def dmp_zz_collins_resultant(f, g, u, K):
"""
Collins's modular resultant algorithm in `Z[X]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_collins_resultant(f, g)
-2*y**2 - 5*y + 1
"""
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < 0 or m < 0:
return dmp_zero(u - 1)
A = dmp_max_norm(f, u, K)
B = dmp_max_norm(g, u, K)
a = dmp_ground_LC(f, u, K)
b = dmp_ground_LC(g, u, K)
v = u - 1
B = K(2)*K.factorial(K(n + m))*A**m*B**n
r, p, P = dmp_zero(v), K.one, K.one
while P <= B:
p = K(nextprime(p))
while not (a % p) or not (b % p):
p = K(nextprime(p))
F = dmp_ground_trunc(f, p, u, K)
G = dmp_ground_trunc(g, p, u, K)
try:
R = dmp_zz_modular_resultant(F, G, p, u, K)
except HomomorphismFailed:
continue
if K.is_one(P):
r = R
else:
r = dmp_apply_pairs(r, R, _collins_crt, (P, p, K), v, K)
P *= p
return r
def dmp_zz_collins_resultant(f, g, u, K):
"""
Collins's modular resultant algorithm in `Z[X]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_collins_resultant
>>> f = ZZ.map([[1], [1, 2]])
>>> g = ZZ.map([[2, 1], [3]])
>>> dmp_zz_collins_resultant(f, g, 1, ZZ)
[-2, -5, 1]
"""
n = dmp_degree(f, u)
m = dmp_degree(g, u)
if n < 0 or m < 0:
return dmp_zero(u-1)
A = dmp_max_norm(f, u, K)
B = dmp_max_norm(g, u, K)
a = dmp_ground_LC(f, u, K)
b = dmp_ground_LC(g, u, K)
v = u - 1
B = K(2)*K.factorial(n+m)*A**m*B**n
r, p, P = dmp_zero(v), K.one, K.one
while P <= B:
p = K(nextprime(p))
while not (a % p) or not (b % p):
p = K(nextprime(p))
F = dmp_ground_trunc(f, p, u, K)
G = dmp_ground_trunc(g, p, u, K)
try:
R = dmp_zz_modular_resultant(F, G, p, u, K)
except HomomorphismFailed:
continue
if K.is_one(P):
r = R
else:
r = dmp_apply_pairs(r, R, _collins_crt, (P, p, K), v, K)
P *= p
return r
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_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_heu_gcd(f, g, u, K):
"""
Heuristic polynomial GCD in ``Z[X]``.
Given univariate polynomials ``f`` and ``g`` in ``Z[X]``, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The evaluation proces reduces f and g variable by
variable into a large integer. The final step is to verify if the
interpolated polynomial is the correct GCD. This gives cofactors of
the input polynomials as a side effect.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_heu_gcd
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0]])
>>> g = ZZ.map([[1], [1, 0], []])
>>> dmp_zz_heu_gcd(f, g, 1, ZZ)
([[1], [1, 0]], [[1], [1, 0]], [[1], []])
**References**
1. [Liao95]_
"""
if not u:
return dup_zz_heu_gcd(f, g, K)
result = _dmp_rr_trivial_gcd(f, g, u, K)
if result is not None:
return result
df = dmp_degree(f, u)
dg = dmp_degree(g, u)
gcd, f, g = dmp_ground_extract(f, g, u, K)
f_norm = dmp_max_norm(f, u, K)
g_norm = dmp_max_norm(g, u, K)
B = 2*min(f_norm, g_norm) + 29
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dmp_ground_LC(f, u, K)),
g_norm // abs(dmp_ground_LC(g, u, K))) + 2)
for i in xrange(0, HEU_GCD_MAX):
ff = dmp_eval(f, x, u, K)
gg = dmp_eval(g, x, u, K)
v = u - 1
if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)):
h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K)
h = _dmp_zz_gcd_interpolate(h, x, v, K)
h = dmp_ground_primitive(h, u, K)[1]
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg_
cff = _dmp_zz_gcd_interpolate(cff, x, v, K)
h, r = dmp_div(f, cff, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff, cfg_
cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K)
h, r = dmp_div(g, cfg, u, K)
if dmp_zero_p(r, u):
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def max_norm(f):
"""Returns maximum norm of `f`. """
return dmp_max_norm(f.rep, f.lev, f.dom)
def test_dmp_max_norm():
assert dmp_max_norm([[[]]], 2, ZZ) == 0
assert dmp_max_norm([[[1]]], 2, ZZ) == 1
assert dmp_max_norm(f_0, 2, ZZ) == 6
def dmp_zz_heu_gcd(f, g, u, K):
"""
Heuristic polynomial GCD in `Z[X]`.
