def dmp_cancel(f, g, u, K, multout=True):
"""
Cancel common factors in a rational function ``f/g``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_cancel
>>> f = ZZ.map([[2], [0], [-2]])
>>> g = ZZ.map([[1], [-2], [1]])
>>> dmp_cancel(f, g, 1, ZZ)
([[2], [2]], [[1], [-1]])
"""
if dmp_zero_p(f, u) or dmp_zero_p(g, u):
if multout:
return f, g
else:
return K.one, K.one, f, g
K0 = None
if K.has_Field and K.has_assoc_Ring:
K0, K = K, K.get_ring()
cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
else:
cp, cq = K.one, K.one
_, p, q = dmp_inner_gcd(f, g, u, K)
if K0 is not None:
p = dmp_convert(p, u, K, K0)
q = dmp_convert(q, u, K, K0)
K = K0
p_neg = K.is_negative(dmp_ground_LC(p, u, K))
q_neg = K.is_negative(dmp_ground_LC(q, u, K))
if p_neg and q_neg:
p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
elif p_neg:
cp, p = -cp, dmp_neg(p, u, K)
elif q_neg:
cp, q = -cp, dmp_neg(q, u, K)
if not multout:
return cp, cq, p, q
p = dmp_mul_ground(p, cp, u, K)
q = dmp_mul_ground(q, cq, u, K)
return p, q
def test_dmp_mul_ground():
assert dmp_mul_ground(f_0, ZZ(2), 2, ZZ) == [
[[ZZ(2),ZZ(4),ZZ(6)], [ZZ(4)]],
[[ZZ(6)]],
[[ZZ(8),ZZ(10),ZZ(12)], [ZZ(2),ZZ(4),ZZ(2)], [ZZ(2)]]
]
assert dmp_mul_ground(F_0, QQ(1,2), 2, QQ) == [
[[QQ(1,14),QQ(2,14),QQ(3,14)], [QQ(2,14)]],
[[QQ(3,14)]],
[[QQ(4,14),QQ(5,14),QQ(6,14)], [QQ(1,14),QQ(2,14),QQ(1,14)], [QQ(1,14)]]
]
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 dmp_cancel(f, g, u, K, include=True):
"""
Cancel common factors in a rational function `f/g`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_cancel(2*x**2 - 2, x**2 - 2*x + 1)
(2*x + 2, x - 1)
"""
K0 = None
if K.has_Field and K.has_assoc_Ring:
K0, K = K, K.get_ring()
cq, f = dmp_clear_denoms(f, u, K0, K, convert=True)
cp, g = dmp_clear_denoms(g, u, K0, K, convert=True)
else:
cp, cq = K.one, K.one
_, p, q = dmp_inner_gcd(f, g, u, K)
if K0 is not None:
_, cp, cq = K.cofactors(cp, cq)
p = dmp_convert(p, u, K, K0)
q = dmp_convert(q, u, K, K0)
K = K0
p_neg = K.is_negative(dmp_ground_LC(p, u, K))
q_neg = K.is_negative(dmp_ground_LC(q, u, K))
if p_neg and q_neg:
p, q = dmp_neg(p, u, K), dmp_neg(q, u, K)
elif p_neg:
cp, p = -cp, dmp_neg(p, u, K)
elif q_neg:
cp, q = -cp, dmp_neg(q, u, K)
if not include:
return cp, cq, p, q
p = dmp_mul_ground(p, cp, u, K)
q = dmp_mul_ground(q, cq, u, K)
return p, q
def dmp_sqf_list_include(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x**5 + 2*x**4*y + x**3*y**2
>>> R.dmp_sqf_list_include(f)
[(1, 1), (x + y, 2), (x, 3)]
>>> R.dmp_sqf_list_include(f, all=True)
[(1, 1), (x + y, 2), (x, 3)]
"""
if not u:
return dup_sqf_list_include(f, K, all=all)
coeff, factors = dmp_sqf_list(f, u, K, all=all)
if factors and factors[0][1] == 1:
g = dmp_mul_ground(factors[0][0], coeff, u, K)
return [(g, 1)] + factors[1:]
else:
g = dmp_ground(coeff, u)
return [(g, 1)] + factors
def dmp_eval(f, a, u, K):
"""
