def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dmp_sqf_list
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0], [], [], []])
>>> dmp_sqf_list(f, 1, ZZ)
(1, [([[1], [1, 0]], 2), ([[1], []], 3)])
>>> dmp_sqf_list(f, 1, ZZ, all=True)
(1, [([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if not K.has_CharacteristicZero:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dmp_sqf_list
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0], [], [], []])
>>> dmp_sqf_list(f, 1, ZZ)
(1, [([[1], [1, 0]], 2), ([[1], []], 3)])
>>> dmp_sqf_list(f, 1, ZZ, all=True)
(1, [([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if not K.has_CharacteristicZero:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x**5 + 2*x**4*y + x**3*y**2
>>> R.dmp_sqf_list(f)
(1, [(x + y, 2), (x, 3)])
>>> R.dmp_sqf_list(f, all=True)
(1, [(1, 1), (x + y, 2), (x, 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if K.is_FiniteField:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.has_Field:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x**5 + 2*x**4*y + x**3*y**2
>>> R.dmp_sqf_list(f)
(1, [(x + y, 2), (x, 3)])
>>> R.dmp_sqf_list(f, all=True)
(1, [(1, 1), (x + y, 2), (x, 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if K.is_FiniteField:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.is_Field:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dmp_ext_factor(f, u, K):
"""Factor multivariate polynomials over algebraic number fields. """
if not u:
return dup_ext_factor(f, K)
lc = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
if all(d <= 0 for d in dmp_degree_list(f, u)):
return lc, []
f, F = dmp_sqf_part(f, u, K), f
s, g, r = dmp_sqf_norm(f, u, K)
factors = dmp_factor_list_include(r, u, K.dom)
if len(factors) == 1:
coeff, factors = lc, [f]
else:
H = dmp_raise([K.one, s * K.unit], u, 0, K)
for i, (factor, _) in enumerate(factors):
h = dmp_convert(factor, u, K.dom, K)
h, _, g = dmp_inner_gcd(h, g, u, K)
h = dmp_compose(h, H, u, K)
factors[i] = h
return lc, dmp_trial_division(F, factors, u, K)
def dmp_ext_factor(f, u, K):
"""Factor multivariate polynomials over algebraic number fields. """
if not u:
return dup_ext_factor(f, K)
lc = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
if all([ d <= 0 for d in dmp_degree_list(f, u) ]):
return lc, []
f, F = dmp_sqf_part(f, u, K), f
s, g, r = dmp_sqf_norm(f, u, K)
factors = dmp_factor_list_include(r, u, K.dom)
if len(factors) == 1:
coeff, factors = lc, [f]
else:
H = dmp_raise([K.one, s*K.unit], u, 0, K)
for i, (factor, _) in enumerate(factors):
h = dmp_convert(factor, u, K.dom, K)
h, _, g = dmp_inner_gcd(h, g, u, K)
h = dmp_compose(h, H, u, K)
factors[i] = h
return lc, dmp_trial_division(F, factors, u, K)
def test_dmp_gcd():
assert dmp_zz_heu_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])
assert dmp_rr_prs_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])
assert dmp_zz_heu_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])
