def test_poly_subresultants():
f = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5
g = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21
assert poly_subresultants(f, g, x) == \
[Poly(f, x), Poly(g, x),
Poly(15*x**4 - 3*x**2 + 9, x),
Poly(65*x**2 + 125*x - 245, x),
Poly(9326*x - 12300, x),
Poly(260708, x)]
assert poly_subresultants((x-1)**2, x**2-1, x) == \
[Poly((x-1)**2, x), Poly(x**2-1, x), Poly(2*x - 2, x)]
def test_poly_subresultants():
f = x**8+x**6-3*x**4-3*x**3+8*x**2+2*x-5
g = 3*x**6+5*x**4-4*x**2-9*x+21
assert poly_subresultants(f, g, x) == \
[Poly(f, x), Poly(g, x),
Poly(15*x**4 - 3*x**2 + 9, x),
Poly(65*x**2 + 125*x - 245, x),
Poly(9326*x - 12300, x),
Poly(260708, x)]
assert poly_subresultants((x-1)**2, x**2-1, x) == \
[Poly((x-1)**2, x), Poly(x**2-1, x), Poly(2*x - 2, x)]
def test_squarefree():
assert Poly(x-1, x).is_squarefree == True
assert Poly((x-1)**2, x).is_squarefree == False
assert Poly(3*x**2, x).as_squarefree() == Poly(3*x, x)
assert Poly(x**2+2*x+1, x).as_squarefree() == Poly(x+1, x)
assert Poly(x**5-x**4-x+1, x).as_squarefree() == Poly(x**4-1, x)
assert poly_sqf(1, x) == [Poly(1, x)]
assert poly_sqf(x, x) == [Poly(x, x)]
assert poly_sqf(3*x**2, x) == [Poly(3, x), Poly(x, x)]
assert poly_sqf(x**2+2*x+1, x) == [Poly(1, x), Poly(x+1, x)]
assert poly_sqf(x**5-x**4-x+1, x) == \
[Poly(x**3 + x**2 + x + 1, x), Poly(x-1, x)]
assert poly_sqf(x**8+6*x**6+12*x**4+8*x**2, x) == \
[Poly(1, x), Poly(x, x), Poly(x**2+2, x)]
# Bronstein, Symbolic Integration, pp. 52
A = Poly(x**4 - 3*x**2 + 6, x)
D = Poly(x**6 - 5*x**4 + 5*x**2 + 4, x)
f, g = D, A - D.diff(x).mul_term(t)
res, R = poly_subresultants(f, g)
S = poly_sqf(Poly(res, t))
assert S == [Poly(45796, t), Poly(1, t), Poly(4*t**2 + 1, t)]
def test_squarefree():
assert Poly(x-1, x).is_squarefree == True
assert Poly((x-1)**2, x).is_squarefree == False
assert Poly(3*x**2, x).as_squarefree() == Poly(3*x, x)
assert Poly(x**2+2*x+1, x).as_squarefree() == Poly(x+1, x)
assert Poly(x**5-x**4-x+1, x).as_squarefree() == Poly(x**4-1, x)
assert poly_sqf(1, x) == [Poly(1, x)]
assert poly_sqf(x, x) == [Poly(x, x)]
assert poly_sqf(3*x**2, x) == [Poly(3, x), Poly(x, x)]
assert poly_sqf(x**2+2*x+1, x) == [Poly(1, x), Poly(x+1, x)]
assert poly_sqf(x**5-x**4-x+1, x) == \
[Poly(x**3 + x**2 + x + 1, x), Poly(x-1, x)]
assert poly_sqf(x**8+6*x**6+12*x**4+8*x**2, x) == \
[Poly(1, x), Poly(x, x), Poly(x**2+2, x)]
# Bronstein, Symbolic Integration, pp. 52
A = Poly(x**4 - 3*x**2 + 6, x)
D = Poly(x**6 - 5*x**4 + 5*x**2 + 4, x)
f, g = D, A - D.diff(x).mul_term(t)
res, R = poly_subresultants(f, g)
S = poly_sqf(Poly(res, t))
assert S == [Poly(45796, t), Poly(1, t), Poly(4*t**2 + 1, t)]
def ratint_logpart(f, g, x, t=None):
"""Lazard-Rioboo-Trager algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime, deg(f) < deg(g) and g is square-free, returns a list
of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
in K[t, x] and q_i in K[t], and:
___ ___
d f d \ ` \ `
-- - = -- ) ) a log(s_i(a, x))
