r = dup_div(f, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS)
and the non-zero scalar subresultants of `f` and `g`.
By [1] Thm. 3, these are the constants '-c' (- to optimize
computation of sign).
The first subdeterminant is set to 1 by convention to match
the polynomial and the scalar subdeterminants.
If 'deg(f) < deg(g)', the subresultants of '(g,f)' are computed.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_subresultants(x**2 + 1, x**2 - 1)
([x**2 + 1, x**2 - 1, -2], [1, 1, 4])
References
==========
[1] W.S. Brown, The Subresultant PRS Algorithm.
ACM Transaction of Mathematical Software 4 (1978) 237-249
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
if not f:
return [], []
if not g:
return [f], [K.one]
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
lc = dup_LC(g, K)
c = lc**d
# Conventional first scalar subdeterminant is 1
S = [K.one, c]
c = -c
while h:
k = dup_degree(h)
R.append(h)
f, g, m, d = g, h, k, m - k
b = -lc * c**d
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
lc = dup_LC(g, K)
if d > 1: # abnormal case
q = c**(d - 1)
c = K.quo((-lc)**d, q)
else:
c = -lc
S.append(-c)
return R, S
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
C = int((n + 1)**(2 * n) * A**(2 * n - 1))
gamma = int(ceil(2 * log(C, 2)))
bound = int(2 * gamma * log(gamma))
for p in xrange(3, bound + 1):
if not isprime(p) or b % p == 0:
continue
p = K.convert(p)
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p, K):
break
l = int(ceil(log(2 * B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p, K)[1]:
modular.append(gf_to_int_poly(ff, p))
g = dup_zz_hensel_lift(p, f, modular, l, K)
T = set(range(len(g)))
factors, s = [], 1
while 2 * s <= len(T):
for S in subsets(T, s):
G, H = [b], [b]
S = set(S)
for i in S:
G = dup_mul(G, g[i], K)
for i in T - S:
H = dup_mul(H, g[i], K)
G = dup_trunc(G, p**l, K)
H = dup_trunc(H, p**l, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm * H_norm <= B:
T = T - S
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def LC(f):
"""Returns the leading coefficent of `f`. """
return dup_LC(f.rep, f.dom)
def dup_inner_subresultants(f, g, K):
"""
Subresultant PRS algorithm in `K[x]`.
Computes the subresultant polynomial remainder sequence (PRS) of `f`
and `g`, and the values for `\beta_i` and `\delta_i`. The last two
sequences of values are necessary for computing the resultant in
:func:`dup_prs_resultant`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_inner_subresultants
>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([1, 0, -1])
>>> dup_inner_subresultants(f, g, ZZ)
([[1, 0, 1], [1, 0, -1], [-2]], [-1, -1], [0, 2])
"""
n = dup_degree(f)
m = dup_degree(g)
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = (-K.one)**(d + 1)
c = -K.one
B, D = [b], [d]
if not f or not g:
return R, B, D
h = dup_prem(f, g, K)
h = dup_mul_ground(h, b, K)
while h:
k = dup_degree(h)
R.append(h)
lc = dup_LC(g, K)
if not d:
q = c
else:
q = c**(d - 1)
c = K.quo((-lc)**d, q)
b = -lc * c**(m - k)
f, g, m, d = g, h, k, m - k
B.append(b)
D.append(d)
h = dup_prem(f, g, K)
h = dup_quo_ground(h, b, K)
return R, B, D
def test_dup_LC():
assert dup_LC([], ZZ) == 0
assert dup_LC([2,3,4,5], ZZ) == 2
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
fc = f[-1]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1))*2**n*A*b))
C = int((n + 1)**(2*n)*A**(2*n - 1))
gamma = int(_ceil(2*_log(C, 2)))
bound = int(2*gamma*_log(gamma))
a = []
# choose a prime number `p` such that `f` be square free in Z_p
# if there are many factors in Z_p, choose among a few different `p`
# the one with fewer factors
for px in xrange(3, bound + 1):
if not isprime(px) or b % px == 0:
continue
px = K.convert(px)
F = gf_from_int_poly(f, px)
if not gf_sqf_p(F, px, K):
continue
fsqfx = gf_factor_sqf(F, px, K)[1]
a.append((px, fsqfx))
if len(fsqfx) < 15 or len(a) > 4:
break
p, fsqf = min(a, key=lambda x: len(x[1]))
