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_transform(f, p, q, K):
"""
Evaluate functional transformation ``q**n * f(p/q)`` in ``K[x]``.
Examples
========
>>> from diofant.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_transform(x**2 - 2*x + 1, x**2 + 1, x - 1)
x**4 - 2*x**3 + 5*x**2 - 4*x + 4
"""
if not f:
return []
n = len(f) - 1
h, Q = [f[0]], [[K.one]]
for i in range(0, n):
Q.append(dup_mul(Q[-1], q, K))
for c, q in zip(f[1:], Q[1:]):
h = dup_mul(h, p, K)
q = dup_mul_ground(q, c, K)
h = dup_add(h, q, K)
return h
def test_dmp_add():
assert dmp_add([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
assert dmp_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
assert dmp_add([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_add([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_add([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_add([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(3)]]]
assert dmp_add([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(3)]]]
assert dmp_add([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_add([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_add([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_add([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
assert dmp_add([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
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 test_dup_add():
assert dup_add([], [], ZZ) == []
assert dup_add([ZZ(1)], [], ZZ) == [ZZ(1)]
assert dup_add([], [ZZ(1)], ZZ) == [ZZ(1)]
assert dup_add([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(2)]
assert dup_add([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(3)]
assert dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(3)]
assert dup_add([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(3)]
assert dup_add([ZZ(1), ZZ(2), ZZ(3)], [ZZ(8), ZZ(9), ZZ(10)],
ZZ) == [ZZ(9), ZZ(11), ZZ(13)]
assert dup_add([], [], QQ) == []
assert dup_add([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
assert dup_add([], [QQ(1, 2)], QQ) == [QQ(1, 2)]
assert dup_add([QQ(1, 4)], [QQ(1, 4)], QQ) == [QQ(1, 2)]
assert dup_add([QQ(1, 4)], [QQ(1, 2)], QQ) == [QQ(3, 4)]
assert dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) == [QQ(1, 2), QQ(5, 3)]
assert dup_add([QQ(1)], [QQ(1, 2), QQ(2, 3)], QQ) == [QQ(1, 2), QQ(5, 3)]
assert dup_add([QQ(1, 7), QQ(2, 7), QQ(3, 7)],
[QQ(8, 7), QQ(9, 7), QQ(10, 7)],
QQ) == [QQ(9, 7), QQ(11, 7), QQ(13, 7)]
def dmp_zz_diophantine(F, c, A, d, p, u, K):
"""Wang/EEZ: Solve multivariate Diophantine equations. """
if not A:
S = [[] for _ in F]
n = dup_degree(c)
for i, coeff in enumerate(c):
if not coeff:
continue
T = dup_zz_diophantine(F, n - i, p, K)
for j, (s, t) in enumerate(zip(S, T)):
t = dup_mul_ground(t, coeff, K)
S[j] = dup_trunc(dup_add(s, t, K), p, K)
else:
n = len(A)
e = dmp_expand(F, u, K)
a, A = A[-1], A[:-1]
B, G = [], []
for f in F:
B.append(dmp_quo(e, f, u, K))
G.append(dmp_eval_in(f, a, n, u, K))
C = dmp_eval_in(c, a, n, u, K)
v = u - 1
S = dmp_zz_diophantine(G, C, A, d, p, v, K)
S = [dmp_raise(s, 1, v, K) for s in S]
for s, b in zip(S, B):
c = dmp_sub_mul(c, s, b, u, K)
c = dmp_ground_trunc(c, p, u, K)
m = dmp_nest([K.one, -a], n, K)
M = dmp_one(n, K)
for k in K.map(range(0, d)):
if dmp_zero_p(c, u):
break
M = dmp_mul(M, m, u, K)
C = dmp_diff_eval_in(c, k + 1, a, n, u, K)
if not dmp_zero_p(C, v):
C = dmp_quo_ground(C, K.factorial(k + 1), v, K)
T = dmp_zz_diophantine(G, C, A, d, p, v, K)
for i, t in enumerate(T):
T[i] = dmp_mul(dmp_raise(t, 1, v, K), M, u, K)
for i, (s, t) in enumerate(zip(S, T)):
S[i] = dmp_add(s, t, u, K)
for t, b in zip(T, B):
c = dmp_sub_mul(c, t, b, u, K)
c = dmp_ground_trunc(c, p, u, K)
S = [dmp_ground_trunc(s, p, u, K) for s in S]
return S