assert dmp_rr_prs_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])
assert dmp_zz_heu_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])
assert dmp_rr_prs_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])
assert dmp_zz_heu_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])
assert dmp_rr_prs_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])
assert dmp_zz_heu_gcd([[]], [[2], [4]], 1, ZZ) == ([[2], [4]], [[]], [[1]])
assert dmp_rr_prs_gcd([[]], [[2], [4]], 1, ZZ) == ([[2], [4]], [[]], [[1]])
assert dmp_zz_heu_gcd([[2], [4]], [[]], 1, ZZ) == ([[2], [4]], [[1]], [[]])
assert dmp_rr_prs_gcd([[2], [4]], [[]], 1, ZZ) == ([[2], [4]], [[1]], [[]])
assert dmp_zz_heu_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])
assert dmp_rr_prs_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])
assert dmp_zz_heu_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])
assert dmp_rr_prs_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])
assert dmp_zz_heu_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])
assert dmp_rr_prs_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])
assert dmp_zz_heu_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])
assert dmp_rr_prs_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])
assert dmp_zz_heu_gcd([[1], [2], [1]], [[1]], 1,
ZZ) == ([[1]], [[1], [2], [1]], [[1]])
assert dmp_rr_prs_gcd([[1], [2], [1]], [[1]], 1,
ZZ) == ([[1]], [[1], [2], [1]], [[1]])
assert dmp_zz_heu_gcd([[1], [2], [1]], [[2]], 1,
ZZ) == ([[1]], [[1], [2], [1]], [[2]])
assert dmp_rr_prs_gcd([[1], [2], [1]], [[2]], 1,
ZZ) == ([[1]], [[1], [2], [1]], [[2]])
assert dmp_zz_heu_gcd([[2], [4], [2]], [[2]], 1,
ZZ) == ([[2]], [[1], [2], [1]], [[1]])
assert dmp_rr_prs_gcd([[2], [4], [2]], [[2]], 1,
ZZ) == ([[2]], [[1], [2], [1]], [[1]])
assert dmp_zz_heu_gcd([[2]], [[2], [4], [2]], 1,
ZZ) == ([[2]], [[1]], [[1], [2], [1]])
assert dmp_rr_prs_gcd([[2]], [[2], [4], [2]], 1,
ZZ) == ([[2]], [[1]], [[1], [2], [1]])
assert dmp_zz_heu_gcd([[2], [4], [2]], [[1], [1]], 1,
ZZ) == ([[1], [1]], [[2], [2]], [[1]])
assert dmp_rr_prs_gcd([[2], [4], [2]], [[1], [1]], 1,
ZZ) == ([[1], [1]], [[2], [2]], [[1]])
assert dmp_zz_heu_gcd([[1], [1]], [[2], [4], [2]], 1,
ZZ) == ([[1], [1]], [[1]], [[2], [2]])
assert dmp_rr_prs_gcd([[1], [1]], [[2], [4], [2]], 1,
ZZ) == ([[1], [1]], [[1]], [[2], [2]])
assert dmp_zz_heu_gcd([[[[1, 2, 1]]]], [[[[2, 2]]]], 3,
ZZ) == ([[[[1, 1]]]], [[[[1, 1]]]], [[[[2]]]])
assert dmp_rr_prs_gcd([[[[1, 2, 1]]]], [[[[2, 2]]]], 3,
ZZ) == ([[[[1, 1]]]], [[[[1, 1]]]], [[[[2]]]])
f, g = [[[[1, 2, 1], [1, 1], []]]], [[[[1, 2, 1]]]]
h, cff, cfg = [[[[1, 1]]]], [[[[1, 1], [1], []]]], [[[[1, 1]]]]
assert dmp_zz_heu_gcd(f, g, 3, ZZ) == (h, cff, cfg)
assert dmp_rr_prs_gcd(f, g, 3, ZZ) == (h, cff, cfg)
assert dmp_zz_heu_gcd(g, f, 3, ZZ) == (h, cfg, cff)
assert dmp_rr_prs_gcd(g, f, 3, ZZ) == (h, cfg, cff)
f, g, h = dmp_fateman_poly_F_1(2, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
