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.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_zz_irreducible_p(f, K):
"""Test irreducibility using Eisenstein's criterion. """
lc = dup_LC(f, K)
tc = dup_TC(f, K)
e_fc = dup_content(f[1:], K)
if e_fc:
e_ff = factorint(int(e_fc))
for p in e_ff.iterkeys():
if (lc % p) and (tc % p**2):
return True
def dup_zz_irreducible_p(f, K):
"""Test irreducibility using Eisenstein's criterion. """
lc = dup_LC(f, K)
tc = dup_TC(f, K)
e_fc = dup_content(f[1:], K)
if e_fc:
e_ff = factorint(int(e_fc))
for p in e_ff.iterkeys():
if (lc % p) and (tc % p**2):
return True
def dup_zz_cyclotomic_factor(f, K):
"""
Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` returns a list of factors
of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for
`n >= 1`. Otherwise returns None.
Factorization is performed using using cyclotomic decomposition of `f`,
which makes this method much faster that any other direct factorization
approach (e.g. Zassenhaus's).
References
==========
1. [Weisstein09]_
"""
lc_f, tc_f = dup_LC(f, K), dup_TC(f, K)
if dup_degree(f) <= 0:
return None
if lc_f != 1 or tc_f not in [-1, 1]:
return None
if any(bool(cf) for cf in f[1:-1]):
return None
n = dup_degree(f)
F = _dup_cyclotomic_decompose(n, K)
if not K.is_one(tc_f):
return F
else:
H = []
for h in _dup_cyclotomic_decompose(2*n, K):
if h not in F:
H.append(h)
return H
def dup_zz_cyclotomic_factor(f, K):
"""
Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` returns a list of factors
of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for
`n >= 1`. Otherwise returns None.
Factorization is performed using using cyclotomic decomposition of `f`,
which makes this method much faster that any other direct factorization
approach (e.g. Zassenhaus's).
References
==========
1. [Weisstein09]_
"""
lc_f, tc_f = dup_LC(f, K), dup_TC(f, K)
if dup_degree(f) <= 0:
return None
if lc_f != 1 or tc_f not in [-1, 1]:
return None
if any(bool(cf) for cf in f[1:-1]):
return None
n = dup_degree(f)
F = _dup_cyclotomic_decompose(n, K)
if not K.is_one(tc_f):
return F
else:
H = []
for h in _dup_cyclotomic_decompose(2 * n, K):
if h not in F:
H.append(h)
return H
def dup_eval(f, a, K):
"""
Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_eval
>>> dup_eval([ZZ(1), ZZ(2), ZZ(3)], 2, ZZ)
11
"""
if not a:
return dup_TC(f, K)
result = K.zero
for c in f:
result *= a
result += c
return result
def dup_eval(f, a, K):
"""
Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_eval
>>> dup_eval([ZZ(1), ZZ(2), ZZ(3)], 2, ZZ)
11
"""
if not a:
return dup_TC(f, K)
result = K.zero
for c in f:
result *= a
result += c
return result
def dup_eval(f, a, K):
"""
Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_eval(x**2 + 2*x + 3, 2)
11
"""
if not a:
return dup_TC(f, K)
result = K.zero
for c in f:
result *= a
result += c
return result
def dup_eval(f, a, K):
"""
Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_eval(x**2 + 2*x + 3, 2)
11
"""
if not a:
return dup_TC(f, K)
result = K.zero
for c in f:
result *= a
result += c
return result
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 TC(f):
"""Returns the trailing coefficent of `f`. """
return dup_TC(f.rep, f.dom)
def test_dup_TC():
assert dup_TC([], ZZ) == 0
assert dup_TC([2,3,4,5], ZZ) == 5
def TC(f):
"""Returns the trailing coefficent of `f`. """
return dup_TC(f.rep, f.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
