def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2+3*x, x)) == [0, -Rational(3, 2)]
assert roots_quadratic(Poly(2*x**2+3, x)) == [I*sqrt(6)/2, -I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2+4*x+3, x)) == \
[-1 + I*sqrt(2)/2, -1 - I*sqrt(2)/2]
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2+3*x, x)) == [0, -Rational(3, 2)]
assert roots_quadratic(Poly(2*x**2+3, x)) == [I*sqrt(6)/2, -I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2+4*x+3, x)) == \
[-1 + I*sqrt(2)/2, -1 - I*sqrt(2)/2]
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
_check(Poly(2*x**2 + 4*x + 3, x).all_roots())
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2))/(a - c),
-e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2))/(a - c)]
# check for simplification
f = Poly(y*x**2 - 2*x - 2*y, x)
assert roots_quadratic(f) == \
[-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
assert roots_quadratic(f) == \
[1,y**2 + 1]
f = Poly(sqrt(2)*x**2 - 1, x)
r = roots_quadratic(f)
assert r == _nsort(r)
# issue 8255
f = Poly(-24*x**2 - 180*x + 264)
assert [w.n(2) for w in f.all_roots(radicals=True)] == \
[w.n(2) for w in f.all_roots(radicals=False)]
for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)):
f = Poly(_a*x**2 + _b*x + _c)
roots = roots_quadratic(f)
assert roots == _nsort(roots)
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [-Rational(3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2),
-e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)]
# check for simplification
f = Poly(y*x**2 - 2*x - 2*y, x)
assert roots_quadratic(f) == \
[-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
assert roots_quadratic(f) == \
[y**2/2 - sqrt(y**4)/2 + 1, y**2/2 + sqrt(y**4)/2 + 1]
f = Poly(sqrt(2)*x**2 - 1, x)
r = roots_quadratic(f)
assert r == _nsort(r)
# issue 8255
f = Poly(-24*x**2 - 180*x + 264)
assert [w.n(2) for w in f.all_roots(radicals=True)] == \
[w.n(2) for w in f.all_roots(radicals=False)]
for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)):
f = Poly(_a*x**2 + _b*x + _c)
roots = roots_quadratic(f)
assert roots == _nsort(roots)
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [-Rational(3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2),
-e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)]
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [-Rational(3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x+3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - ((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)**S.Half,
-e*(a + c)/(a - c) + ((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)**S.Half]
def test_roots_binomial():
assert roots_binomial(Poly(5 * x, x)) == [0]
assert roots_binomial(Poly(5 * x**4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5 * x + 2, x)) == [Rational(-2, 5)]
A = 10**Rational(3, 4) / 10
assert roots_binomial(Poly(5*x**4 + 2, x)) == \
[-A - A*I, -A + A*I, A - A*I, A + A*I]
_check(roots_binomial(Poly(x**8 - 2)))
a1 = Symbol('a1', nonnegative=True)
b1 = Symbol('b1', nonnegative=True)
r0 = roots_quadratic(Poly(a1 * x**2 + b1, x))
r1 = roots_binomial(Poly(a1 * x**2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
if a == b and a != 1: # a == b == 1 is sufficient
continue
p = Poly(a * x**n + s * b)
ans = roots_binomial(p)
assert ans == _nsort(ans)
# issue 8813
assert roots(Poly(2 * x**3 - 16 * y**3, x)) == {
2 * y * (Rational(-1, 2) - sqrt(3) * I / 2): 1,
2 * y: 1,
2 * y * (Rational(-1, 2) + sqrt(3) * I / 2): 1
}
def __new__(cls, expr, func=None, x=None, auto=True, quadratic=False):
"""Construct a new ``RootSum`` instance carrying all roots of a polynomial. """
coeff, poly = cls._transform(expr, x)
if not poly.is_univariate:
raise MultivariatePolynomialError("only univariate polynomials are allowed")
if func is None:
func = Lambda(poly.gen, poly.gen)
else:
try:
is_func = func.is_Function