f, g, h = dmp_fateman_poly_F_1(4, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 4, ZZ)
assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
and dmp_mul(H, cfg, 4, ZZ) == g
f, g, h = dmp_fateman_poly_F_1(6, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 6, ZZ)
assert H == h and dmp_mul(H, cff, 6, ZZ) == f \
and dmp_mul(H, cfg, 6, ZZ) == g
f, g, h = dmp_fateman_poly_F_1(8, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 8, ZZ)
assert H == h and dmp_mul(H, cff, 8, ZZ) == f \
and dmp_mul(H, cfg, 8, ZZ) == g
f, g, h = dmp_fateman_poly_F_2(2, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
f, g, h = dmp_fateman_poly_F_3(2, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
f, g, h = dmp_fateman_poly_F_3(4, ZZ)
H, cff, cfg = dmp_inner_gcd(f, g, 4, ZZ)
assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
and dmp_mul(H, cfg, 4, ZZ) == g
f = [[QQ(1, 2)], [QQ(1)], [QQ(1, 2)]]
g = [[QQ(1, 2)], [QQ(1, 2)]]
h = [[QQ(1)], [QQ(1)]]
assert dmp_qq_heu_gcd(f, g, 1, QQ) == (h, g, [[QQ(1, 2)]])
assert dmp_ff_prs_gcd(f, g, 1, QQ) == (h, g, [[QQ(1, 2)]])
f = [[RR(2.1), RR(-2.2), RR(2.1)], []]
g = [[RR(1.0)], [], [], []]
assert dmp_ff_prs_gcd(f, g, 1, RR) == \
([[RR(1.0)], []], [[RR(2.1), RR(-2.2), RR(2.1)]], [[RR(1.0)], [], []])
def cofactors(f, g):
"""Returns GCD of `f` and `g` and their cofactors. """
lev, dom, per, F, G = f.unify(g)
h, cff, cfg = dmp_inner_gcd(F, G, lev, dom)
return per(h), per(cff), per(cfg)
def test_dmp_gcd():
assert dmp_zz_heu_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])
assert dmp_rr_prs_gcd([[]], [[]], 1, ZZ) == ([[]], [[]], [[]])
assert dmp_zz_heu_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])
assert dmp_rr_prs_gcd([[2]], [[]], 1, ZZ) == ([[2]], [[1]], [[]])
assert dmp_zz_heu_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])
assert dmp_rr_prs_gcd([[-2]], [[]], 1, ZZ) == ([[2]], [[-1]], [[]])
assert dmp_zz_heu_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])
assert dmp_rr_prs_gcd([[]], [[-2]], 1, ZZ) == ([[2]], [[]], [[-1]])
assert dmp_zz_heu_gcd([[]], [[2],[4]], 1, ZZ) == ([[2],[4]], [[]], [[1]])
assert dmp_rr_prs_gcd([[]], [[2],[4]], 1, ZZ) == ([[2],[4]], [[]], [[1]])
assert dmp_zz_heu_gcd([[2],[4]], [[]], 1, ZZ) == ([[2],[4]], [[1]], [[]])
assert dmp_rr_prs_gcd([[2],[4]], [[]], 1, ZZ) == ([[2],[4]], [[1]], [[]])
assert dmp_zz_heu_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])
assert dmp_rr_prs_gcd([[2]], [[2]], 1, ZZ) == ([[2]], [[1]], [[1]])
assert dmp_zz_heu_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])
assert dmp_rr_prs_gcd([[-2]], [[2]], 1, ZZ) == ([[2]], [[-1]], [[1]])
assert dmp_zz_heu_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])
assert dmp_rr_prs_gcd([[2]], [[-2]], 1, ZZ) == ([[2]], [[1]], [[-1]])
assert dmp_zz_heu_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])
assert dmp_rr_prs_gcd([[-2]], [[-2]], 1, ZZ) == ([[2]], [[-1]], [[-1]])
assert dmp_zz_heu_gcd([[1],[2],[1]], [[1]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[1]])
