def dup_zz_hensel_step(m, f, g, h, s, t, K):
"""
One step in Hensel lifting in `Z[x]`.
Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s`
and `t` such that::
f == g*h (mod m)
s*g + t*h == 1 (mod m)
lc(f) is not a zero divisor (mod m)
lc(h) == 1
deg(f) == deg(g) + deg(h)
deg(s) < deg(h)
deg(t) < deg(g)
returns polynomials `G`, `H`, `S` and `T`, such that::
f == G*H (mod m**2)
S*G + T**H == 1 (mod m**2)
References
==========
1. [Gathen99]_
"""
M = m**2
e = dup_sub_mul(f, g, h, K)
e = dup_trunc(e, M, K)
q, r = dup_div(dup_mul(s, e, K), h, K)
q = dup_trunc(q, M, K)
r = dup_trunc(r, M, K)
u = dup_add(dup_mul(t, e, K), dup_mul(q, g, K), K)
G = dup_trunc(dup_add(g, u, K), M, K)
H = dup_trunc(dup_add(h, r, K), M, K)
u = dup_add(dup_mul(s, G, K), dup_mul(t, H, K), K)
b = dup_trunc(dup_sub(u, [K.one], K), M, K)
c, d = dup_div(dup_mul(s, b, K), H, K)
c = dup_trunc(c, M, K)
d = dup_trunc(d, M, K)
u = dup_add(dup_mul(t, b, K), dup_mul(c, G, K), K)
S = dup_trunc(dup_sub(s, d, K), M, K)
T = dup_trunc(dup_sub(t, u, K), M, K)
return G, H, S, T
def dup_zz_hensel_step(m, f, g, h, s, t, K):
"""
One step in Hensel lifting in `Z[x]`.
Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s`
and `t` such that::
f == g*h (mod m)
s*g + t*h == 1 (mod m)
lc(f) is not a zero divisor (mod m)
lc(h) == 1
deg(f) == deg(g) + deg(h)
deg(s) < deg(h)
deg(t) < deg(g)
returns polynomials `G`, `H`, `S` and `T`, such that::
f == G*H (mod m**2)
S*G + T**H == 1 (mod m**2)
References
==========
1. [Gathen99]_
"""
M = m**2
e = dup_sub_mul(f, g, h, K)
e = dup_trunc(e, M, K)
q, r = dup_div(dup_mul(s, e, K), h, K)
q = dup_trunc(q, M, K)
r = dup_trunc(r, M, K)
u = dup_add(dup_mul(t, e, K), dup_mul(q, g, K), K)
G = dup_trunc(dup_add(g, u, K), M, K)
H = dup_trunc(dup_add(h, r, K), M, K)
u = dup_add(dup_mul(s, G, K), dup_mul(t, H, K), K)
b = dup_trunc(dup_sub(u, [K.one], K), M, K)
c, d = dup_div(dup_mul(s, b, K), H, K)
c = dup_trunc(c, M, K)
d = dup_trunc(d, M, K)
u = dup_add(dup_mul(t, b, K), dup_mul(c, G, K), K)
S = dup_trunc(dup_sub(s, d, K), M, K)
T = dup_trunc(dup_sub(t, u, K), M, K)
return G, H, S, T
def test_dmp_sub():
assert dmp_sub([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
assert dmp_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == dup_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_sub([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_sub([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(-1, 2)]]]
assert dmp_sub([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(1, 7)]]]
assert dmp_sub([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(-1, 7)]]]
def dup_revert(f, n, K):
"""
Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.
This function computes first ``2**n`` terms of a polynomial that
is a result of inversion of a polynomial modulo ``x**n``. This is
useful to efficiently compute series expansion of ``1/f``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densetools import dup_revert
>>> f = [-QQ(1,720), QQ(0), QQ(1,24), QQ(0), -QQ(1,2), QQ(0), QQ(1)]
>>> dup_revert(f, 8, QQ)
[61/720, 0/1, 5/24, 0/1, 1/2, 0/1, 1/1]
"""
g = [K.revert(dup_TC(f, K))]
h = [K.one, K.zero, K.zero]
N = int(_ceil(_log(n, 2)))
for i in xrange(1, N + 1):
a = dup_mul_ground(g, K(2), K)
b = dup_mul(f, dup_sqr(g, K), K)
g = dup_rem(dup_sub(a, b, K), h, K)
h = dup_lshift(h, dup_degree(h), K)
return g
def dup_revert(f, n, K):
"""
Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.
