def roots(f, *gens, **flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
Examples
========
>>> from sympy import Poly, roots
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}
>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}
>>> roots([1, 0, -1])
{-1: 1, 1: 1}
References
==========
.. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
trig = flags.pop('trig', False)
quartics = flags.pop('quartics', True)
quintics = flags.pop('quintics', False)
multiple = flags.pop('multiple', False)
filter = flags.pop('filter', None)
predicate = flags.pop('predicate', None)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
F = Poly(f, *gens, **flags)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
raise PolynomialError("generator must be a Symbol")
else:
f = F
if f.length == 2 and f.degree() != 1:
# check for foo**n factors in the constant
n = f.degree()
npow_bases = []
others = []
expr = f.as_expr()
con = expr.as_independent(*gens)[0]
for p in Mul.make_args(con):
if p.is_Pow and not p.exp % n:
npow_bases.append(p.base**(p.exp/n))
else:
others.append(p)
if npow_bases:
b = Mul(*npow_bases)
B = Dummy()
d = roots(Poly(expr - con + B**n*Mul(*others), *gens,
**flags), *gens, **flags)
rv = {}
for k, v in d.items():
rv[k.subs(B, b)] = v
return rv
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, currentroot, k):
if currentroot in result:
result[currentroot] += k
else:
result[currentroot] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for currentroot in _try_heuristics(factors[0]):
roots.append(currentroot)
for currentfactor in factors[1:]:
previous, roots = list(roots), []
for currentroot in previous:
g = currentfactor - Poly(currentroot, f.gen)
for currentroot in _try_heuristics(g):
roots.append(currentroot)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S.Zero]*f.degree()
if f.length() == 2:
if f.degree() == 1:
return list(map(cancel, roots_linear(f)))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break
n = f.degree()
if n == 1:
result += list(map(cancel, roots_linear(f)))
elif n == 2:
result += list(map(cancel, roots_quadratic(f)))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f, trig=trig)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
# Convert the generators to symbols
dumgens = symbols('x:%d' % len(f.gens), cls=Dummy)
f = f.per(f.rep, dumgens)
(k,), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S.Zero: k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().is_Ring:
f = f.to_field()
# Use EX instead of ZZ_I or QQ_I
if f.get_domain().is_QQ_I:
f = f.per(f.rep.convert(EX))
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
dom = f.get_domain()
if not dom.is_Exact and dom.is_Numerical:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
else:
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for currentroot in _try_decompose(f):
_update_dict(result, currentroot, 1)
else:
for r in _try_heuristics(f):
_update_dict(result, r, 1)
else:
for currentroot in _try_decompose(f):
_update_dict(result, currentroot, 1)
else:
for currentfactor, k in factors:
for r in _try_heuristics(Poly(currentfactor, f.gen, field=True)):
_update_dict(result, r, k)
if coeff is not S.One:
_result, result, = result, {}
for currentroot, k in _result.items():
result[coeff*currentroot] = k
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: all(a.is_real for a in r.as_numer_denom()),
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k*rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
# adding zero roots after non-trivial roots have been translated
result.update(zeros)
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero]*result[zero])
return zeros
def roots(f,
*gens,
auto=True,
cubics=True,
trig=False,
quartics=True,
quintics=False,
multiple=False,
filter=None,
predicate=None,
strict=False,
**flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
If the ``strict`` flag is True, ``UnsolvableFactorError`` will be
raised if the roots found are known to be incomplete (because
some roots are not expressible in radicals).
Examples
========
>>> from sympy import Poly, roots, degree
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}
>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}
>>> roots([1, 0, -1])
{-1: 1, 1: 1}
``roots`` will only return roots expressible in radicals. If
the given polynomial has some or all of its roots inexpressible in
radicals, the result of ``roots`` will be incomplete or empty
respectively.
Example where result is incomplete:
>>> roots((x-1)*(x**5-x+1), x)
{1: 1}
In this case, the polynomial has an unsolvable quintic factor
whose roots cannot be expressed by radicals. The polynomial has a
rational root (due to the factor `(x-1)`), which is returned since
``roots`` always finds all rational roots.