Evaluate a polynomial at ``x_0 = a`` in ``K[X]`` using the Horner scheme.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_eval(2*x*y + 3*x + y + 2, 2)
5*y + 8
"""
if not u:
return dup_eval(f, a, K)
if not a:
return dmp_TC(f, K)
result, v = dmp_LC(f, K), u - 1
for coeff in f[1:]:
result = dmp_mul_ground(result, a, v, K)
result = dmp_add(result, coeff, v, K)
return result
def dmp_sqf_list_include(f, u, 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 dmp_sqf_list_include
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0], [], [], []])
>>> dmp_sqf_list_include(f, 1, ZZ)
[([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)]
>>> dmp_sqf_list_include(f, 1, ZZ, all=True)
[([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)]
"""
if not u:
return dup_sqf_list_include(f, K, all=all)
coeff, factors = dmp_sqf_list(f, u, K, all=all)
if factors and factors[0][1] == 1:
g = dmp_mul_ground(factors[0][0], coeff, u, K)
return [(g, 1)] + factors[1:]
else:
g = dmp_ground(coeff, u)
return [(g, 1)] + factors
def dmp_rr_lcm(f, g, u, K):
"""
Computes polynomial LCM over a ring in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_rr_lcm(f, g)
x**3 + 2*x**2*y + x*y**2
"""
fc, f = dmp_ground_primitive(f, u, K)
gc, g = dmp_ground_primitive(g, u, K)
c = K.lcm(fc, gc)
h = dmp_quo(dmp_mul(f, g, u, K),
dmp_gcd(f, g, u, K), u, K)
return dmp_mul_ground(h, c, u, K)
def dmp_discriminant(f, u, K):
"""
Computes discriminant of a polynomial in `K[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y,z,t = ring("x,y,z,t", ZZ)
>>> R.dmp_discriminant(x**2*y + x*z + t)
-4*y*t + z**2
"""
if not u:
return dup_discriminant(f, K)
d, v = dmp_degree(f, u), u - 1
if d <= 0:
return dmp_zero(v)
else:
s = (-1)**((d*(d - 1)) // 2)
c = dmp_LC(f, K)
r = dmp_resultant(f, dmp_diff(f, 1, u, K), u, K)
c = dmp_mul_ground(c, K(s), v, K)
return dmp_quo(r, c, v, K)
def dmp_discriminant(f, u, K):
"""
Computes discriminant of a polynomial in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_discriminant
>>> f = ZZ.map([[[[1]], [[]]], [[[1], []]], [[[1, 0]]]])
>>> dmp_discriminant(f, 3, ZZ)
[[[-4, 0]], [[1], [], []]]
"""
if not u:
return dup_discriminant(f, K)
d, v = dmp_degree(f, u), u-1
if d <= 0:
return dmp_zero(v)
else:
s = (-1)**((d*(d-1)) // 2)
c = dmp_LC(f, K)
r = dmp_resultant(f, dmp_diff(f, 1, u, K), u, K)
c = dmp_mul_ground(c, K(s), v, K)
return dmp_exquo(r, c, v, K)
def dmp_eval(f, a, u, K):
"""
Evaluate a polynomial at ``x_0 = a`` in ``K[X]`` using the Horner scheme.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_eval
>>> f = ZZ.map([[2, 3], [1, 2]])
>>> dmp_eval(f, 2, 1, ZZ)
[5, 8]
"""
if not u:
return dup_eval(f, a, K)
if not a:
return dmp_TC(f, K)
result, v = dmp_LC(f, K), u - 1
for coeff in f[1:]:
result = dmp_mul_ground(result, a, v, K)
result = dmp_add(result, coeff, v, K)
return result
def dmp_clear_denoms(f, u, K0, K1=None, convert=False):
"""
Clear denominators, i.e. transform ``K_0`` to ``K_1``.
