def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_gcdex(f, g)
(-1/5*x + 3/5, 1/5*x**2 - 6/5*x + 2, x + 1)
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in ``F[x]``.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
**Examples**
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_half_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_half_gcdex(f, g, QQ)
([-1/5, 3/5], [1/1, 1/1])
"""
if not (K.has_Field or not K.is_Exact):
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_exquo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in `F[x]`.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_half_gcdex(f, g)
(-1/5*x + 3/5, x + 1)
"""
if not K.has_Field:
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_quo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in `F[x]`.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_half_gcdex(f, g)
(-1/5*x + 3/5, x + 1)
"""
if not K.has_Field:
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_quo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_gcdex(f, g, QQ)
([-1/5, 3/5], [1/5, -6/5, 2/1], [1/1, 1/1])
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> R, x = ring("x", QQ)
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> R.dup_gcdex(f, g)
(-1/5*x + 3/5, 1/5*x**2 - 6/5*x + 2, x + 1)
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_gcdex(f, g, K):
"""
Extended Euclidean algorithm in `F[x]`.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_gcdex(f, g, QQ)
([-1/5, 3/5], [1/5, -6/5, 2/1], [1/1, 1/1])
"""
s, h = dup_half_gcdex(f, g, K)
F = dup_sub_mul(h, s, f, K)
t = dup_quo(F, g, K)
return s, t, h
def dup_half_gcdex(f, g, K):
"""
Half extended Euclidean algorithm in `F[x]`.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.euclidtools import dup_half_gcdex
>>> f = QQ.map([1, -2, -6, 12, 15])
>>> g = QQ.map([1, 1, -4, -4])
>>> dup_half_gcdex(f, g, QQ)
([-1/5, 3/5], [1/1, 1/1])
"""
if not (K.has_Field or not K.is_Exact):
raise DomainError("can't compute half extended GCD over %s" % K)
a, b = [K.one], []
while g:
q, r = dup_div(f, g, K)
f, g = g, r
a, b = b, dup_sub_mul(a, q, b, K)
a = dup_quo_ground(a, dup_LC(f, K), K)
f = dup_monic(f, K)
return a, f
def test_dup_sub_mul():
assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [
ZZ(-3),
ZZ(-7),
ZZ(-3),
ZZ(1),
]
def test_dup_sub_mul():
assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)],
[ZZ(3), ZZ(2), ZZ(1)], [ZZ(1), ZZ(2)],
ZZ) == [ZZ(-3), ZZ(-7), ZZ(-3),
ZZ(1)]
assert dmp_sub_mul([[ZZ(1), ZZ(2)], [ZZ(3)]],
[[ZZ(3)], [ZZ(2), ZZ(1)]], [[ZZ(1)], [ZZ(2)]], 1,
ZZ) == [[ZZ(-3)], [ZZ(-1), ZZ(-5)], [ZZ(-4),
ZZ(1)]]
def dup_zz_hensel_step(m, f, g, h, s, t, K):
"""
One step in Hensel lifting in `Z[x]`.
Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s`
and `t` such that::
f == g*h (mod m)
s*g + t*h == 1 (mod m)
lc(f) is not a zero divisor (mod m)
lc(h) == 1
deg(f) == deg(g) + deg(h)
deg(s) < deg(h)
deg(t) < deg(g)
returns polynomials `G`, `H`, `S` and `T`, such that::
f == G*H (mod m**2)
S*G + T**H == 1 (mod m**2)
References
==========
1. [Gathen99]_
"""
M = m**2
e = dup_sub_mul(f, g, h, K)
e = dup_trunc(e, M, K)
q, r = dup_div(dup_mul(s, e, K), h, K)
q = dup_trunc(q, M, K)
r = dup_trunc(r, M, K)
u = dup_add(dup_mul(t, e, K), dup_mul(q, g, K), K)
G = dup_trunc(dup_add(g, u, K), M, K)
H = dup_trunc(dup_add(h, r, K), M, K)
u = dup_add(dup_mul(s, G, K), dup_mul(t, H, K), K)
b = dup_trunc(dup_sub(u, [K.one], K), M, K)
c, d = dup_div(dup_mul(s, b, K), H, K)
c = dup_trunc(c, M, K)
d = dup_trunc(d, M, K)
u = dup_add(dup_mul(t, b, K), dup_mul(c, G, K), K)
S = dup_trunc(dup_sub(s, d, K), M, K)
T = dup_trunc(dup_sub(t, u, K), M, K)
return G, H, S, T
def dup_zz_hensel_step(m, f, g, h, s, t, K):
"""
One step in Hensel lifting in `Z[x]`.
Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s`
and `t` such that::
f == g*h (mod m)
s*g + t*h == 1 (mod m)
lc(f) is not a zero divisor (mod m)
lc(h) == 1
deg(f) == deg(g) + deg(h)
deg(s) < deg(h)
deg(t) < deg(g)
returns polynomials `G`, `H`, `S` and `T`, such that::
f == G*H (mod m**2)
S*G + T**H == 1 (mod m**2)
References
==========
1. [Gathen99]_
"""
M = m**2
e = dup_sub_mul(f, g, h, K)
e = dup_trunc(e, M, K)
q, r = dup_div(dup_mul(s, e, K), h, K)
q = dup_trunc(q, M, K)
r = dup_trunc(r, M, K)
u = dup_add(dup_mul(t, e, K), dup_mul(q, g, K), K)
G = dup_trunc(dup_add(g, u, K), M, K)
H = dup_trunc(dup_add(h, r, K), M, K)
u = dup_add(dup_mul(s, G, K), dup_mul(t, H, K), K)
b = dup_trunc(dup_sub(u, [K.one], K), M, K)
c, d = dup_div(dup_mul(s, b, K), H, K)
c = dup_trunc(c, M, K)
d = dup_trunc(d, M, K)
u = dup_add(dup_mul(t, b, K), dup_mul(c, G, K), K)
S = dup_trunc(dup_sub(s, d, K), M, K)
T = dup_trunc(dup_sub(t, u, K), M, K)
return G, H, S, T
def test_dup_sub_mul():
assert dup_sub_mul([ZZ(1),ZZ(2),ZZ(3)], [ZZ(3),ZZ(2),ZZ(1)],
[ZZ(1),ZZ(2)], ZZ) == [ZZ(-3),ZZ(-7),ZZ(-3), ZZ(1)]
def test_dup_sub_mul():
assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
[ZZ(1), ZZ(2)], ZZ) == [ZZ(-3), ZZ(-7), ZZ(-3), ZZ(1)]
assert dmp_sub_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
[[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(-3)], [ZZ(-1), ZZ(-5)], [ZZ(-4), ZZ(1)]]