def test_roots_cubic():
assert roots_cubic(Poly(2 * x ** 3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x ** 3 - 3 * x ** 2 + 3 * x - 1, x)) == [1, 1, 1]
assert roots_cubic(Poly(x ** 3 + 1, x)) == [
-1,
S.Half - I * sqrt(3) / 2,
S.Half + I * sqrt(3) / 2,
]
assert (
roots_cubic(Poly(2 * x ** 3 - 3 * x ** 2 - 3 * x - 1, x))[0]
== S.Half + 3 ** Rational(1, 3) / 2 + 3 ** Rational(2, 3) / 2
)
eq = -(x ** 3) + 2 * x ** 2 + 3 * x - 2
assert (
roots(eq, trig=True, multiple=True)
== roots_cubic(Poly(eq, x), trig=True)
== [
Rational(2, 3) + 2 * sqrt(13) * cos(acos(8 * sqrt(13) / 169) / 3) / 3,
-2 * sqrt(13) * sin(-acos(8 * sqrt(13) / 169) / 3 + pi / 6) / 3
+ Rational(2, 3),
-2 * sqrt(13) * cos(-acos(8 * sqrt(13) / 169) / 3 + pi / 3) / 3
+ Rational(2, 3),
]
)
def test_roots_cubic():
assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3*x**2 + 3*x - 1, x)) == [1, 1, 1]
assert roots_cubic(Poly(x**3 + 1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
def test_issue_8438():
p = Poly([1, y, -2, -3], x).as_expr()
roots = roots_cubic(Poly(p, x), x)
z = Rational(-3, 2) - I*Rational(7, 2) # this will fail in code given in commit msg
post = [r.subs(y, z) for r in roots]
assert set(post) == \
set(roots_cubic(Poly(p.subs(y, z), x)))
# /!\ if p is not made an expression, this is *very* slow
assert all(p.subs({y: z, x: i}).n(2, chop=True) == 0 for i in post)
def test_issue_8438():
p = Poly([1, y, -2, -3], x).as_expr()
roots = roots_cubic(Poly(p, x), x)
z = -S(3)/2 - 7*I/2 # this will fail in code given in commit msg
post = [r.subs(y, z) for r in roots]
assert set(post) == \
set(roots_cubic(Poly(p.subs(y, z), x)))
# /!\ if p is not made an expression, this is *very* slow
assert all(p.subs({y: z, x: i}).n(2, chop=True) == 0 for i in post)
def test_roots_cubic():
assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3*x**2 + 3*x - 1, x)) == [1, 1, 1]
assert roots_cubic(Poly(x**3 + 1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
eq = -x**3 + 2*x**2 + 3*x - 2
assert roots(eq, trig=True, multiple=True) == \
roots_cubic(Poly(eq, x), trig=True) == [
S(2)/3 + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3,
-2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + S(2)/3,
-2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + S(2)/3,
]
def test_roots_cubic():
assert roots_cubic(Poly(2 * x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3 * x**2 + 3 * x - 1, x)) == [1, 1, 1]
assert roots_cubic(Poly(x**3 + 1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
eq = -x**3 + 2 * x**2 + 3 * x - 2
assert list(sorted((
roots(eq, trig=True, multiple=True)))) == \
roots_cubic(Poly(eq, x), trig=True) == [
-2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + S(2)/3,
-2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + S(2)/3,
S(2)/3 + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3]
def test_roots_cubic():
assert roots_cubic(Poly(2 * x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3 * x**2 + 3 * x - 1, x)) == [1, 1, 1]
# valid for arbitrary y (issue 21263)
r = root(y, 3)
assert roots_cubic(Poly(x**3 - y, x)) == [
r, r * (-S.Half + sqrt(3) * I / 2), r * (-S.Half - sqrt(3) * I / 2)
]
# simpler form when y is negative
assert roots_cubic(Poly(x**3 - -1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
eq = -x**3 + 2 * x**2 + 3 * x - 2
assert roots(eq, trig=True, multiple=True) == \
roots_cubic(Poly(eq, x), trig=True) == [
Rational(2, 3) + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3,
-2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + Rational(2, 3),
-2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + Rational(2, 3),
]
def test_roots_cubic():
assert roots_cubic(Poly(2 * x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3 * x**2 + 3 * x - 1, x)) == [1, 1, 1]
assert roots_cubic(Poly(x**3 + 1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
def test_roots_cubic():
assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3-3*x**2+3*x-1, x)) == [1, 1, 1]
assert roots_cubic(Poly(x**3+1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]