Examples
========
>>> from sympy.polys.domains import QQ, ZZ
>>> from sympy.polys.densetools import dmp_clear_denoms
>>> f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]
>>> dmp_clear_denoms(f, 1, QQ, convert=False)
(6, [[3/1], [2/1, 6/1]])
>>> f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]
>>> dmp_clear_denoms(f, 1, QQ, convert=True)
(6, [[3], [2, 6]])
"""
if not u:
return dup_clear_denoms(f, K0, K1, convert=convert)
if K1 is None:
K1 = K0.get_ring()
common = _rec_clear_denoms(f, u, K0, K1)
if not K1.is_one(common):
f = dmp_mul_ground(f, common, u, K0)
if not convert:
return common, f
else:
return common, dmp_convert(f, u, K0, K1)
def dmp_rr_lcm(f, g, u, K):
"""
Computes polynomial LCM over a ring in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_rr_lcm
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0]])
>>> g = ZZ.map([[1], [1, 0], []])
>>> dmp_rr_lcm(f, g, 1, ZZ)
[[1], [2, 0], [1, 0, 0], []]
"""
fc, f = dmp_ground_primitive(f, u, K)
gc, g = dmp_ground_primitive(g, u, K)
c = K.lcm(fc, gc)
h = dmp_exquo(dmp_mul(f, g, u, K),
dmp_gcd(f, g, u, K), u, K)
return dmp_mul_ground(h, c, u, K)
def dmp_qq_heu_gcd(f, g, u, K0):
"""
Heuristic polynomial GCD in `Q[X]`.
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,y, = ring("x,y", QQ)
>>> f = QQ(1,4)*x**2 + x*y + y**2
>>> g = QQ(1,2)*x**2 + x*y
>>> R.dmp_qq_heu_gcd(f, g)
(x + 2*y, 1/4*x + 1/2*y, 1/2*x)
"""
result = _dmp_ff_trivial_gcd(f, g, u, K0)
if result is not None:
return result
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)
h = dmp_convert(h, u, K1, K0)
c = dmp_ground_LC(h, u, K0)
h = dmp_ground_monic(h, u, K0)
cff = dmp_convert(cff, u, K1, K0)
cfg = dmp_convert(cfg, u, K1, K0)
cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)
return h, cff, cfg
def dmp_qq_heu_gcd(f, g, u, K0):
"""
Heuristic polynomial GCD in `Q[X]`.
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,y, = ring("x,y", QQ)
>>> f = QQ(1,4)*x**2 + x*y + y**2
>>> g = QQ(1,2)*x**2 + x*y
>>> R.dmp_qq_heu_gcd(f, g)
(x + 2*y, 1/4*x + 1/2*y, 1/2*x)
"""
result = _dmp_ff_trivial_gcd(f, g, u, K0)
if result is not None:
return result
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)
h = dmp_convert(h, u, K1, K0)
c = dmp_ground_LC(h, u, K0)
h = dmp_ground_monic(h, u, K0)
cff = dmp_convert(cff, u, K1, K0)
cfg = dmp_convert(cfg, u, K1, K0)
cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)
return h, cff, cfg
def dmp_qq_heu_gcd(f, g, u, K0):
"""
Heuristic polynomial GCD in `Q[X]`.
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 dmp_qq_heu_gcd
>>> f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]
>>> g = [[QQ(1,2)], [QQ(1), QQ(0)], []]
>>> dmp_qq_heu_gcd(f, g, 1, QQ)
([[1/1], [2/1, 0/1]], [[1/4], [1/2, 0/1]], [[1/2], []])
"""
result = _dmp_ff_trivial_gcd(f, g, u, K0)
if result is not None:
return result
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)
h = dmp_convert(h, u, K1, K0)
c = dmp_ground_LC(h, u, K0)
h = dmp_ground_monic(h, u, K0)
cff = dmp_convert(cff, u, K1, K0)
cfg = dmp_convert(cfg, u, K1, K0)
cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)
return h, cff, cfg
def dmp_qq_heu_gcd(f, g, u, K0):
"""
Heuristic polynomial GCD in ``Q[X]``.