except AttributeError:
is_func = False
if is_func and (func.nargs == 1 or 1 in func.nargs):
if not isinstance(func, Lambda):
func = Lambda(poly.gen, func(poly.gen))
else:
raise ValueError("expected a univariate function, got %s" % func)
var, expr = func.args
if coeff is not S.One:
expr = expr.subs(var, coeff*var)
deg = poly.degree()
if not expr.has(var):
return deg*expr
if expr.is_Add:
add_const, expr = expr.as_independent(var)
else:
add_const = S.Zero
if expr.is_Mul:
mul_const, expr = expr.as_independent(var)
else:
mul_const = S.One
func = Lambda(var, expr)
rational = cls._is_func_rational(poly, func)
(_, factors), terms = poly.factor_list(), []
for poly, k in factors:
if poly.is_linear:
term = func(roots_linear(poly)[0])
elif quadratic and poly.is_quadratic:
term = sum(map(func, roots_quadratic(poly)))
else:
if not rational or not auto:
term = cls._new(poly, func, auto)
else:
term = cls._rational_case(poly, func)
terms.append(k*term)
return mul_const*Add(*terms) + deg*add_const
def test_roots_binomial():
assert roots_binomial(Poly(5*x, x)) == [0]
assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5*x + 2, x)) == [-Rational(2, 5)]
A = 10**Rational(3, 4)/10
assert roots_binomial(Poly(5*x**4 + 2, x)) == \
[-A - A*I, -A + A*I, A - A*I, A + A*I]
a1 = Symbol('a1', nonnegative=True)
b1 = Symbol('b1', nonnegative=True)
r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
r1 = roots_binomial(Poly(a1*x**2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
if a == b and a != 1: # a == b == 1 is sufficient
continue
p = Poly(a*x**n + s*b)
ans = roots_binomial(p)
assert ans == _nsort(ans)
# issue 8813
assert roots(Poly(2*x**3 - 16*y**3, x)) == {
2*y*(-S(1)/2 - sqrt(3)*I/2): 1,
2*y: 1,
2*y*(-S(1)/2 + sqrt(3)*I/2): 1}
def __new__(cls, expr, func=None, x=None, auto=True, quadratic=False):
"""Construct a new ``RootSum`` instance of roots of a polynomial."""
coeff, poly = cls._transform(expr, x)
if not poly.is_univariate:
raise MultivariatePolynomialError(
"only univariate polynomials are allowed")
if func is None:
func = Lambda(poly.gen, poly.gen)
else:
is_func = getattr(func, 'is_Function', False)
if is_func and 1 in func.nargs:
if not isinstance(func, Lambda):
func = Lambda(poly.gen, func(poly.gen))
else:
raise ValueError("expected a univariate function, got %s" %
func)
var, expr = func.variables[0], func.expr
if coeff is not S.One:
expr = expr.subs(var, coeff * var)
deg = poly.degree()
if not expr.has(var):
return deg * expr
if expr.is_Add:
add_const, expr = expr.as_independent(var)
else:
add_const = S.Zero
if expr.is_Mul:
mul_const, expr = expr.as_independent(var)
else:
mul_const = S.One
func = Lambda(var, expr)
rational = cls._is_func_rational(poly, func)
factors, terms = _pure_factors(poly), []
for poly, k in factors:
if poly.is_linear:
term = func(roots_linear(poly)[0])
elif quadratic and poly.is_quadratic:
term = sum(map(func, roots_quadratic(poly)))
else:
if not rational or not auto:
term = cls._new(poly, func, auto)
else:
term = cls._rational_case(poly, func)
terms.append(k * term)
return mul_const * Add(*terms) + deg * add_const
def _roots_trivial(cls, poly, radicals):
"""Compute roots in linear, quadratic and binomial cases. """
if poly.degree() == 1:
return roots_linear(poly)
if not radicals:
return None
if poly.degree() == 2:
return roots_quadratic(poly)
elif poly.length() == 2 and poly.TC():
return roots_binomial(poly)
else:
return None
def _roots_trivial(cls, poly, radicals):
"""Compute roots in linear, quadratic and binomial cases. """
if poly.degree() == 1:
return roots_linear(poly)
if not radicals:
return None
if poly.degree() == 2:
return roots_quadratic(poly)
elif poly.length() == 2 and poly.TC():
return roots_binomial(poly)
else:
return None
def test_roots_binomial():
assert roots_binomial(Poly(5 * x, x)) == [0]