Example where result is empty:
>>> roots(x**7-3*x**2+1, x)
{}
Here, the polynomial has no roots expressible in radicals, so
``roots`` returns an empty dictionary.
The result produced by ``roots`` is complete if and only if the
sum of the multiplicity of each root is equal to the degree of
the polynomial. If strict=True, UnsolvableFactorError will be
raised if the result is incomplete.
The result can be be checked for completeness as follows:
>>> f = x**3-2*x**2+1
>>> sum(roots(f, x).values()) == degree(f, x)
True
>>> f = (x-1)*(x**5-x+1)
>>> sum(roots(f, x).values()) == degree(f, x)
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
F = Poly(f, *gens, **flags)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
raise PolynomialError("generator must be a Symbol")
f = F
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
else:
n = f.degree()
if f.length() == 2 and n > 2:
# check for foo**n in constant if dep is c*gen**m
con, dep = f.as_expr().as_independent(*f.gens)
fcon = -(-con).factor()
if fcon != con:
con = fcon
bases = []
for i in Mul.make_args(con):
if i.is_Pow:
b, e = i.as_base_exp()
if e.is_Integer and b.is_Add:
bases.append((b, Dummy(positive=True)))
if bases:
rv = roots(Poly((dep + con).xreplace(dict(bases)),
*f.gens),
*F.gens,
auto=auto,
cubics=cubics,
trig=trig,
quartics=quartics,
quintics=quintics,
multiple=multiple,
filter=filter,
predicate=predicate,
**flags)
return {
factor_terms(k.xreplace({v: k
for k, v in bases})): v
for k, v in rv.items()
}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, zeros, currentroot, k):
if currentroot == S.Zero:
if S.Zero in zeros:
zeros[S.Zero] += k
else:
zeros[S.Zero] = k
if currentroot in result:
result[currentroot] += k
else:
result[currentroot] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for currentroot in _try_heuristics(factors[0]):
roots.append(currentroot)
for currentfactor in factors[1:]:
previous, roots = list(roots), []
for currentroot in previous:
g = currentfactor - Poly(currentroot, f.gen)
for currentroot in _try_heuristics(g):
roots.append(currentroot)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S.Zero] * f.degree()
if f.length() == 2:
if f.degree() == 1:
return list(map(cancel, roots_linear(f)))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break
n = f.degree()
if n == 1:
result += list(map(cancel, roots_linear(f)))
elif n == 2:
result += list(map(cancel, roots_quadratic(f)))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f, trig=trig)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
# Convert the generators to symbols
dumgens = symbols('x:%d' % len(f.gens), cls=Dummy)
f = f.per(f.rep, dumgens)
(k, ), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S.Zero: k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().is_Ring:
f = f.to_field()
# Use EX instead of ZZ_I or QQ_I
if f.get_domain().is_QQ_I:
f = f.per(f.rep.convert(EX))
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
dom = f.get_domain()
if not dom.is_Exact and dom.is_Numerical:
for r in f.nroots():
_update_dict(result, zeros, r, 1)
elif f.degree() == 1:
_update_dict(result, zeros, roots_linear(f)[0], 1)
elif f.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, zeros, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, zeros, r, 1)
else:
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for currentroot in _try_decompose(f):
_update_dict(result, zeros, currentroot, 1)
else:
for r in _try_heuristics(f):
_update_dict(result, zeros, r, 1)
else:
for currentroot in _try_decompose(f):
_update_dict(result, zeros, currentroot, 1)
else:
for currentfactor, k in factors:
for r in _try_heuristics(
Poly(currentfactor, f.gen, field=True)):
_update_dict(result, zeros, r, k)
if coeff is not S.One:
_result, result, = result, {}
for currentroot, k in _result.items():
result[coeff * currentroot] = k
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: all(a.is_real for a in r.as_numer_denom()),
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k * rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
# adding zero roots after non-trivial roots have been translated
result.update(zeros)
if strict and sum(result.values()) < f.degree():
raise UnsolvableFactorError(
filldedent('''
Strict mode: some factors cannot be solved in radicals, so
a complete list of solutions cannot be returned. Call
roots with strict=False to get solutions expressible in
radicals (if there are any).
'''))
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero] * result[zero])
return zeros