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 dmp_qq_heu_gcd
>>> f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]
>>> g = [[QQ(1,2)], [QQ(1), QQ(0)], []]
>>> dmp_qq_heu_gcd(f, g, 1, QQ)
([[1/1], [2/1, 0/1]], [[1/4], [1/2, 0/1]], [[1/2], []])
"""
result = _dmp_ff_trivial_gcd(f, g, u, K0)
if result is not None:
return result
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1)
h = dmp_convert(h, u, K1, K0)
c = dmp_ground_LC(h, u, K0)
h = dmp_ground_monic(h, u, K0)
cff = dmp_convert(cff, u, K1, K0)
cfg = dmp_convert(cfg, u, K1, K0)
cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0)
cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0)
return h, cff, cfg
def test_dmp_ground_primitive():
assert dmp_ground_primitive([[]], 1, ZZ) == (ZZ(0), [[]])
assert dmp_ground_primitive(f_0, 2, ZZ) == (ZZ(1), f_0)
assert dmp_ground_primitive(dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2,
ZZ) == (ZZ(2), f_0)
assert dmp_ground_primitive(f_1, 2, ZZ) == (ZZ(1), f_1)
assert dmp_ground_primitive(dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2,
ZZ) == (ZZ(3), f_1)
assert dmp_ground_primitive(f_2, 2, ZZ) == (ZZ(1), f_2)
assert dmp_ground_primitive(dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2,
ZZ) == (ZZ(4), f_2)
assert dmp_ground_primitive(f_3, 2, ZZ) == (ZZ(1), f_3)
assert dmp_ground_primitive(dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2,
ZZ) == (ZZ(5), f_3)
assert dmp_ground_primitive(f_4, 2, ZZ) == (ZZ(1), f_4)
assert dmp_ground_primitive(dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2,
ZZ) == (ZZ(6), f_4)
assert dmp_ground_primitive(f_5, 2, ZZ) == (ZZ(1), f_5)
assert dmp_ground_primitive(dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2,
ZZ) == (ZZ(7), f_5)
assert dmp_ground_primitive(f_6, 3, ZZ) == (ZZ(1), f_6)
assert dmp_ground_primitive(dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3,
ZZ) == (ZZ(8), f_6)
assert dmp_ground_primitive([[ZZ(2)]], 1, ZZ) == (ZZ(2), [[ZZ(1)]])
assert dmp_ground_primitive([[QQ(2)]], 1, QQ) == (QQ(1), [[QQ(2)]])
def dmp_diff(f, m, u, K):
"""
``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_diff
>>> f = ZZ.map([[1, 2, 3], [2, 3, 1]])
>>> dmp_diff(f, 1, 1, ZZ)
[[1, 2, 3]]
>>> dmp_diff(f, 2, 1, ZZ)
[[]]
"""
if not u:
return dup_diff(f, m, K)
if m <= 0:
return f
n = dmp_degree(f, u)
if n < m:
return dmp_zero(u)
deriv, v = [], u - 1
if m == 1:
for coeff in f[:-m]:
deriv.append(dmp_mul_ground(coeff, K(n), v, K))
n -= 1
else:
for coeff in f[:-m]:
k = n
for i in xrange(n - 1, n - m, -1):
k *= i
deriv.append(dmp_mul_ground(coeff, K(k), v, K))
n -= 1
return dmp_strip(deriv, u)
def dmp_diff(f, m, u, K):
"""
``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1
>>> R.dmp_diff(f, 1)
y**2 + 2*y + 3
>>> R.dmp_diff(f, 2)
0
"""
if not u:
return dup_diff(f, m, K)
if m <= 0:
return f
n = dmp_degree(f, u)
if n < m:
return dmp_zero(u)
deriv, v = [], u - 1
if m == 1:
for coeff in f[:-m]:
deriv.append(dmp_mul_ground(coeff, K(n), v, K))
n -= 1
else:
for coeff in f[:-m]:
k = n
for i in range(n - 1, n - m, -1):
k *= i
deriv.append(dmp_mul_ground(coeff, K(k), v, K))
n -= 1
return dmp_strip(deriv, u)
def dmp_diff(f, m, u, K):
"""
``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1
>>> R.dmp_diff(f, 1)
y**2 + 2*y + 3
>>> R.dmp_diff(f, 2)
0
"""
if not u:
return dup_diff(f, m, K)
if m <= 0:
return f
n = dmp_degree(f, u)
if n < m:
return dmp_zero(u)
deriv, v = [], u - 1
