def dup_ff_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a field.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_ff_prs_gcd
>>> f = QQ.map([1, 0, -1])
>>> g = QQ.map([1, -3, 2])
>>> dup_ff_prs_gcd(f, g, QQ)
([1/1, -1/1], [1/1, 1/1], [1/1, -2/1])
"""
result = _dup_ff_trivial_gcd(f, g, K)
if result is not None:
return result
h = dup_subresultants(f, g, K)[-1]
h = dup_monic(h, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_ff_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a field.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_ff_prs_gcd
>>> f = QQ.map([1, 0, -1])
>>> g = QQ.map([1, -3, 2])
>>> dup_ff_prs_gcd(f, g, QQ)
([1/1, -1/1], [1/1, 1/1], [1/1, -2/1])
"""
result = _dup_ff_trivial_gcd(f, g, K)
if result is not None:
return result
h = dup_subresultants(f, g, K)[-1]
h = dup_monic(h, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_ff_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a field.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> R.dup_ff_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
result = _dup_ff_trivial_gcd(f, g, K)
if result is not None:
return result
h = dup_subresultants(f, g, K)[-1]
h = dup_monic(h, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_ff_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a field.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> R.dup_ff_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
result = _dup_ff_trivial_gcd(f, g, K)
if result is not None:
return result
h = dup_subresultants(f, g, K)[-1]
h = dup_monic(h, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
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_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2)
[(x, 1), (x + 2, 4)]
"""
if not f:
raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial")
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
def test_dup_div():
f, g, q, r = [5,4,3,2,1], [1,2,3], [5,-6,0], [20,1]
assert dup_div(f, g, ZZ) == (q, r)
assert dup_quo(f, g, ZZ) == q
assert dup_rem(f, g, ZZ) == r
raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
f, g, q, r = [5,4,3,2,1,0], [1,2,0,0,9], [5,-6], [15,2,-44,54]
assert dup_div(f, g, ZZ) == (q, r)
assert dup_quo(f, g, ZZ) == q
assert dup_rem(f, g, ZZ) == r
raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
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_gcdex(f, g)
(-1/5*x + 3/5, 1/5*x**2 - 6/5*x + 2, x + 1)
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
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_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_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2)
[(x, 1), (x + 2, 4)]
"""
if not f:
raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial")
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_gcdex(f, g, QQ)
([-1/5, 3/5], [1/5, -6/5, 2/1], [1/1, 1/1])
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_rr_lcm(f, g, K):
"""
Computes polynomial LCM over a ring in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_rr_lcm
>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -3, 2])
>>> dup_rr_lcm(f, g, ZZ)
[1, -2, -1, 2]
"""
fc, f = dup_primitive(f, K)
gc, g = dup_primitive(g, K)
c = K.lcm(fc, gc)
h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)
return dup_mul_ground(h, c, K)
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.is_Field:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_gcdex(f, g, QQ)
([-1/5, 3/5], [1/5, -6/5, 2/1], [1/1, 1/1])
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_rr_lcm(f, g, K):
"""
Computes polynomial LCM over a ring in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_rr_lcm
>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -3, 2])
>>> dup_rr_lcm(f, g, ZZ)
[1, -2, -1, 2]
"""
fc, f = dup_primitive(f, K)
gc, g = dup_primitive(g, K)
c = K.lcm(fc, gc)
h = dup_quo(dup_mul(f, g, K),
dup_gcd(f, g, K), K)
return dup_mul_ground(h, c, K)
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
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_gcdex(f, g)
(-1/5*x + 3/5, 1/5*x**2 - 6/5*x + 2, x + 1)
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_zz_cyclotomic_poly(n, K):
"""Efficiently generate n-th cyclotomic polnomial. """
h = [K.one,-K.one]
for p, k in factorint(n).iteritems():
h = dup_quo(dup_inflate(h, p, K), h, K)
h = dup_inflate(h, p**(k-1), K)
return h
def dup_zz_cyclotomic_poly(n, K):
"""Efficiently generate n-th cyclotomic polnomial. """
h = [K.one, -K.one]
for p, k in factorint(n).iteritems():
h = dup_quo(dup_inflate(h, p, K), h, K)
h = dup_inflate(h, p**(k - 1), K)
return h
def dup_rr_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a ring.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_rr_prs_gcd
>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -3, 2])
>>> dup_rr_prs_gcd(f, g, ZZ)
([1, -1], [1, 1], [1, -2])
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
