def dmp_ground_extract(f, g, u, K):
"""
Extract common content from a pair of polynomials in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_ground_extract
>>> f = ZZ.map([[6, 12], [18]])
>>> g = ZZ.map([[4, 8], [12]])
>>> dmp_ground_extract(f, g, 1, ZZ)
(2, [[3, 6], [9]], [[2, 4], [6]])
"""
fc = dmp_ground_content(f, u, K)
gc = dmp_ground_content(g, u, K)
gcd = K.gcd(fc, gc)
if not K.is_one(gcd):
f = dmp_exquo_ground(f, gcd, u, K)
g = dmp_exquo_ground(g, gcd, u, K)
return gcd, f, g
def test_dmp_exquo_ground():
f = dmp_normal([[6],[2],[8]], 1, ZZ)
assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f
assert dmp_exquo_ground(f, ZZ(2), 1, ZZ) == dmp_normal([[3],[1],[4]], 1, ZZ)
assert dmp_normal(dmp_exquo_ground(f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2],[],[2]], 1, ZZ)
def dmp_ground_extract(f, g, u, K):
"""
Extract common content from a pair of polynomials in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_ground_extract
>>> f = ZZ.map([[6, 12], [18]])
>>> g = ZZ.map([[4, 8], [12]])
>>> dmp_ground_extract(f, g, 1, ZZ)
(2, [[3, 6], [9]], [[2, 4], [6]])
"""
fc = dmp_ground_content(f, u, K)
gc = dmp_ground_content(g, u, K)
gcd = K.gcd(fc, gc)
if not K.is_one(gcd):
f = dmp_exquo_ground(f, gcd, u, K)
g = dmp_exquo_ground(g, gcd, u, K)
return gcd, f, g
def test_dmp_exquo_ground():
f = dmp_normal([[6],[2],[8]], 1, ZZ)
assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f
assert dmp_exquo_ground(f, ZZ(2), 1, ZZ) == dmp_normal([[3],[1],[4]], 1, ZZ)
assert dmp_normal(dmp_exquo_ground(f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2],[],[2]], 1, ZZ)
def dmp_ground_primitive(f, u, K):
"""
Compute content and the primitive form of ``f`` in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densetools import dmp_ground_primitive
>>> f = ZZ.map([[2, 6], [4, 12]])
>>> g = QQ.map([[2, 6], [4, 12]])
>>> dmp_ground_primitive(f, 1, ZZ)
(2, [[1, 3], [2, 6]])
>>> dmp_ground_primitive(g, 1, QQ)
(1/1, [[2/1, 6/1], [4/1, 12/1]])
"""
if not u:
return dup_primitive(f, K)
if dmp_zero_p(f, u):
return K.zero, f
cont = dmp_ground_content(f, u, K)
if K.is_one(cont):
return cont, f
else:
return cont, dmp_exquo_ground(f, cont, u, K)
def dmp_ground_monic(f, u, K):
"""
Divide all coefficients by ``LC(f)`` in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ, QQ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y + 6*x**2 + 3*x*y + 9*y + 3
>>> R.dmp_ground_monic(f)
x**2*y + 2*x**2 + x*y + 3*y + 1
>>> R, x,y = ring("x,y", QQ)
>>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3
>>> R.dmp_ground_monic(f)
x**2*y + 8/3*x**2 + 5/3*x*y + 2*x + 2/3*y + 1
"""
if not u:
return dup_monic(f, K)
if dmp_zero_p(f, u):
return f
lc = dmp_ground_LC(f, u, K)
if K.is_one(lc):
return f
else:
return dmp_exquo_ground(f, lc, u, K)
def dmp_ground_primitive(f, u, K):
"""
Compute content and the primitive form of ``f`` in ``K[X]``.