if m == 1:
for coeff in f[:-m]:
deriv.append(dmp_mul_ground(coeff, K(n), v, K))
n -= 1
else:
for coeff in f[:-m]:
k = n
for i in range(n - 1, n - m, -1):
k *= i
deriv.append(dmp_mul_ground(coeff, K(k), v, K))
n -= 1
return dmp_strip(deriv, u)
def test_dmp_ground_content():
assert dmp_ground_content([[]], 1, ZZ) == ZZ(0)
assert dmp_ground_content([[]], 1, QQ) == QQ(0)
assert dmp_ground_content([[1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[-1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[1], [1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[2], [2]], 1, ZZ) == ZZ(2)
assert dmp_ground_content([[1], [2], [1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[2], [4], [2]], 1, ZZ) == ZZ(2)
assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == QQ(1)
assert dmp_ground_content(f_0, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == ZZ(2)
assert dmp_ground_content(f_1, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == ZZ(3)
assert dmp_ground_content(f_2, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == ZZ(4)
assert dmp_ground_content(f_3, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == ZZ(5)
assert dmp_ground_content(f_4, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == ZZ(6)
assert dmp_ground_content(f_5, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == ZZ(7)
assert dmp_ground_content(f_6, 3, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == ZZ(8)
def test_dmp_ground_content():
assert dmp_ground_content([[]], 1, ZZ) == ZZ(0)
assert dmp_ground_content([[]], 1, QQ) == QQ(0)
assert dmp_ground_content([[1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[-1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[1],[1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[2],[2]], 1, ZZ) == ZZ(2)
assert dmp_ground_content([[1],[2],[1]], 1, ZZ) == ZZ(1)
assert dmp_ground_content([[2],[4],[2]], 1, ZZ) == ZZ(2)
assert dmp_ground_content([[QQ(2,3)],[QQ(4,9)]], 1, QQ) == QQ(2,9)
assert dmp_ground_content([[QQ(2,3)],[QQ(4,5)]], 1, QQ) == QQ(2,15)
assert dmp_ground_content(f_0, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == ZZ(2)
assert dmp_ground_content(f_1, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == ZZ(3)
assert dmp_ground_content(f_2, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == ZZ(4)
assert dmp_ground_content(f_3, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == ZZ(5)
assert dmp_ground_content(f_4, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == ZZ(6)
assert dmp_ground_content(f_5, 2, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == ZZ(7)
assert dmp_ground_content(f_6, 3, ZZ) == ZZ(1)
assert dmp_ground_content(dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == ZZ(8)
def dmp_diff(f, m, u, K):
"""
``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_diff
>>> f = ZZ.map([[1, 2, 3], [2, 3, 1]])
>>> dmp_diff(f, 1, 1, ZZ)
[[1, 2, 3]]
>>> dmp_diff(f, 2, 1, ZZ)
[[]]
"""
if not u:
return dup_diff(f, m, K)
if m <= 0:
return f
n = dmp_degree(f, u)
if n < m:
return dmp_zero(u)
deriv, v = [], u - 1
if m == 1:
for coeff in f[:-m]:
deriv.append(dmp_mul_ground(coeff, K(n), v, K))
n -= 1
else:
for coeff in f[:-m]:
k = n
for i in xrange(n - 1, n - m, -1):
k *= i
deriv.append(dmp_mul_ground(coeff, K(k), v, K))
n -= 1
return dmp_strip(deriv, u)
def test_dmp_ground_primitive():
assert dmp_ground_primitive([[]], 1, ZZ) == (ZZ(0), [[]])
