def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K):
"""Wang/EEZ: Compute correct leading coefficients. """
C, J, v = [], [0] * len(E), u - 1
for h in H:
c = dmp_one(v, K)
d = dup_LC(h, K) * cs
for i in reversed(range(len(E))):
k, e, (t, _) = 0, E[i], T[i]
while not (d % e):
d, k = d // e, k + 1
if k != 0:
c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1
C.append(c)
if any(not j for j in J):
raise ExtraneousFactors # pragma: no cover
CC, HH = [], []
for c, h in zip(C, H):
d = dmp_eval_tail(c, A, v, K)
lc = dup_LC(h, K)
if K.is_one(cs):
cc = lc // d
else:
g = K.gcd(lc, d)
d, cc = d // g, lc // g
h, cs = dup_mul_ground(h, d, K), cs // d
c = dmp_mul_ground(c, cc, v, K)
CC.append(c)
HH.append(h)
if K.is_one(cs):
return f, HH, CC
CCC, HHH = [], []
for c, h in zip(CC, HH):
CCC.append(dmp_mul_ground(c, cs, v, K))
HHH.append(dmp_mul_ground(h, cs, 0, K))
f = dmp_mul_ground(f, cs**(len(H) - 1), u, K)
return f, HHH, CCC
def dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K):
"""Wang/EEZ: Parallel Hensel lifting algorithm. """
S, n, v = [f], len(A), u - 1
H = list(H)
for i, a in enumerate(reversed(A[1:])):
s = dmp_eval_in(S[0], a, n - i, u - i, K)
S.insert(0, dmp_ground_trunc(s, p, v - i, K))
d = max(dmp_degree_list(f, u)[1:])
for j, s, a in zip(range(2, n + 2), S, A):
G, w = list(H), j - 1
I, J = A[:j - 2], A[j - 1:]
for i, (h, lc) in enumerate(zip(H, LC)):
lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w - 1, K)
H[i] = [lc] + dmp_raise(h[1:], 1, w - 1, K)
m = dmp_nest([K.one, -a], w, K)
M = dmp_one(w, K)
c = dmp_sub(s, dmp_expand(H, w, K), w, K)
dj = dmp_degree_in(s, w, w)
for k in K.map(range(0, dj)):
if dmp_zero_p(c, w):
break
M = dmp_mul(M, m, w, K)
C = dmp_diff_eval_in(c, k + 1, a, w, w, K)
if not dmp_zero_p(C, w - 1):
C = dmp_quo_ground(C, K.factorial(k + 1), w - 1, K)
T = dmp_zz_diophantine(G, C, I, d, p, w - 1, K)
for i, (h, t) in enumerate(zip(H, T)):
h = dmp_add_mul(h, dmp_raise(t, 1, w - 1, K), M, w, K)
H[i] = dmp_ground_trunc(h, p, w, K)
h = dmp_sub(s, dmp_expand(H, w, K), w, K)
c = dmp_ground_trunc(h, p, w, K)
if dmp_expand(H, u, K) != f:
raise ExtraneousFactors # pragma: no cover
else:
return H
def dmp_zz_wang_test_points(f, T, ct, A, u, K):
"""Wang/EEZ: Test evaluation points for suitability. """
if not dmp_eval_tail(dmp_LC(f, K), A, u - 1, K):
raise EvaluationFailed('no luck')
g = dmp_eval_tail(f, A, u, K)
if not dup_sqf_p(g, K):
raise EvaluationFailed('no luck')
c, h = dup_primitive(g, K)
if K.is_negative(dup_LC(h, K)):
c, h = -c, dup_neg(h, K)
v = u - 1
E = [dmp_eval_tail(t, A, v, K) for t, _ in T]
D = dmp_zz_wang_non_divisors(E, c, ct, K)
if D is not None:
return c, h, E
else:
raise EvaluationFailed('no luck')
def test_dmp_eval_tail():
assert dmp_eval_tail([[]], [1], 1, ZZ) == []
assert dmp_eval_tail([[[]]], [1], 2, ZZ) == [[]]
assert dmp_eval_tail([[[]]], [1, 2], 2, ZZ) == []
assert dmp_eval_tail(f_0, [], 2, ZZ) == f_0
assert dmp_eval_tail(f_0, [1, -17, 8], 2, ZZ) == 84496
assert dmp_eval_tail(f_0, [-17, 8], 2, ZZ) == [-1409, 3, 85902]
assert dmp_eval_tail(f_0, [8], 2, ZZ) == [[83, 2], [3], [302, 81, 1]]
assert dmp_eval_tail(f_1, [-17, 8], 2, ZZ) == [-136, 15699, 9166, -27144]
assert dmp_eval_tail(
f_2, [-12, 3], 2, ZZ) == [-1377, 0, -702, -1224, 0, -624]
assert dmp_eval_tail(
f_3, [-12, 3], 2, ZZ) == [144, 82, -5181, -28872, -14868, -540]
assert dmp_eval_tail(
f_4, [25, -1], 2, ZZ) == [152587890625, 9765625, -59605407714843750,
-3839159765625, -1562475, 9536712644531250, 610349546750, -4, 24414375000, 1562520]
assert dmp_eval_tail(f_5, [25, -1], 2, ZZ) == [-1, -78, -2028, -17576]
assert dmp_eval_tail(f_6, [0, 2, 4], 3, ZZ) == [5040, 0, 0, 4480]