dx g dx /__, /__,
i=1..n a | q_i(a) = 0
"""
f, g = Poly(f, x), Poly(g, x)
t = t or Symbol('t', dummy=True)
a, b = g, f - g.diff().mul_term(t)
res, R = poly_subresultants(a, b)
Q = poly_sqf(Poly(res, t))
R_map, H, i = {}, [], 1
for r in R:
R_map[r.degree] = r
for q in Q:
if q.degree > 0:
_, q = q.as_primitive()
if g.degree == i:
H.append((g, q))
else:
h = R_map[i]
A = poly_sqf(h.LC, t)
for j in xrange(0, len(A)):
T = poly_gcd(A[j], q)**(j+1)
h = poly_div(h, Poly(T, x))[0]
# NOTE: h.LC is always invertible in K[t]
inv, coeffs = Poly(h.LC, t).invert(q), [S(1)]
for coeff in h.coeffs[1:]:
T = poly_div(inv*coeff, q)[1]
coeffs.append(T.as_basic())
h = Poly(zip(coeffs, h.monoms), x)
H.append((h, q))
i += 1
return H
def ratint_logpart(f, g, x, t=None):
"""Lazard-Rioboo-Trager algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime, deg(f) < deg(g) and g is square-free, returns a list
of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
in K[t, x] and q_i in K[t], and:
___ ___
d f d \ ` \ `
-- - = -- ) ) a log(s_i(a, x))
dx g dx /__, /__,
i=1..n a | q_i(a) = 0
"""
f, g = Poly(f, x), Poly(g, x)
t = t or Symbol('t', dummy=True)
a, b = g, f - g.diff().mul_term(t)
res, R = poly_subresultants(a, b)
Q = poly_sqf(Poly(res, t))
R_map, H, i = {}, [], 1
for r in R:
R_map[r.degree] = r
for q in Q:
if q.degree > 0:
_, q = q.as_primitive()
if g.degree == i:
H.append((g, q))
else:
h = R_map[i]
A = poly_sqf(h.LC, t)
for j in xrange(0, len(A)):
T = poly_gcd(A[j], q)**(j+1)
h = poly_div(h, Poly(T, x))[0]
# NOTE: h.LC is always invertible in K[t]
inv, coeffs = Poly(h.LC, t).invert(q), [S(1)]
for coeff in h.coeffs[1:]:
T = poly_div(inv*coeff, q)[1]
coeffs.append(T.as_basic())
h = Poly(zip(coeffs, h.monoms), x)
H.append((h, q))
i += 1
return H
def test_poly_subresultants():
f = x**8+x**6-3*x**4-3*x**3+8*x**2+2*x-5
g = 3*x**6+5*x**4-4*x**2-9*x+21
assert poly_subresultants(f, g, x) == \
(Poly(260708, x), [Poly(f, x), Poly(g, x),
Poly(15*x**4 - 3*x**2 + 9, x),
Poly(65*x**2 + 125*x - 245, x),
Poly(9326*x - 12300, x),
Poly(260708, x)])
assert poly_subresultants((x-1)**2, x**2-1, x) == \
(Poly(0, x), [Poly((x-1)**2, x),
Poly(x**2-1, x),
Poly(2*x - 2, x)])
f = Poly(-x**3 + 5, x)
g = Poly((1 + 3*t)*x**2, x)
assert poly_subresultants(f, g) == \
(Poly(25 + 225*t + 675*t**2 + 675*t**3, x), [Poly(-x**3 + 5, x),
Poly((1 + 3*t)*x**2, x),
Poly(5 + 30*t + 45*t**2, x)])
def test_poly_subresultants():
f = x**8+x**6-3*x**4-3*x**3+8*x**2+2*x-5
g = 3*x**6+5*x**4-4*x**2-9*x+21
assert poly_subresultants(f, g, x) == \
(Poly(260708, x), [Poly(f, x), Poly(g, x),
Poly(15*x**4 - 3*x**2 + 9, x),
Poly(65*x**2 + 125*x - 245, x),
Poly(9326*x - 12300, x),
Poly(260708, x)])
assert poly_subresultants((x-1)**2, x**2-1, x) == \
(Poly(0, x), [Poly((x-1)**2, x),
Poly(x**2-1, x),
Poly(2*x - 2, x)])
f = Poly(-x**3 + 5, x)
g = Poly((1 + 3*t)*x**2, x)
assert poly_subresultants(f, g) == \
(Poly(25 + 225*t + 675*t**2 + 675*t**3, x), [Poly(-x**3 + 5, x),
Poly((1 + 3*t)*x**2, x),
Poly(5 + 30*t + 45*t**2, x)])