**Examples**
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densetools import dmp_ground_primitive
>>> f = ZZ.map([[2, 6], [4, 12]])
>>> g = QQ.map([[2, 6], [4, 12]])
>>> dmp_ground_primitive(f, 1, ZZ)
(2, [[1, 3], [2, 6]])
>>> dmp_ground_primitive(g, 1, QQ)
(1/1, [[2/1, 6/1], [4/1, 12/1]])
"""
if not u:
return dup_primitive(f, K)
if dmp_zero_p(f, u):
return K.zero, f
cont = dmp_ground_content(f, u, K)
if K.is_one(cont):
return cont, f
else:
return cont, dmp_exquo_ground(f, cont, u, K)
def dmp_ground_monic(f, u, K):
"""
Divide all coefficients by ``LC(f)`` in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ, QQ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y + 6*x**2 + 3*x*y + 9*y + 3
>>> R.dmp_ground_monic(f)
x**2*y + 2*x**2 + x*y + 3*y + 1
>>> R, x,y = ring("x,y", QQ)
>>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3
>>> R.dmp_ground_monic(f)
x**2*y + 8/3*x**2 + 5/3*x*y + 2*x + 2/3*y + 1
"""
if not u:
return dup_monic(f, K)
if dmp_zero_p(f, u):
return f
lc = dmp_ground_LC(f, u, K)
if K.is_one(lc):
return f
else:
return dmp_exquo_ground(f, lc, u, K)
def dmp_ground_monic(f, u, K):
"""
Divides all coefficients by ``LC(f)`` in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densetools import dmp_ground_monic
>>> f = ZZ.map([[3, 6], [3, 0], [9, 3]])
>>> g = QQ.map([[3, 8], [5, 6], [2, 3]])
>>> dmp_ground_monic(f, 1, ZZ)
[[1, 2], [1, 0], [3, 1]]
>>> dmp_ground_monic(g, 1, QQ)
[[1/1, 8/3], [5/3, 2/1], [2/3, 1/1]]
"""
if not u:
return dup_monic(f, K)
if dmp_zero_p(f, u):
return f
lc = dmp_ground_LC(f, u, K)
if K.is_one(lc):
return f
else:
return dmp_exquo_ground(f, lc, u, K)
def dmp_ground_monic(f, u, K):
"""
Divides all coefficients by ``LC(f)`` in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densetools import dmp_ground_monic
>>> f = ZZ.map([[3, 6], [3, 0], [9, 3]])
>>> g = QQ.map([[3, 8], [5, 6], [2, 3]])
>>> dmp_ground_monic(f, 1, ZZ)
[[1, 2], [1, 0], [3, 1]]
>>> dmp_ground_monic(g, 1, QQ)
[[1/1, 8/3], [5/3, 2/1], [2/3, 1/1]]
"""
if not u:
return dup_monic(f, K)
if dmp_zero_p(f, u):
return f
lc = dmp_ground_LC(f, u, K)
if K.is_one(lc):
return f
else:
return dmp_exquo_ground(f, lc, u, K)
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(xrange(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 xrange(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_exquo_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_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(xrange(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 xrange(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_exquo_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_gcd_interpolate(h, x, v, K):
"""Interpolate polynomial GCD from integer GCD. """
f = []
while not dmp_zero_p(h, v):
g = dmp_ground_trunc(h, x, v, K)
f.insert(0, g)
h = dmp_sub(h, g, v, K)
h = dmp_exquo_ground(h, x, v, K)
if K.is_negative(dmp_ground_LC(f, v+1, K)):
return dmp_neg(f, v+1, K)
else:
return f
def _dmp_zz_gcd_interpolate(h, x, v, K):
"""Interpolate polynomial GCD from integer GCD. """
f = []
while not dmp_zero_p(h, v):
g = dmp_ground_trunc(h, x, v, K)
f.insert(0, g)
h = dmp_sub(h, g, v, K)
h = dmp_exquo_ground(h, x, v, K)
if K.is_negative(dmp_ground_LC(f, v + 1, K)):
return dmp_neg(f, v + 1, K)
else:
return f
def dmp_qq_collins_resultant(f, g, u, K0):
"""
Collins's modular resultant algorithm in ``Q[X]``.
**Examples**
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dmp_qq_collins_resultant
>>> f = [[QQ(1,2)], [QQ(1), QQ(2,3)]]
>>> g = [[QQ(2), QQ(1)], [QQ(3)]]
>>> dmp_qq_collins_resultant(f, g, 1, QQ)
[-2/1, -7/3, 5/6]
"""
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < 0 or m < 0:
return dmp_zero(u-1)
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
r = dmp_zz_collins_resultant(f, g, u, K1)
r = dmp_convert(r, u-1, K1, K0)
c = K0.convert(cf**m * cg**n, K1)
return dmp_exquo_ground(r, c, u-1, K0)
def dmp_qq_collins_resultant(f, g, u, K0):
"""
Collins's modular resultant algorithm in ``Q[X]``.
**Examples**
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dmp_qq_collins_resultant
>>> f = [[QQ(1,2)], [QQ(1), QQ(2,3)]]
>>> g = [[QQ(2), QQ(1)], [QQ(3)]]
>>> dmp_qq_collins_resultant(f, g, 1, QQ)
[-2/1, -7/3, 5/6]
"""
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < 0 or m < 0:
return dmp_zero(u - 1)
K1 = K0.get_ring()
cf, f = dmp_clear_denoms(f, u, K0, K1)
cg, g = dmp_clear_denoms(g, u, K0, K1)
f = dmp_convert(f, u, K0, K1)
g = dmp_convert(g, u, K0, K1)
r = dmp_zz_collins_resultant(f, g, u, K1)
r = dmp_convert(r, u - 1, K1, K0)
c = K0.convert(cf**m * cg**n, K1)
return dmp_exquo_ground(r, c, u - 1, K0)
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_exquo(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 xrange(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_exquo_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
def exquo_ground(f, c):
"""Exact quotient of ``f`` by a an element of the ground domain. """
return f.per(dmp_exquo_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
def exquo_ground(f, c):
"""Exact quotient of ``f`` by a an element of the ground domain. """
return f.per(dmp_exquo_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
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_exquo(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 xrange(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_exquo_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
