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 diofant.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 diofant.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 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
pytest.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
pytest.raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
def dup_euclidean_prs(f, g, K):
"""
Euclidean polynomial remainder sequence (PRS) in `K[x]`.
Examples
========
>>> from diofant.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