assert dmp_ground_primitive(f_0, 2, ZZ) == (ZZ(1), f_0)
assert dmp_ground_primitive(dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == (ZZ(2), f_0)
assert dmp_ground_primitive(f_1, 2, ZZ) == (ZZ(1), f_1)
assert dmp_ground_primitive(dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == (ZZ(3), f_1)
assert dmp_ground_primitive(f_2, 2, ZZ) == (ZZ(1), f_2)
assert dmp_ground_primitive(dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == (ZZ(4), f_2)
assert dmp_ground_primitive(f_3, 2, ZZ) == (ZZ(1), f_3)
assert dmp_ground_primitive(dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == (ZZ(5), f_3)
assert dmp_ground_primitive(f_4, 2, ZZ) == (ZZ(1), f_4)
assert dmp_ground_primitive(dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == (ZZ(6), f_4)
assert dmp_ground_primitive(f_5, 2, ZZ) == (ZZ(1), f_5)
assert dmp_ground_primitive(dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == (ZZ(7), f_5)
assert dmp_ground_primitive(f_6, 3, ZZ) == (ZZ(1), f_6)
assert dmp_ground_primitive(dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == (ZZ(8), f_6)
assert dmp_ground_primitive([[ZZ(2)]], 1, ZZ) == (ZZ(2), [[ZZ(1)]])
assert dmp_ground_primitive([[QQ(2)]], 1, QQ) == (QQ(1), [[QQ(2)]])
def dmp_factor_list_include(f, u, K):
"""Factor polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list_include(f, K)
coeff, factors = dmp_factor_list(f, u, K)
if not factors:
return [(dmp_ground(coeff, u), 1)]
else:
g = dmp_mul_ground(factors[0][0], coeff, u, K)
return [(g, factors[0][1])] + factors[1:]
def dmp_factor_list_include(f, u, K):
"""Factor polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list_include(f, K)
coeff, factors = dmp_factor_list(f, u, K)
if not factors:
return [(dmp_ground(coeff, u), 1)]
else:
g = dmp_mul_ground(factors[0][0], coeff, u, K)
return [(g, factors[0][1])] + factors[1:]
def dmp_clear_denoms(f, u, K0, K1=None, convert=False):
"""
Clear denominators, i.e. transform ``K_0`` to ``K_1``.
Examples
========
>>> from sympy.polys.domains import QQ, ZZ
>>> from sympy.polys.densetools import dmp_clear_denoms
>>> f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]
>>> dmp_clear_denoms(f, 1, QQ, convert=False)
(6, [[3/1], [2/1, 6/1]])
>>> f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]
>>> dmp_clear_denoms(f, 1, QQ, convert=True)
(6, [[3], [2, 6]])
"""
if not u:
return dup_clear_denoms(f, K0, K1, convert=convert)
if K1 is None:
if K0.has_assoc_Ring:
K1 = K0.get_ring()
else:
K1 = K0
common = _rec_clear_denoms(f, u, K0, K1)
if not K1.is_one(common):
f = dmp_mul_ground(f, common, u, K0)
if not convert:
return common, f
else:
return common, dmp_convert(f, u, K0, K1)
def dmp_clear_denoms(f, u, K0, K1=None, convert=False):
"""
Clear denominators, i.e. transform ``K_0`` to ``K_1``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x,y = ring("x,y", QQ)
>>> f = QQ(1,2)*x + QQ(1,3)*y + 1
>>> R.dmp_clear_denoms(f, convert=False)
(6, 3*x + 2*y + 6)
>>> R.dmp_clear_denoms(f, convert=True)
(6, 3*x + 2*y + 6)
"""
if not u:
return dup_clear_denoms(f, K0, K1, convert=convert)
if K1 is None:
if K0.has_assoc_Ring:
K1 = K0.get_ring()
else:
K1 = K0
common = _rec_clear_denoms(f, u, K0, K1)
if not K1.is_one(common):
f = dmp_mul_ground(f, common, u, K0)
if not convert:
return common, f
else:
return common, dmp_convert(f, u, K0, K1)
def dmp_clear_denoms(f, u, K0, K1=None, convert=False):
"""