l = int(_ceil(_log(2*B + 1, p)))
modular = [gf_to_int_poly(ff, p) for ff in fsqf]
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
pl = p**l
while 2*s <= len(T):
for S in subsets(sorted_T, s):
# lift the constant coefficient of the product `G` of the factors
# in the subset `S`; if it is does not divide `fc`, `G` does
# not divide the input polynomial
if b == 1:
q = 1
for i in S:
q = q*g[i][-1]
q = q % pl
if not _test_pl(fc, q, pl):
continue
else:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
G1 = dup_primitive(G, K)[1]
q = G1[-1]
if q and fc % q != 0:
continue
H = [b]
S = set(S)
T_S = T - S
if b == 1:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
for i in T_S:
H = dup_mul(H, g[i], K)
H = dup_trunc(H, pl, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm*H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def dup_zz_factor(f, K):
"""
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.domains 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**
1. [Gathen99]_
"""
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, []
elif n == 1:
return cont, [(g, 1)]
if query('USE_IRREDUCIBLE_IN_FACTOR'):
if dup_zz_irreducible_p(g, K):
return cont, [(g, 1)]
g = dup_sqf_part(g, K)
H, factors = None, []
if query('USE_CYCLOTOMIC_FACTOR'):
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_cyclotomic_p(f, K, irreducible=False):
"""
Efficiently test if ``f`` is a cyclotomic polynomial.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1
>>> R.dup_cyclotomic_p(f)
False
>>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1
>>> R.dup_cyclotomic_p(g)
True
"""
if K.is_QQ:
try:
K0, K = K, K.get_ring()
f = dup_convert(f, K0, K)
except CoercionFailed:
return False
elif not K.is_ZZ:
return False
lc = dup_LC(f, K)
tc = dup_TC(f, K)
if lc != 1 or (tc != -1 and tc != 1):
return False
if not irreducible:
coeff, factors = dup_factor_list(f, K)
if coeff != K.one or factors != [(f, 1)]:
return False
n = dup_degree(f)
g, h = [], []
for i in range(n, -1, -2):
g.insert(0, f[i])
for i in range(n - 1, -1, -2):
h.insert(0, f[i])
g = dup_sqr(dup_strip(g), K)
h = dup_sqr(dup_strip(h), K)
F = dup_sub(g, dup_lshift(h, 1, K), K)
if K.is_negative(dup_LC(F, K)):
F = dup_neg(F, K)
if F == f:
return True
g = dup_mirror(f, K)
if K.is_negative(dup_LC(g, K)):
g = dup_neg(g, K)
if F == g and dup_cyclotomic_p(g, K):
return True
G = dup_sqf_part(F, K)
if dup_sqr(G, K) == F and dup_cyclotomic_p(G, K):
return True
return False
def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [dup_trunc(F, p**l, K)]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
# print("g: %s * %s" % (g,f_i))
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
# print("g: %s" % g)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, q = gf_gcdex(g, h, p, K)
# print("gcdex %s %s %d = %s %s %s)" % (g,h,p,q,s,t))
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
# print("h: %s" % f_list[k])
for _ in range(1, d + 1):
# print("go %d %s %s %s %s %s" % (m, f, g, h, s, t))
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
# print("go %d %s %s %s %s %s" % (m, f, g, h, s, t))
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def dup_zz_factor(f, K):
"""
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))
Examples
========
Consider the polynomial `f = 2*x**4 - 2`::
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_zz_factor(2*x**4 - 2)
(2, [(x - 1, 1), (x + 1, 1), (x**2 + 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
==========
.. [1] [Gathen99]_
"""
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, []
elif n == 1:
return cont, [(g, 1)]
if query('USE_IRREDUCIBLE_IN_FACTOR'):
if dup_zz_irreducible_p(g, K):
return cont, [(g, 1)]
g = dup_sqf_part(g, K)
H = None
if query('USE_CYCLOTOMIC_FACTOR'):
H = dup_zz_cyclotomic_factor(g, K)
if H is None:
H = dup_zz_zassenhaus(g, K)
factors = dup_trial_division(f, H, K)
return cont, factors