assert dmp_rr_prs_gcd([[1],[2],[1]], [[1]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[1]])
assert dmp_zz_heu_gcd([[1],[2],[1]], [[2]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[2]])
assert dmp_rr_prs_gcd([[1],[2],[1]], [[2]], 1, ZZ) == ([[1]], [[1], [2], [1]], [[2]])
assert dmp_zz_heu_gcd([[2],[4],[2]], [[2]], 1, ZZ) == ([[2]], [[1], [2], [1]], [[1]])
assert dmp_rr_prs_gcd([[2],[4],[2]], [[2]], 1, ZZ) == ([[2]], [[1], [2], [1]], [[1]])
assert dmp_zz_heu_gcd([[2]], [[2],[4],[2]], 1, ZZ) == ([[2]], [[1]], [[1], [2], [1]])
assert dmp_rr_prs_gcd([[2]], [[2],[4],[2]], 1, ZZ) == ([[2]], [[1]], [[1], [2], [1]])
assert dmp_zz_heu_gcd([[2],[4],[2]], [[1],[1]], 1, ZZ) == ([[1], [1]], [[2], [2]], [[1]])
assert dmp_rr_prs_gcd([[2],[4],[2]], [[1],[1]], 1, ZZ) == ([[1], [1]], [[2], [2]], [[1]])
assert dmp_zz_heu_gcd([[1],[1]], [[2],[4],[2]], 1, ZZ) == ([[1], [1]], [[1]], [[2], [2]])
assert dmp_rr_prs_gcd([[1],[1]], [[2],[4],[2]], 1, ZZ) == ([[1], [1]], [[1]], [[2], [2]])
assert dmp_zz_heu_gcd([[[[1,2,1]]]], [[[[2,2]]]], 3, ZZ) == ([[[[1,1]]]], [[[[1,1]]]], [[[[2]]]])
assert dmp_rr_prs_gcd([[[[1,2,1]]]], [[[[2,2]]]], 3, ZZ) == ([[[[1,1]]]], [[[[1,1]]]], [[[[2]]]])
f, g = [[[[1,2,1],[1,1],[]]]], [[[[1,2,1]]]]
h, cff, cfg = [[[[1,1]]]], [[[[1,1],[1],[]]]], [[[[1,1]]]]
assert dmp_zz_heu_gcd(f, g, 3, ZZ) == (h, cff, cfg)
assert dmp_rr_prs_gcd(f, g, 3, ZZ) == (h, cff, cfg)
assert dmp_zz_heu_gcd(g, f, 3, ZZ) == (h, cfg, cff)
assert dmp_rr_prs_gcd(g, f, 3, ZZ) == (h, cfg, cff)
f, g, h = dmp_fateman_poly_F_1(2, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
f, g, h = dmp_fateman_poly_F_1(4, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 4, ZZ)
assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
and dmp_mul(H, cfg, 4, ZZ) == g
f, g, h = dmp_fateman_poly_F_1(6, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 6, ZZ)
assert H == h and dmp_mul(H, cff, 6, ZZ) == f \
and dmp_mul(H, cfg, 6, ZZ) == g
f, g, h = dmp_fateman_poly_F_1(8, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 8, ZZ)
assert H == h and dmp_mul(H, cff, 8, ZZ) == f \
and dmp_mul(H, cfg, 8, ZZ) == g
f, g, h = dmp_fateman_poly_F_2(2, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
f, g, h = dmp_fateman_poly_F_3(2, ZZ)
H, cff, cfg = dmp_zz_heu_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
H, cff, cfg = dmp_rr_prs_gcd(f, g, 2, ZZ)
assert H == h and dmp_mul(H, cff, 2, ZZ) == f \
and dmp_mul(H, cfg, 2, ZZ) == g
f, g, h = dmp_fateman_poly_F_3(4, ZZ)
H, cff, cfg = dmp_inner_gcd(f, g, 4, ZZ)
assert H == h and dmp_mul(H, cff, 4, ZZ) == f \
and dmp_mul(H, cfg, 4, ZZ) == g
f = [[QQ(1,2)],[QQ(1)],[QQ(1,2)]]
g = [[QQ(1,2)],[QQ(1,2)]]
h = [[QQ(1)],[QQ(1)]]
assert dmp_qq_heu_gcd(f, g, 1, QQ) == (h, g, [[QQ(1,2)]])
assert dmp_ff_prs_gcd(f, g, 1, QQ) == (h, g, [[QQ(1,2)]])
def cofactors(f, g):
"""Returns GCD of `f` and `g` and their cofactors. """
lev, dom, per, F, G = f.unify(g)
h, cff, cfg = dmp_inner_gcd(F, G, lev, dom)
return per(h), per(cff), per(cfg)