Clear denominators, i.e. transform ``K_0`` to ``K_1``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x,y = ring("x,y", QQ)
>>> f = QQ(1,2)*x + QQ(1,3)*y + 1
>>> R.dmp_clear_denoms(f, convert=False)
(6, 3*x + 2*y + 6)
>>> R.dmp_clear_denoms(f, convert=True)
(6, 3*x + 2*y + 6)
"""
if not u:
return dup_clear_denoms(f, K0, K1, convert=convert)
if K1 is None:
if K0.has_assoc_Ring:
K1 = K0.get_ring()
else:
K1 = K0
common = _rec_clear_denoms(f, u, K0, K1)
if not K1.is_one(common):
f = dmp_mul_ground(f, common, u, K0)
if not convert:
return common, f
else:
return common, dmp_convert(f, u, K0, K1)
def dmp_rr_lcm(f, g, u, K):
"""
Computes polynomial LCM over a ring in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_rr_lcm
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0]])
>>> g = ZZ.map([[1], [1, 0], []])
>>> dmp_rr_lcm(f, g, 1, ZZ)
[[1], [2, 0], [1, 0, 0], []]
"""
fc, f = dmp_ground_primitive(f, u, K)
gc, g = dmp_ground_primitive(g, u, K)
c = K.lcm(fc, gc)
h = dmp_quo(dmp_mul(f, g, u, K), dmp_gcd(f, g, u, K), u, K)
return dmp_mul_ground(h, c, u, K)
def dmp_factor_list(f, u, K0, **args):
"""Factor polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list(f, K0, **args)
if not K0.has_CharacteristicZero: # pragma: no cover
raise DomainError('only characteristic zero allowed')
if 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_ground_to_ring(f, u, K0, K)
f = dmp_convert(f, u, K0, K)
else:
K = K0
if K.is_ZZ:
coeff, factors = _dmp_inner_factor(f, u, K)
elif K.is_Poly:
f, v = dmp_inject(f, u, K)
coeff, factors = dmp_factor_list(f, v, K.dom, **args)
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)
if not args.get('include', False):
return coeff, factors
else:
if not factors:
return [(dmp_ground(coeff, u), 1)]
else:
g = dmp_mul_ground(factors[0][0], coeff, u, K)
return [(g, factors[0][1])] + factors[1:]
def mul_ground(f, c):
"""Multiply ``f`` by a an element of the ground domain. """
return f.per(dmp_mul_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def dmp_zz_heu_gcd(f, g, u, 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 evaluation proces reduces f and g variable by
variable into a large integer. The final step is to verify if the
interpolated polynomial is the correct GCD. This gives cofactors of
the input polynomials as a side effect.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_zz_heu_gcd(f, g)
(x + y, x + y, x)
References
==========
1. [Liao95]_
"""
if not u:
return dup_zz_heu_gcd(f, g, K)
result = _dmp_rr_trivial_gcd(f, g, u, K)
if result is not None:
return result
gcd, f, g = dmp_ground_extract(f, g, u, K)
f_norm = dmp_max_norm(f, u, K)
g_norm = dmp_max_norm(g, u, K)
B = K(2*min(f_norm, g_norm) + 29)
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dmp_ground_LC(f, u, K)),
g_norm // abs(dmp_ground_LC(g, u, K))) + 2)
for i in xrange(0, HEU_GCD_MAX):
ff = dmp_eval(f, x, u, K)
gg = dmp_eval(g, x, u, K)
v = u - 1
if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)):
h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K)
h = _dmp_zz_gcd_interpolate(h, x, v, K)
h = dmp_ground_primitive(h, u, K)[1]
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg_
cff = _dmp_zz_gcd_interpolate(cff, x, v, K)
h, r = dmp_div(f, cff, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff, cfg_
cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K)
h, r = dmp_div(g, cfg, u, K)
if dmp_zero_p(r, u):
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def mul_ground(f, c):