This function computes first ``2**n`` terms of a polynomial that
is a result of inversion of a polynomial modulo ``x**n``. This is
useful to efficiently compute series expansion of ``1/f``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densetools import dup_revert
>>> f = [-QQ(1,720), QQ(0), QQ(1,24), QQ(0), -QQ(1,2), QQ(0), QQ(1)]
>>> dup_revert(f, 8, QQ)
[61/720, 0/1, 5/24, 0/1, 1/2, 0/1, 1/1]
"""
g = [K.revert(dup_TC(f, K))]
h = [K.one, K.zero, K.zero]
N = int(_ceil(_log(n, 2)))
for i in xrange(1, N + 1):
a = dup_mul_ground(g, K(2), K)
b = dup_mul(f, dup_sqr(g, K), K)
g = dup_rem(dup_sub(a, b, K), h, K)
h = dup_lshift(h, dup_degree(h), K)
return g
def dup_revert(f, n, K):
"""
Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.
This function computes first ``2**n`` terms of a polynomial that
is a result of inversion of a polynomial modulo ``x**n``. This is
useful to efficiently compute series expansion of ``1/f``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1
>>> R.dup_revert(f, 8)
61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1
"""
g = [K.revert(dup_TC(f, K))]
h = [K.one, K.zero, K.zero]
N = int(_ceil(_log(n, 2)))
for i in range(1, N + 1):
a = dup_mul_ground(g, K(2), K)
b = dup_mul(f, dup_sqr(g, K), K)
g = dup_rem(dup_sub(a, b, K), h, K)
h = dup_lshift(h, dup_degree(h), K)
return g
def dup_revert(f, n, K):
"""
Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.
This function computes first ``2**n`` terms of a polynomial that
is a result of inversion of a polynomial modulo ``x**n``. This is
useful to efficiently compute series expansion of ``1/f``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1
>>> R.dup_revert(f, 8)
61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1
"""
g = [K.revert(dup_TC(f, K))]
h = [K.one, K.zero, K.zero]
N = int(_ceil(_log(n, 2)))
for i in range(1, N + 1):
a = dup_mul_ground(g, K(2), K)
b = dup_mul(f, dup_sqr(g, K), K)
g = dup_rem(dup_sub(a, b, K), h, K)
h = dup_lshift(h, dup_degree(h), K)
return g
def test_dmp_sub():
assert dmp_sub([ZZ(1),ZZ(2)], [ZZ(1)], 0, ZZ) == \
dup_sub([ZZ(1),ZZ(2)], [ZZ(1)], ZZ)
assert dmp_sub([QQ(1,2),QQ(2,3)], [QQ(1)], 0, QQ) == \
dup_sub([QQ(1,2),QQ(2,3)], [QQ(1)], QQ)
assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_sub([[[QQ(1,2)]]], [[[]]], 2, QQ) == [[[QQ(1,2)]]]
assert dmp_sub([[[]]], [[[QQ(1,2)]]], 2, QQ) == [[[QQ(-1,2)]]]
assert dmp_sub([[[QQ(2,7)]]], [[[QQ(1,7)]]], 2, QQ) == [[[QQ(1,7)]]]
assert dmp_sub([[[QQ(1,7)]]], [[[QQ(2,7)]]], 2, QQ) == [[[QQ(-1,7)]]]
def dup_chebyshevt(n, K):
"""Low-level implementation of Chebyshev polynomials of the 1st kind. """
seq = [[K.one], [K.one, K.zero]]
for i in xrange(2, n + 1):
a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2), K)
seq.append(dup_sub(a, seq[-2], K))
return seq[n]
def dup_spherical_bessel_fn_minus(n, K):
""" Low-level implementation of fn(-n, x) """
seq = [[K.one, K.zero], [K.zero]]
for i in xrange(2, n + 1):
a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(3 - 2 * i), K)
seq.append(dup_sub(a, seq[-2], K))
return seq[n]
def dup_chebyshevt(n, K):
"""Low-level implementation of Chebyshev polynomials of the 1st kind. """
seq = [[K.one], [K.one, K.zero]]
for i in range(2, n + 1):
a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2), K)
seq.append(dup_sub(a, seq[-2], K))
return seq[n]
def dup_spherical_bessel_fn_minus(n, K):
""" Low-level implementation of fn(-n, x) """
seq = [[K.one, K.zero], [K.zero]]
for i in range(2, n + 1):
a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(3 - 2*i), K)
seq.append(dup_sub(a, seq[-2], K))
return seq[n]
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_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_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_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.is_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_laguerre(n, K):
"""Low-level implementation of Laguerre polynomials. """
seq = [[K.one], [-K.one, K.one]]
for i in xrange(2, n + 1):
a = dup_mul(seq[-1], [-K(1, i), K(2 * i - 1, i)], K)
b = dup_mul_ground(seq[-2], K(i - 1, i), K)