fc, F = dup_primitive(f, K)
gc, G = dup_primitive(g, K)
c = K.gcd(fc, gc)
h = dup_subresultants(F, G, K)[-1]
_, h = dup_primitive(h, K)
if K.is_negative(dup_LC(h, K)):
c = -c
h = dup_mul_ground(h, c, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_rr_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a ring.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_rr_prs_gcd
>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -3, 2])
>>> dup_rr_prs_gcd(f, g, ZZ)
([1, -1], [1, 1], [1, -2])
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
fc, F = dup_primitive(f, K)
gc, G = dup_primitive(g, K)
c = K.gcd(fc, gc)
h = dup_subresultants(F, G, K)[-1]
_, h = dup_primitive(h, K)
if K.is_negative(dup_LC(h, K)):
c = -c
h = dup_mul_ground(h, c, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def _dup_cyclotomic_decompose(n, K):
H = [[K.one, -K.one]]
for p, k in factorint(n).iteritems():
Q = [dup_quo(dup_inflate(h, p, K), h, K) for h in H]
H.extend(Q)
for i in xrange(1, k):
Q = [dup_inflate(q, p, K) for q in Q]
H.extend(Q)
return H
def _dup_cyclotomic_decompose(n, K):
H = [[K.one,-K.one]]
for p, k in factorint(n).iteritems():
Q = [ dup_quo(dup_inflate(h, p, K), h, K) for h in H ]
H.extend(Q)
for i in xrange(1, k):
Q = [ dup_inflate(q, p, K) for q in Q ]
H.extend(Q)
return H
def dup_rr_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a ring.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
fc, F = dup_primitive(f, K)
gc, G = dup_primitive(g, K)
c = K.gcd(fc, gc)
h = dup_subresultants(F, G, K)[-1]
_, h = dup_primitive(h, K)
if K.is_negative(dup_LC(h, K)):
c = -c
h = dup_mul_ground(h, c, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_rr_prs_gcd(f, g, K):
"""
Computes polynomial GCD using subresultants over a ring.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,
and ``cfg = quo(g, h)``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_prs_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
fc, F = dup_primitive(f, K)
gc, G = dup_primitive(g, K)
c = K.gcd(fc, gc)
h = dup_subresultants(F, G, K)[-1]
_, h = dup_primitive(h, K)
if K.is_negative(dup_LC(h, K)):
c = -c
h = dup_mul_ground(h, c, K)
cff = dup_quo(f, h, K)
cfg = dup_quo(g, h, K)
return h, cff, cfg
def dup_ff_lcm(f, g, K):
"""
Computes polynomial LCM over a field in ``K[x]``.
**Examples**
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_ff_lcm
>>> f = [QQ(1,2), QQ(7,4), QQ(3,2)]
>>> g = [QQ(1,2), QQ(1), QQ(0)]
>>> dup_ff_lcm(f, g, QQ)
[1/1, 7/2, 3/1, 0/1]
"""
h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)
return dup_monic(h, K)
def dup_ff_lcm(f, g, K):
"""
Computes polynomial LCM over a field in ``K[x]``.
**Examples**
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_ff_lcm
>>> f = [QQ(1,2), QQ(7,4), QQ(3,2)]
>>> g = [QQ(1,2), QQ(1), QQ(0)]
>>> dup_ff_lcm(f, g, QQ)
[1/1, 7/2, 3/1, 0/1]
"""
h = dup_quo(dup_mul(f, g, K),
dup_gcd(f, g, K), K)
return dup_monic(h, K)
def dup_ff_lcm(f, g, K):
"""
Computes polynomial LCM over a field in `K[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = QQ(1,2)*x**2 + QQ(7,4)*x + QQ(3,2)
>>> g = QQ(1,2)*x**2 + x
>>> R.dup_ff_lcm(f, g)
x**3 + 7/2*x**2 + 3*x
"""
h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)
return dup_monic(h, K)
def dup_ff_lcm(f, g, K):
"""
Computes polynomial LCM over a field in `K[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = QQ(1,2)*x**2 + QQ(7,4)*x + QQ(3,2)
>>> g = QQ(1,2)*x**2 + x
>>> R.dup_ff_lcm(f, g)
x**3 + 7/2*x**2 + 3*x
"""
h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)
return dup_monic(h, K)
def dup_rr_lcm(f, g, K):
"""
Computes polynomial LCM over a ring in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_lcm(x**2 - 1, x**2 - 3*x + 2)
x**3 - 2*x**2 - x + 2
"""
fc, f = dup_primitive(f, K)
gc, g = dup_primitive(g, K)
c = K.lcm(fc, gc)
h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)
return dup_mul_ground(h, c, K)
def dup_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_gff_list
>>> f = ZZ.map([1, 2, -1, -2, 0, 0])
>>> dup_gff_list(f, ZZ)
[([1, 0], 1), ([1, 2], 4)]
"""
if not f:
raise ValueError(
"greatest factorial factorization doesn't exist for a zero polynomial"
)
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
def dup_rr_lcm(f, g, K):
"""
Computes polynomial LCM over a ring in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_rr_lcm(x**2 - 1, x**2 - 3*x + 2)
x**3 - 2*x**2 - x + 2
"""
fc, f = dup_primitive(f, K)
gc, g = dup_primitive(g, K)
c = K.lcm(fc, gc)
h = dup_quo(dup_mul(f, g, K), dup_gcd(f, g, K), K)
return dup_mul_ground(h, c, K)
def dup_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_gff_list
>>> f = ZZ.map([1, 2, -1, -2, 0, 0])
>>> dup_gff_list(f, ZZ)
[([1, 0], 1), ([1, 2], 4)]
"""
if not f:
raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial")
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