"""Multiply ``f`` by a an element of the ground domain. """
return f.per(dmp_mul_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def dmp_zz_heu_gcd(f, g, u, 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 evaluation proces reduces f and g variable by
variable into a large integer. The final step is to verify if the
interpolated polynomial is the correct GCD. This gives cofactors of
the input polynomials as a side effect.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_heu_gcd
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0]])
>>> g = ZZ.map([[1], [1, 0], []])
>>> dmp_zz_heu_gcd(f, g, 1, ZZ)
([[1], [1, 0]], [[1], [1, 0]], [[1], []])
**References**
1. [Liao95]_
"""
if not u:
return dup_zz_heu_gcd(f, g, K)
result = _dmp_rr_trivial_gcd(f, g, u, K)
if result is not None:
return result
df = dmp_degree(f, u)
dg = dmp_degree(g, u)
gcd, f, g = dmp_ground_extract(f, g, u, K)
f_norm = dmp_max_norm(f, u, K)
g_norm = dmp_max_norm(g, u, K)
B = 2*min(f_norm, g_norm) + 29
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dmp_ground_LC(f, u, K)),
g_norm // abs(dmp_ground_LC(g, u, K))) + 2)
for i in xrange(0, HEU_GCD_MAX):
ff = dmp_eval(f, x, u, K)
gg = dmp_eval(g, x, u, K)
v = u - 1
if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)):
h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K)
h = _dmp_zz_gcd_interpolate(h, x, v, K)
h = dmp_ground_primitive(h, u, K)[1]
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg_
cff = _dmp_zz_gcd_interpolate(cff, x, v, K)
h, r = dmp_div(f, cff, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff, cfg_
cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K)
h, r = dmp_div(g, cfg, u, K)
if dmp_zero_p(r, u):
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
dmp_div, dmp_rem, dmp_quo, dmp_exquo,
dup_max_norm, dmp_max_norm,
dup_l1_norm, dmp_l1_norm,
dup_expand, dmp_expand,
)
from sympy.polys.polyerrors import (
ExactQuotientFailed,
)
from sympy.polys.specialpolys import f_0
from sympy.polys.domains import FF, ZZ, QQ
from sympy.utilities.pytest import raises
F_0 = dmp_mul_ground(dmp_normal(f_0, 2, QQ), QQ(1,7), 2, QQ)
def test_dup_add_term():
f = dup_normal([], ZZ)
assert dup_add_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1], ZZ)
assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 0], ZZ)
assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 0, 0], ZZ)
f = dup_normal([1,1,1], ZZ)
assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1, 1, 2], ZZ)
assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 2, 1], ZZ)
assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([2, 1, 1], ZZ)
def dmp_factor_list(f, u, K0, **args):
"""Factor polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list(f, K0, **args)
if not K0.has_CharacteristicZero: # pragma: no cover
raise DomainError('only characteristic zero allowed')
if 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_ground_to_ring(f, u, K0, K)
f = dmp_convert(f, u, K0, K)
else:
K = K0
if K.is_ZZ:
coeff, factors = _dmp_inner_factor(f, u, K)
elif K.is_Poly:
f, v = dmp_inject(f, u, K)
coeff, factors = dmp_factor_list(f, v, K.dom, **args)
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)
if not args.get('include', False):
return coeff, factors
else:
if not factors:
return [(dmp_ground(coeff, u), 1)]
else:
g = dmp_mul_ground(factors[0][0], coeff, u, K)
return [(g, factors[0][1])] + factors[1:]
