def _simplify(expr):
u"""Simplifie une expression.
Alias de simplify (sympy 0.6.4).
Mais simplify n'est pas garanti d'être stable dans le temps.
(Cf. simplify.__doc__)."""
return together(expand(Poly.cancel(powsimp(expr))))
def _simplify(expr):
u"""Simplifie une expression.
Alias de simplify (sympy 0.6.4).
Mais simplify n'est pas garanti d'être stable dans le temps.
(Cf. simplify.__doc__)."""
return together(expand(Poly.cancel(powsimp(expr))))
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def test_poly_cancel():
assert Poly.cancel(x) == x
assert Poly.cancel(x+1) == x+1
assert Poly.cancel((x+1)/(1-x)) == (x+1)/(1-x)
assert Poly.cancel((x**2-1)/(x-1)) == x+1
assert Poly.cancel((x**2-y**2)/(x-y)) == x+y
assert Poly.cancel((x**2-y)/(x-y)) == (x**2 - y)/(x - y)
assert Poly.cancel((x**2-2)/(x+sqrt(2))) == x - sqrt(2)
assert Poly.cancel((x**2-y**2)/(x-y), x) == x+y
assert Poly.cancel((x, S.One), x) == x
assert Poly.cancel((x+1, S.One), x) == x+1
assert Poly.cancel((x+1, x-1), x) == (x+1)/(x-1)
assert Poly.cancel((x**2-1, x-1), x) == x+1
assert Poly.cancel((x**2-y**2, x-y), x, y) == x+y
assert Poly.cancel((x**2-y, x-y), x, y) == (x**2 - y)/(x - y)
assert Poly.cancel((x**2-2, x+sqrt(2)), x) == x - sqrt(2)
assert Poly.cancel((x**2-y**2, x-y), x) == x+y
assert Poly.cancel(((x**2-y**2).as_poly(x), (x-y).as_poly(x))) == x+y
f = -1/(3 + 2*sqrt(2))*(1 + 1/(3 + 2*sqrt(2))*(7 + 5*sqrt(2)))
assert Poly.cancel(f) == -2 + sqrt(2)
raises(SymbolsError, "Poly.cancel((x**2-y**2, x-y))")
def test_poly_cancel():
assert Poly.cancel(x) == x
assert Poly.cancel(x+1) == x+1
assert Poly.cancel((x+1)/(1-x)) == (x+1)/(1-x)
assert Poly.cancel((x**2-1)/(x-1)) == x+1
assert Poly.cancel((x**2-y**2)/(x-y)) == x+y
assert Poly.cancel((x**2-y)/(x-y)) == (x**2 - y)/(x - y)
assert Poly.cancel((x**2-2)/(x+sqrt(2))) == x - sqrt(2)
assert Poly.cancel((x**2-y**2)/(x-y), x) == x+y
assert Poly.cancel((x, S.One), x) == x
assert Poly.cancel((x+1, S.One), x) == x+1
assert Poly.cancel((x+1, x-1), x) == (x+1)/(x-1)
assert Poly.cancel((x**2-1, x-1), x) == x+1
assert Poly.cancel((x**2-y**2, x-y), x, y) == x+y
assert Poly.cancel((x**2-y, x-y), x, y) == (x**2 - y)/(x - y)
assert Poly.cancel((x**2-2, x+sqrt(2)), x) == x - sqrt(2)
assert Poly.cancel((x**2-y**2, x-y), x) == x+y
assert Poly.cancel(((x**2-y**2).as_poly(x), (x-y).as_poly(x))) == x+y
f = -1/(3 + 2*sqrt(2))*(1 + 1/(3 + 2*sqrt(2))*(7 + 5*sqrt(2)))
assert Poly.cancel(f) == -2 + sqrt(2)
a, b = x, 1/(1/y + 1/(x+y))
assert Poly.cancel(y/(x+y) * b/(a+b), x, y) == y**2/(x**2 + 3*x*y + y**2 )
raises(SymbolsError, "Poly.cancel((x**2-y**2, x-y))")