Given univariate polynomials `f` and `g` in `Z[X]`, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The evaluation proces reduces f and g variable by
variable into a large integer. The final step is to verify if the
interpolated polynomial is the correct GCD. This gives cofactors of
the input polynomials as a side effect.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y, = ring("x,y", ZZ)
>>> f = x**2 + 2*x*y + y**2
>>> g = x**2 + x*y
>>> R.dmp_zz_heu_gcd(f, g)
(x + y, x + y, x)
References
==========
1. [Liao95]_
"""
if not u:
return dup_zz_heu_gcd(f, g, K)
result = _dmp_rr_trivial_gcd(f, g, u, K)
if result is not None:
return result
gcd, f, g = dmp_ground_extract(f, g, u, K)
f_norm = dmp_max_norm(f, u, K)
g_norm = dmp_max_norm(g, u, K)
B = K(2*min(f_norm, g_norm) + 29)
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dmp_ground_LC(f, u, K)),
g_norm // abs(dmp_ground_LC(g, u, K))) + 2)
for i in xrange(0, HEU_GCD_MAX):
ff = dmp_eval(f, x, u, K)
gg = dmp_eval(g, x, u, K)
v = u - 1
if not (dmp_zero_p(ff, v) or dmp_zero_p(gg, v)):
h, cff, cfg = dmp_zz_heu_gcd(ff, gg, v, K)
h = _dmp_zz_gcd_interpolate(h, x, v, K)
h = dmp_ground_primitive(h, u, K)[1]
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg_
cff = _dmp_zz_gcd_interpolate(cff, x, v, K)
h, r = dmp_div(f, cff, u, K)
if dmp_zero_p(r, u):
cfg_, r = dmp_div(g, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff, cfg_
cfg = _dmp_zz_gcd_interpolate(cfg, x, v, K)
h, r = dmp_div(g, cfg, u, K)
if dmp_zero_p(r, u):
cff_, r = dmp_div(f, h, u, K)
if dmp_zero_p(r, u):
h = dmp_mul_ground(h, gcd, u, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def dmp_factor_list(f, u, K0):
"""Factor multivariate polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list(f, K0)
J, f = dmp_terms_gcd(f, u, K0)
cont, f = dmp_ground_primitive(f, u, K0)
if K0.is_FiniteField: # pragma: no cover
coeff, factors = dmp_gf_factor(f, u, K0)
elif K0.is_Algebraic:
coeff, factors = dmp_ext_factor(f, u, K0)
# elif K0.is_GaussianRing:
# coeff, factors = dmp_zz_i_factor(f, u, K0)
# elif K0.is_GaussianField:
# coeff, factors = dmp_qq_i_factor(f, u, K0)
else:
if not K0.is_Exact:
K0_inexact, K0 = K0, K0.get_exact()
f = dmp_convert(f, u, K0_inexact, K0)
else:
K0_inexact = None
if K0.is_Field:
K = K0.get_ring()
denom, f = dmp_clear_denoms(f, u, K0, K)
f = dmp_convert(f, u, K0, K)
else:
K = K0
if K.is_ZZ:
levels, f, v = dmp_exclude(f, u, K)
coeff, factors = dmp_zz_factor(f, v, K)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_include(f, levels, v, K), k)
elif K.is_Poly:
f, v = dmp_inject(f, u, K)
coeff, factors = dmp_factor_list(f, v, K.dom)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_eject(f, v, K), k)
coeff = K.convert(coeff, K.dom)
else: # pragma: no cover
raise DomainError('factorization not supported over %s' % K0)
if K0.is_Field:
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_convert(f, u, K, K0), k)
coeff = K0.convert(coeff, K)
coeff = K0.quo(coeff, denom)
if K0_inexact:
for i, (f, k) in enumerate(factors):
max_norm = dmp_max_norm(f, u, K0)
f = dmp_quo_ground(f, max_norm, u, K0)
f = dmp_convert(f, u, K0, K0_inexact)
factors[i] = (f, k)
coeff = K0.mul(coeff, K0.pow(max_norm, k))
coeff = K0_inexact.convert(coeff, K0)
K0 = K0_inexact
for i, j in enumerate(reversed(J)):
if not j:
continue
term = {(0, ) * (u - i) + (1, ) + (0, ) * i: K0.one}
factors.insert(0, (dmp_from_dict(term, u, K0), j))
return coeff * cont, _sort_factors(factors)
def max_norm(f):
"""Returns maximum norm of `f`. """
return dmp_max_norm(f.rep, f.lev, f.dom)
def dmp_factor_list(f, u, K0):
"""Factor multivariate polynomials into irreducibles in `K[X]`. """
if not u:
return dup_factor_list(f, K0)
J, f = dmp_terms_gcd(f, u, K0)
cont, f = dmp_ground_primitive(f, u, K0)
if K0.is_FiniteField: # pragma: no cover
coeff, factors = dmp_gf_factor(f, u, K0)
elif K0.is_Algebraic:
coeff, factors = dmp_ext_factor(f, u, K0)
else:
if not K0.is_Exact:
K0_inexact, K0 = K0, K0.get_exact()
f = dmp_convert(f, u, K0_inexact, K0)
else:
K0_inexact = None
if K0.is_Field:
K = K0.get_ring()
denom, f = dmp_clear_denoms(f, u, K0, K)
f = dmp_convert(f, u, K0, K)
else:
K = K0
if K.is_ZZ:
levels, f, v = dmp_exclude(f, u, K)
coeff, factors = dmp_zz_factor(f, v, K)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_include(f, levels, v, K), k)
elif K.is_Poly:
f, v = dmp_inject(f, u, K)
coeff, factors = dmp_factor_list(f, v, K.dom)
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_eject(f, v, K), k)
coeff = K.convert(coeff, K.dom)
else: # pragma: no cover
raise DomainError('factorization not supported over %s' % K0)
if K0.is_Field:
for i, (f, k) in enumerate(factors):
factors[i] = (dmp_convert(f, u, K, K0), k)
coeff = K0.convert(coeff, K)
coeff = K0.quo(coeff, denom)
if K0_inexact:
for i, (f, k) in enumerate(factors):
max_norm = dmp_max_norm(f, u, K0)
f = dmp_quo_ground(f, max_norm, u, K0)
f = dmp_convert(f, u, K0, K0_inexact)
factors[i] = (f, k)
coeff = K0.mul(coeff, K0.pow(max_norm, k))
coeff = K0_inexact.convert(coeff, K0)
K0 = K0_inexact
for i, j in enumerate(reversed(J)):
if not j:
continue
term = {(0,)*(u - i) + (1,) + (0,)*i: K0.one}
factors.insert(0, (dmp_from_dict(term, u, K0), j))
return coeff*cont, _sort_factors(factors)