assert roots_binomial(Poly(5 * x ** 4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5 * x + 2, x)) == [-Rational(2, 5)]
A = 10 ** Rational(3, 4) / 10
assert roots_binomial(Poly(5 * x ** 4 + 2, x)) == [-A - A * I, -A + A * I, A - A * I, A + A * I]
a1 = Symbol("a1", nonnegative=True)
b1 = Symbol("b1", nonnegative=True)
r0 = roots_quadratic(Poly(a1 * x ** 2 + b1, x))
r1 = roots_binomial(Poly(a1 * x ** 2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
def test_roots_binomial():
assert roots_binomial(Poly(5 * x, x)) == [0]
assert roots_binomial(Poly(5 * x**4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5 * x + 2, x)) == [-Rational(2, 5)]
A = 10**Rational(3, 4) / 10
assert roots_binomial(Poly(5*x**4 + 2, x)) == \
[-A - A*I, -A + A*I, A - A*I, A + A*I]
a1 = Symbol('a1', nonnegative=True)
b1 = Symbol('b1', nonnegative=True)
r0 = roots_quadratic(Poly(a1 * x**2 + b1, x))
r1 = roots_binomial(Poly(a1 * x**2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
def roots_trivial(poly, radicals=True):
"""Compute roots in linear, quadratic and binomial cases. """
if poly.degree() == 1:
return roots_linear(poly)
else:
if not radicals:
return None
if poly in _rootof_trivial_cache:
roots = _rootof_trivial_cache[poly]
else:
if radicals and poly.degree() == 2:
roots = roots_quadratic(poly)
elif radicals and poly.length() == 2 and poly.TC():
roots = roots_binomial(poly)
else:
return None
_rootof_trivial_cache[poly] = roots
return roots
def roots_trivial(poly, radicals=True):
"""Compute roots in linear, quadratic and binomial cases. """
if poly.degree() == 1:
return roots_linear(poly)
else:
if not radicals:
return None
if poly in _rootof_trivial_cache:
roots = _rootof_trivial_cache[poly]
else:
if radicals and poly.degree() == 2:
roots = roots_quadratic(poly)
elif radicals and poly.length() == 2 and poly.TC():
roots = roots_binomial(poly)
else:
return None
_rootof_trivial_cache[poly] = roots
return roots
def _roots_trivial(cls, poly, radicals):
"""Compute roots in linear, quadratic and binomial cases. """
if poly.degree() == 1:
return roots_linear(poly)
if not radicals:
return None
free = len(poly.free_symbols)
sort = isinstance(poly, PurePoly) and free or free > 1
if poly.degree() == 2:
roots = roots_quadratic(poly, sort=sort)
elif poly.length() == 2 and poly.TC():
roots = roots_binomial(poly, sort=sort)
else:
return None
# put roots in same order as RootOf instances
if not sort:
key = [r.n(2) for r in roots]
key = [(1 if not r.is_real else 0, C.re(r), C.im(r))
for r in key]
_, roots = zip(*sorted(zip(key, roots)))
return roots
def test_roots_binomial():
assert roots_binomial(Poly(5 * x, x)) == [0]
assert roots_binomial(Poly(5 * x ** 4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5 * x + 2, x)) == [-Rational(2, 5)]
A = 10 ** Rational(3, 4) / 10
assert roots_binomial(Poly(5 * x ** 4 + 2, x)) == [-A - A * I, -A + A * I, A - A * I, A + A * I]
a1 = Symbol("a1", nonnegative=True)
b1 = Symbol("b1", nonnegative=True)
r0 = roots_quadratic(Poly(a1 * x ** 2 + b1, x))
r1 = roots_binomial(Poly(a1 * x ** 2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
if a == b and a != 1: # a == b == 1 is sufficient
continue
p = Poly(a * x ** n + s * b)
roots = roots_binomial(p)
assert roots == _nsort(roots)
def test_roots_binomial():
assert roots_binomial(Poly(5*x, x)) == [0]
assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5*x + 2, x)) == [-Rational(2, 5)]
A = 10**Rational(3, 4)/10
assert roots_binomial(Poly(5*x**4 + 2, x)) == \
[-A - A*I, -A + A*I, A - A*I, A + A*I]
a1 = Symbol('a1', nonnegative=True)
b1 = Symbol('b1', nonnegative=True)
r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
r1 = roots_binomial(Poly(a1*x**2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
if a == b and a != 1: # a == b == 1 is sufficient
continue
p = Poly(a*x**n + s*b)
roots = roots_binomial(p)
assert roots == _nsort(roots)
