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 = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
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)]):
if i == 2:
c = -a*(a**2/S(8) - b/S(2))
elif i == 3:
d = a*(a*(3*a**2/S(256) - b/S(16)) + c/S(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).n(chop=True) == 0 for ai in ans])
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 = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
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)]):
if i == 2:
c = -a*(a**2/S(8) - b/S(2))
elif i == 3:
d = a*(a*(3*a**2/S(256) - b/S(16)) + c/S(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).n(chop=True) == 0 for ai in ans])
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 = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
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/S(8) - b/S(2))
elif i == 3:
d = a*(a*(3*a**2/S(256) - b/S(16)) + c/S(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).n(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 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 = (
dict(y=-Rational(1, 3), z=-Rational(1, 4)), # 4 real
dict(y=-Rational(1, 3), z=-Rational(1, 2)), # 2 real
dict(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 = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
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/S(8) - b/S(2))
elif i == 3:
d = a*(a*(3*a**2/S(256) - b/S(16)) + c/S(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).n(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(test_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([test_numerically(eq.subs(((x, i), (z, -1))), 0) for i in zans])
# but some are (see also issue 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 = (
dict(y=-Rational(1, 3), z=-Rational(1, 4)), # 4 real
dict(y=-Rational(1, 3), z=-Rational(1, 2)), # 2 real
dict(y=-Rational(1, 3), z=-2)) # 0 real
for rep in reps:
sol = roots_quartic(Poly(eq.subs(rep), x))
assert all([test_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 = [S.Half + I * sqrt(3) / 2, S.Half - I * sqrt(3) / 2, S.Zero, -S.One]
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 / S(8) - b / S(2))
elif i == 3:
d = a * (a * (3 * a**2 / S(256) - b / S(16)) + c / S(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).n(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(test_numerically(eq.subs(x, i), 0) for i in sol)
# but some are (see also iss 1890)
raises(PolynomialError, lambda: roots_quartic(Poly(y * x**4 + x + z, x)))
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 = [S.Half + I * sqrt(3) / 2, S.Half - I * sqrt(3) / 2, S.Zero, -S.One]
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 / S(8) - b / S(2))
elif i == 3:
d = a * (a * (3 * a ** 2 / S(256) - b / S(16)) + c / S(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).n(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(test_numerically(eq.subs(x, i), 0) for i in sol)
# but some are (see also iss 1890)
raises(PolynomialError, lambda: roots_quartic(Poly(y * x ** 4 + x + z, x)))
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 = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
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 = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
def test_issue_21287():
assert not any(
isinstance(i, Piecewise) for i in roots_quartic(
Poly(x**4 - x**2 * (3 + 5 * I) + 2 * x * (-1 + I) - 1 + 3 * I, x)))
def test_issue_15076():
sol = roots_quartic(Poly(t**4 - 6 * t**2 + t / x - 3, t))
assert sol[0].has(x)
def test_issue_15076():
sol = roots_quartic(Poly(t**4 - 6*t**2 + t/x - 3, t))
assert sol[0].has(x)