seq.append(dup_sub(a, b, K))
return seq[n]
def dup_legendre(n, K):
"""Low-level implementation of Legendre polynomials. """
seq = [[K.one], [K.one, K.zero]]
for i in xrange(2, n + 1):
a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2 * i - 1, i), K)
b = dup_mul_ground(seq[-2], K(i - 1, i), K)
seq.append(dup_sub(a, b, K))
return seq[n]
def dup_laguerre(n, alpha, K):
"""Low-level implementation of Laguerre polynomials. """
seq = [[K.zero], [K.one]]
for i in xrange(1, n + 1):
a = dup_mul(seq[-1], [-K.one / i, alpha / i + K(2 * i - 1) / i], K)
b = dup_mul_ground(seq[-2], alpha / i + K(i - 1) / i, K)
seq.append(dup_sub(a, b, K))
return seq[-1]
def dup_laguerre(n, K):
"""Low-level implementation of Laguerre polynomials. """
seq = [[K.one], [-K.one, K.one]]
for i in xrange(2, n+1):
a = dup_mul(seq[-1], [-K(1, i), K(2*i-1, i)], K)
b = dup_mul_ground(seq[-2], K(i-1, i), K)
seq.append(dup_sub(a, b, K))
return seq[n]
def dup_laguerre(n, alpha, K):
"""Low-level implementation of Laguerre polynomials. """
seq = [[K.zero], [K.one]]
for i in range(1, n + 1):
a = dup_mul(seq[-1], [-K.one/i, alpha/i + K(2*i - 1)/i], K)
b = dup_mul_ground(seq[-2], alpha/i + K(i - 1)/i, K)
seq.append(dup_sub(a, b, K))
return seq[-1]
def dup_legendre(n, K):
"""Low-level implementation of Legendre polynomials. """
seq = [[K.one], [K.one, K.zero]]
for i in range(2, n + 1):
a = dup_mul_ground(dup_lshift(seq[-1], 1, K), K(2*i - 1, i), K)
b = dup_mul_ground(seq[-2], K(i - 1, i), K)
seq.append(dup_sub(a, b, K))
return seq[n]
def dup_gegenbauer(n, a, K):
"""Low-level implementation of Gegenbauer polynomials. """
seq = [[K.one], [K(2)*a, K.zero]]
for i in range(2, n + 1):
f1 = K(2) * (i + a - K.one) / i
f2 = (i + K(2)*a - K(2)) / i
p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K)
p2 = dup_mul_ground(seq[-2], f2, K)
seq.append(dup_sub(p1, p2, K))
return seq[n]
def dup_gegenbauer(n, a, K):
"""Low-level implementation of Gegenbauer polynomials. """
seq = [[K.one], [K(2) * a, K.zero]]
for i in xrange(2, n + 1):
f1 = K(2) * (i + a - K.one) / i
f2 = (i + K(2) * a - K(2)) / i
p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K)
p2 = dup_mul_ground(seq[-2], f2, K)
seq.append(dup_sub(p1, p2, K))
return seq[n]
def dup_hermite(n, K):
"""Low-level implementation of Hermite polynomials. """
seq = [[K.one], [K(2), K.zero]]
for i in range(2, n + 1):
a = dup_lshift(seq[-1], 1, K)
b = dup_mul_ground(seq[-2], K(i - 1), K)
c = dup_mul_ground(dup_sub(a, b, K), K(2), K)
seq.append(c)
return seq[n]
def dup_hermite(n, K):
"""Low-level implementation of Hermite polynomials. """
seq = [[K.one], [K(2), K.zero]]
for i in xrange(2, n + 1):
a = dup_lshift(seq[-1], 1, K)
b = dup_mul_ground(seq[-2], K(i - 1), K)
c = dup_mul_ground(dup_sub(a, b, K), K(2), K)
seq.append(c)
return seq[n]
def dup_jacobi(n, a, b, K):
"""Low-level implementation of Jacobi polynomials. """
seq = [[K.one], [(a + b + K(2))/K(2), (a - b)/K(2)]]
for i in range(2, n + 1):
den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2))
f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den)
f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den)
f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den
p0 = dup_mul_ground(seq[-1], f0, K)
p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K)
p2 = dup_mul_ground(seq[-2], f2, K)
seq.append(dup_sub(dup_add(p0, p1, K), p2, K))
return seq[n]
def dup_jacobi(n, a, b, K):
"""Low-level implementation of Jacobi polynomials. """
seq = [[K.one], [(a + b + K(2))/K(2), (a - b)/K(2)]]
for i in range(2, n + 1):
den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2))
f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den)
f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den)
f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den
p0 = dup_mul_ground(seq[-1], f0, K)
p1 = dup_mul_ground(dup_lshift(seq[-1], 1, K), f1, K)
p2 = dup_mul_ground(seq[-2], f2, K)
seq.append(dup_sub(dup_add(p0, p1, K), p2, K))
return seq[n]
def dup_cyclotomic_p(f, K, irreducible=False):
"""
Efficiently test if ``f`` is a cyclotomic polnomial.
Examples
========
>>> from sympy.polys.factortools import dup_cyclotomic_p
>>> from sympy.polys.domains import ZZ
>>> f = [1, 0, 1, 0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 1, 0, 1]
>>> dup_cyclotomic_p(f, ZZ)
False
>>> g = [1, 0, 1, 0, 0, 0,-1, 0,-1, 0,-1, 0, 0, 0, 1, 0, 1]
>>> dup_cyclotomic_p(g, ZZ)
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 xrange(n, -1, -2):
g.insert(0, f[i])
for i in xrange(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 sub(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_sub(F, G, dom))
def test_dup_sub():
assert dup_sub([], [], ZZ) == []
assert dup_sub([ZZ(1)], [], ZZ) == [ZZ(1)]
assert dup_sub([], [ZZ(1)], ZZ) == [ZZ(-1)]
assert dup_sub([ZZ(1)], [ZZ(1)], ZZ) == []
assert dup_sub([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(-1)]
assert dup_sub([ZZ(1),ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1),ZZ(1)]
assert dup_sub([ZZ(1)], [ZZ(1),ZZ(2)], ZZ) == [ZZ(-1),ZZ(-1)]
assert dup_sub([ZZ(3),ZZ(2),ZZ(1)], [ZZ(8),ZZ(9),ZZ(10)], ZZ) == [ZZ(-5),ZZ(-7),ZZ(-9)]
assert dup_sub([], [], QQ) == []
assert dup_sub([QQ(1,2)], [], QQ) == [QQ(1,2)]
assert dup_sub([], [QQ(1,2)], QQ) == [QQ(-1,2)]
assert dup_sub([QQ(1,3)], [QQ(1,3)], QQ) == []
assert dup_sub([QQ(1,3)], [QQ(2,3)], QQ) == [QQ(-1,3)]
assert dup_sub([QQ(1,7),QQ(2,7)], [QQ(1)], QQ) == [QQ(1,7),QQ(-5,7)]
assert dup_sub([QQ(1)], [QQ(1,7),QQ(2,7)], QQ) == [QQ(-1,7),QQ(5,7)]
assert dup_sub([QQ(3,7),QQ(2,7),QQ(1,7)], [QQ(8,7),QQ(9,7),QQ(10,7)], QQ) == [QQ(-5,7),QQ(-7,7),QQ(-9,7)]
def sub(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_sub(F, G, dom))
def dup_zz_cyclotomic_p(f, K, irreducible=False):
"""
Efficiently test if ``f`` is a cyclotomic polnomial.
**Examples**
>>> from sympy.polys.factortools import dup_zz_cyclotomic_p
>>> from sympy.polys.domains import ZZ
>>> f = [1, 0, 1, 0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 1, 0, 1]
>>> dup_zz_cyclotomic_p(f, ZZ)
False
>>> g = [1, 0, 1, 0, 0, 0,-1, 0,-1, 0,-1, 0, 0, 0, 1, 0, 1]
>>> dup_zz_cyclotomic_p(g, ZZ)
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 xrange(n, -1, -2):
g.insert(0, f[i])
for i in xrange(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_zz_cyclotomic_p(g, K):
return True
G = dup_sqf_part(F, K)
if dup_sqr(G, K) == F and dup_zz_cyclotomic_p(G, K):
return True
return False
