def test_roots_quartic():
assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
assert roots_quartic(Poly(x**4 + x**3,
x)) in [[-1, 0, 0, 0], [0, -1, 0, 0],
[0, 0, -1, 0], [0, 0, 0, -1]]
assert roots_quartic(Poly(x**4 - x**3, x)) in [[1, 0, 0, 0], [0, 1, 0, 0],
[0, 0, 1, 0], [0, 0, 0, 1]]
lhs = roots_quartic(Poly(x**4 + x, x))
rhs = [
Rational(1, 2) + I * sqrt(3) / 2,
Rational(1, 2) - I * sqrt(3) / 2, 0, -1
]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
# test of all branches of roots quartic
for i, (a, b, c, d) in enumerate([(1, 2, 3, 0), (3, -7, -9, 9),
(1, 2, 3, 4), (1, 2, 3, 4),
(-7, -3, 3, -6), (-3, 5, -6, -4),
(6, -5, -10, -3)]):
if i == 2:
c = -a * (a**2 / Integer(8) - b / Integer(2))
elif i == 3:
d = a * (a * (3 * a**2 / Integer(256) - b / Integer(16)) +
c / Integer(4))
eq = x**4 + a * x**3 + b * x**2 + c * x + d
ans = roots_quartic(Poly(eq, x))
assert all(eq.subs({x: ai}).evalf(chop=True) == 0 for ai in ans)
# not all symbolic quartics are unresolvable
eq = Poly(q * x + q / 4 + x**4 + x**3 + 2 * x**2 - Rational(1, 3), x)
sol = roots_quartic(eq)
assert all(verify_numerically(eq.subs({x: i}), 0) for i in sol)
z = symbols('z', negative=True)
eq = x**4 + 2 * x**3 + 3 * x**2 + x * (z + 11) + 5
zans = roots_quartic(Poly(eq, x))
assert all(verify_numerically(eq.subs({x: i, z: -1}), 0) for i in zans)
# but some are (see also issue sympy/sympy#4989)
# it's ok if the solution is not Piecewise, but the tests below should pass
eq = Poly(y * x**4 + x**3 - x + z, x)
ans = roots_quartic(eq)
assert all(type(i) == Piecewise for i in ans)
reps = (
{
y: -Rational(1, 3),
z: -Rational(1, 4)
}, # 4 real
{
y: -Rational(1, 3),
z: -Rational(1, 2)
}, # 2 real
{
y: -Rational(1, 3),
z: -2
}) # 0 real
for rep in reps:
sol = roots_quartic(Poly(eq.subs(rep), x))
assert all(
verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol))
def test_roots_quartic():
assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
assert roots_quartic(Poly(x**4 + x**3, x)) in [
[-1, 0, 0, 0],
[0, -1, 0, 0],
[0, 0, -1, 0],
[0, 0, 0, -1]
]
assert roots_quartic(Poly(x**4 - x**3, x)) in [
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]
]
lhs = roots_quartic(Poly(x**4 + x, x))
rhs = [Rational(1, 2) + I*sqrt(3)/2, Rational(1, 2) - I*sqrt(3)/2, 0, -1]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
# test of all branches of roots quartic
for i, (a, b, c, d) in enumerate([(1, 2, 3, 0),
(3, -7, -9, 9),
(1, 2, 3, 4),
(1, 2, 3, 4),
(-7, -3, 3, -6),
(-3, 5, -6, -4),
(6, -5, -10, -3)]):
if i == 2:
c = -a*(a**2/Integer(8) - b/Integer(2))
elif i == 3:
d = a*(a*(3*a**2/Integer(256) - b/Integer(16)) + c/Integer(4))
eq = x**4 + a*x**3 + b*x**2 + c*x + d
ans = roots_quartic(Poly(eq, x))
assert all(eq.subs({x: ai}).evalf(chop=True) == 0 for ai in ans)
# not all symbolic quartics are unresolvable
eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x)
sol = roots_quartic(eq)
assert all(verify_numerically(eq.subs({x: i}), 0) for i in sol)
z = symbols('z', negative=True)
eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5
zans = roots_quartic(Poly(eq, x))
assert all(verify_numerically(eq.subs({x: i, z: -1}), 0) for i in zans)
# but some are (see also issue sympy/sympy#4989)
# it's ok if the solution is not Piecewise, but the tests below should pass
eq = Poly(y*x**4 + x**3 - x + z, x)
ans = roots_quartic(eq)
assert all(type(i) == Piecewise for i in ans)
reps = ({y: -Rational(1, 3), z: -Rational(1, 4)}, # 4 real
{y: -Rational(1, 3), z: -Rational(1, 2)}, # 2 real
{y: -Rational(1, 3), z: -2}) # 0 real
for rep in reps:
sol = roots_quartic(Poly(eq.subs(rep), x))
assert all(verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol))