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_invert(f, g, K):
"""
Compute multiplicative inverse of `f` modulo `g` in `F[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**2 - 1
>>> g = 2*x - 1
>>> h = x - 1
>>> R.dup_invert(f, g)
-4/3
>>> R.dup_invert(f, h)
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
s, h = dup_half_gcdex(f, g, K)
if h == [K.one]:
return dup_rem(s, g, K)
else:
raise NotInvertible("zero divisor")
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_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_invert(f, g, K):
"""
Compute multiplicative inverse of `f` modulo `g` in `F[x]`.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_invert
>>> f = QQ.map([1, 0, -1])
>>> g = QQ.map([2, -1])
>>> h = QQ.map([1, -1])
>>> dup_invert(f, g, QQ)
[-4/3]
>>> dup_invert(f, h, QQ)
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
s, h = dup_half_gcdex(f, g, K)
if h == [K.one]:
return dup_rem(s, g, K)
else:
raise NotInvertible("zero divisor")
def dup_invert(f, g, K):
"""
Compute multiplicative inverse of ``f`` modulo ``g`` in ``F[x]``.
**Examples**
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_invert
>>> f = QQ.map([1, 0, -1])
>>> g = QQ.map([2, -1])
>>> h = QQ.map([1, -1])
>>> dup_invert(f, g, QQ)
[-4/3]
>>> dup_invert(f, h, QQ)
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
s, h = dup_half_gcdex(f, g, K)
if h == [K.one]:
return dup_rem(s, g, K)
else:
raise NotInvertible("zero divisor")
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_invert(f, g, K):
"""
Compute multiplicative inverse of `f` modulo `g` in `F[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**2 - 1
>>> g = 2*x - 1
>>> h = x - 1
>>> R.dup_invert(f, g)
-4/3
>>> R.dup_invert(f, h)
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
s, h = dup_half_gcdex(f, g, K)
if h == [K.one]:
return dup_rem(s, g, K)
else:
raise NotInvertible("zero divisor")
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 pow(f, n):
"""Raise `f` to a non-negative power `n`. """
if isinstance(n, int):
if n < 0:
F, n = dup_invert(f.rep, f.mod, f.dom), -n
else:
F = f.rep
return f.per(dup_rem(dup_pow(F, n, f.dom), f.mod, f.dom))
else:
raise TypeError("`int` expected, got %s" % type(n))
def pow(f, n):
"""Raise `f` to a non-negative power `n`. """
if isinstance(n, int):
if n < 0:
F, n = dup_invert(f.rep, f.mod, f.dom), -n
else:
F = f.rep
return f.per(dup_rem(dup_pow(F, n, f.dom), f.mod, f.dom))
else:
raise TypeError("`int` expected, got %s" % type(n))
def dup_euclidean_prs(f, g, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs = R.dup_euclidean_prs(f, g)
>>> prs[0]
x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> prs[1]
3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs[2]
-5/9*x**4 + 1/9*x**2 - 1/3
>>> prs[3]
-117/25*x**2 - 9*x + 441/25
>>> prs[4]
233150/19773*x - 102500/6591
>>> prs[5]
-1288744821/543589225
"""
prs = [f, g]
h = dup_rem(f, g, K)
while h:
prs.append(h)
f, g = g, h
h = dup_rem(f, g, K)
return prs
def dup_euclidean_prs(f, g, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_euclidean_prs
>>> f = QQ.map([1, 0, 1, 0, -3, -3, 8, 2, -5])
>>> g = QQ.map([3, 0, 5, 0, -4, -9, 21])
>>> prs = dup_euclidean_prs(f, g, QQ)
>>> prs[0]
[1/1, 0/1, 1/1, 0/1, -3/1, -3/1, 8/1, 2/1, -5/1]
>>> prs[1]
[3/1, 0/1, 5/1, 0/1, -4/1, -9/1, 21/1]
>>> prs[2]
[-5/9, 0/1, 1/9, 0/1, -1/3]
>>> prs[3]
[-117/25, -9/1, 441/25]
>>> prs[4]
[233150/19773, -102500/6591]
>>> prs[5]
[-1288744821/543589225]
"""
prs = [f, g]
h = dup_rem(f, g, K)
while h:
prs.append(h)
f, g = g, h
h = dup_rem(f, g, K)
return prs
def dup_euclidean_prs(f, g, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs = R.dup_euclidean_prs(f, g)
>>> prs[0]
x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
>>> prs[1]
3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
>>> prs[2]
-5/9*x**4 + 1/9*x**2 - 1/3
>>> prs[3]
-117/25*x**2 - 9*x + 441/25
>>> prs[4]
233150/19773*x - 102500/6591
>>> prs[5]
-1288744821/543589225
"""
prs = [f, g]
h = dup_rem(f, g, K)
while h:
prs.append(h)
f, g = g, h
h = dup_rem(f, g, K)
return prs
def dup_euclidean_prs(f, g, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_euclidean_prs
>>> f = QQ.map([1, 0, 1, 0, -3, -3, 8, 2, -5])
>>> g = QQ.map([3, 0, 5, 0, -4, -9, 21])
>>> prs = dup_euclidean_prs(f, g, QQ)
>>> prs[0]
[1/1, 0/1, 1/1, 0/1, -3/1, -3/1, 8/1, 2/1, -5/1]
>>> prs[1]
[3/1, 0/1, 5/1, 0/1, -4/1, -9/1, 21/1]
>>> prs[2]
[-5/9, 0/1, 1/9, 0/1, -1/3]
>>> prs[3]
[-117/25, -9/1, 441/25]
>>> prs[4]
[233150/19773, -102500/6591]
>>> prs[5]
[-1288744821/543589225]
"""
prs = [f, g]
h = dup_rem(f, g, K)
while h:
prs.append(h)
f, g = g, h
h = dup_rem(f, g, K)
return prs
def mul(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_rem(dup_mul(F, G, dom), mod, dom))
def quo(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom))
def div(f, g):
dom, per, F, G, mod = f.unify(g)
return (per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom)), self.zero(mod, dom))
def mul(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_rem(dup_mul(F, G, dom), mod, dom))
def quo(f, g):
dom, per, F, G, mod = f.unify(g)
return per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom))
def div(f, g):
dom, per, F, G, mod = f.unify(g)
return (per(dup_rem(dup_mul(F, dup_invert(G, mod, dom), dom), mod, dom)), self.zero(mod, dom))
