def test_dmp_degree_list():
assert dmp_degree_list([[[[ ]]]], 3) == (-1,-1,-1,-1)
assert dmp_degree_list([[[[1]]]], 3) == ( 0, 0, 0, 0)
assert dmp_degree_list(f_0, 2) == (2, 2, 2)
assert dmp_degree_list(f_1, 2) == (3, 3, 3)
assert dmp_degree_list(f_2, 2) == (5, 3, 3)
assert dmp_degree_list(f_3, 2) == (5, 4, 7)
assert dmp_degree_list(f_4, 2) == (9, 12, 8)
assert dmp_degree_list(f_5, 2) == (3, 3, 3)
assert dmp_degree_list(f_6, 3) == (4, 4, 6, 3)
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_zz_mignotte_bound(f, u, K):
"""Mignotte bound for multivariate polynomials in `K[X]`. """
a = dmp_max_norm(f, u, K)
b = abs(dmp_ground_LC(f, u, K))
n = sum(dmp_degree_list(f, u))
return K.sqrt(K(n+1))*2**n*a*b
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_zz_mignotte_bound(f, u, K):
"""Mignotte bound for multivariate polynomials in `K[X]`. """
a = dmp_max_norm(f, u, K)
b = abs(dmp_ground_LC(f, u, K))
n = sum(dmp_degree_list(f, u))
return K.sqrt(K(n + 1)) * 2**n * a * b
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_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_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_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_factor(f, u, K):
"""
Factor (non square-free) polynomials in `Z[X]`.
Given a multivariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
is used to recover the multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*(x**2 - y**2)`::
>>> from sympy.polys.factortools import dmp_zz_factor
>>> from sympy.polys.domains import ZZ
>>> dmp_zz_factor([[2], [], [-2, 0, 0]], 1, ZZ)
(2, [([[1], [-1, 0]], 1), ([[1], [1, 0]], 1)])
In result we got the following factorization::
f = 2 (x - y) (x + y)
**References**
1. [Gathen99]_
"""
if not u:
return dup_zz_factor(f, K)
if dmp_zero_p(f, u):
return K.zero, []
cont, g = dmp_ground_primitive(f, u, K)
if dmp_ground_LC(g, u, K) < 0:
cont, g = -cont, dmp_neg(g, u, K)
if all([ d <= 0 for d in dmp_degree_list(g, u) ]):
return cont, []
G, g = dmp_primitive(g, u, K)
factors = []
if dmp_degree(g, u) > 0:
g = dmp_sqf_part(g, u, K)
H = dmp_zz_wang(g, u, K)
for h in H:
k = 0
while True:
q, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
f, k = q, k+1
else:
break
factors.append((h, k))
for g, k in dmp_zz_factor(G, u-1, K)[1]:
factors.insert(0, ([g], k))
return cont, _sort_factors(factors)
def degree_list(f):
"""Returns a list of degrees of `f`. """
return dmp_degree_list(f.rep, f.lev)
def degree_list(f):
"""Returns a list of degrees of `f`. """
return dmp_degree_list(f.rep, f.lev)
def is_ground(f):
"""Returns `True` if `f` is an element of the ground domain. """
return all(d <= 0 for d in dmp_degree_list(f.rep, f.lev))
def dmp_zz_factor(f, u, K):
"""
Factor (non square-free) polynomials in `Z[X]`.
Given a multivariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
is used to recover the multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*(x**2 - y**2)`::
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_zz_factor(2*x**2 - 2*y**2)
(2, [(x - y, 1), (x + y, 1)])
In result we got the following factorization::
f = 2 (x - y) (x + y)
References
==========
1. [Gathen99]_
"""
if not u:
return dup_zz_factor(f, K)
if dmp_zero_p(f, u):
return K.zero, []
cont, g = dmp_ground_primitive(f, u, K)
if dmp_ground_LC(g, u, K) < 0:
cont, g = -cont, dmp_neg(g, u, K)
if all(d <= 0 for d in dmp_degree_list(g, u)):
return cont, []
G, g = dmp_primitive(g, u, K)
factors = []
if dmp_degree(g, u) > 0:
g = dmp_sqf_part(g, u, K)
H = dmp_zz_wang(g, u, K)
factors = dmp_trial_division(f, H, u, K)
for g, k in dmp_zz_factor(G, u - 1, K)[1]:
factors.insert(0, ([g], k))
return cont, _sort_factors(factors)
def dmp_zz_factor(f, u, K):
"""
Factor (non square-free) polynomials in `Z[X]`.
Given a multivariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
is used to recover the multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*(x**2 - y**2)`::
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_zz_factor(2*x**2 - 2*y**2)
(2, [(x - y, 1), (x + y, 1)])
In result we got the following factorization::
f = 2 (x - y) (x + y)
References
==========
.. [1] [Gathen99]_
"""
if not u:
return dup_zz_factor(f, K)
if dmp_zero_p(f, u):
return K.zero, []
cont, g = dmp_ground_primitive(f, u, K)
if dmp_ground_LC(g, u, K) < 0:
cont, g = -cont, dmp_neg(g, u, K)
if all(d <= 0 for d in dmp_degree_list(g, u)):
return cont, []
G, g = dmp_primitive(g, u, K)
factors = []
if dmp_degree(g, u) > 0:
g = dmp_sqf_part(g, u, K)
H = dmp_zz_wang(g, u, K)
factors = dmp_trial_division(f, H, u, K)
for g, k in dmp_zz_factor(G, u - 1, K)[1]:
factors.insert(0, ([g], k))
return cont, _sort_factors(factors)
def total_degree(f):
"""Returns the total degree of `f`. """
return sum(dmp_degree_list(f.rep, f.lev))
def total_degree(f):
"""Returns the total degree of `f`. """
return sum(dmp_degree_list(f.rep, f.lev))
def dmp_zz_factor(f, u, K):
"""
Factor (non square-free) polynomials in `Z[X]`.
Given a multivariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
is used to recover the multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*(x**2 - y**2)`::
>>> from sympy.polys.factortools import dmp_zz_factor
>>> from sympy.polys.domains import ZZ
>>> dmp_zz_factor([[2], [], [-2, 0, 0]], 1, ZZ)
(2, [([[1], [-1, 0]], 1), ([[1], [1, 0]], 1)])
In result we got the following factorization::
f = 2 (x - y) (x + y)
References
==========
1. [Gathen99]_
"""
if not u:
return dup_zz_factor(f, K)
if dmp_zero_p(f, u):
return K.zero, []
cont, g = dmp_ground_primitive(f, u, K)
if dmp_ground_LC(g, u, K) < 0:
cont, g = -cont, dmp_neg(g, u, K)
if all(d <= 0 for d in dmp_degree_list(g, u)):
return cont, []
G, g = dmp_primitive(g, u, K)
factors = []
if dmp_degree(g, u) > 0:
g = dmp_sqf_part(g, u, K)
H = dmp_zz_wang(g, u, K)
for h in H:
k = 0
while True:
q, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
f, k = q, k + 1
else:
break
factors.append((h, k))
for g, k in dmp_zz_factor(G, u - 1, K)[1]:
factors.insert(0, ([g], k))
return cont, _sort_factors(factors)
def is_ground(f):
"""Returns `True` if `f` is an element of the ground domain. """
return all(d <= 0 for d in dmp_degree_list(f.rep, f.lev))