def test_subterm(u, u_diff):
substituted = simplify(integrand / u_diff)
if symbol not in substituted.free_symbols:
# replaced everything already
return False
substituted_ = substituted.subs(u, u_var).cancel()
if symbol not in substituted_.free_symbols:
_, denom = integrand.as_numer_denom()
_, denom2 = substituted_.as_numer_denom()
denom2 = denom2.subs(u_var, u)
constant = denom/denom2
if symbol not in constant.free_symbols:
return (constant, substituted_ / constant)
if u.is_polynomial(symbol):
h = Poly(u, symbol)
if h.degree() == 1 and h.nth(0) != 0:
t = (u_var - h.nth(0)) / h.nth(1)
substituted_ = simplify(integrand.subs(symbol, t))
args = list(get_args(substituted_, u_var))
if symbol not in substituted_.free_symbols and len(args) == 1 and args[0] == u_var:
return (1/h.nth(1), substituted_)
return False
def average_tuple_frequency(modulus_poly, coeff_count, irreducible_count,
tuple_length):
"""
Calculates the average tuple frequency - the frequency you would expect if the
residues were uniformly distributed.
"""
return irreducible_count / (
(coeff_count**Poly.degree(modulus_poly) - 1)**(tuple_length))
def _minimal_polynomial_sq(p, n, x):
"""
Returns the minimal polynomial for the ``nth-root`` of a sum of surds
or ``None`` if it fails.
Parameters
==========
p : sum of surds
n : positive integer
x : variable of the returned polynomial
Examples
========
>>> from sympy.polys.numberfields import _minimal_polynomial_sq
>>> from sympy import sqrt
>>> from sympy.abc import x
>>> q = 1 + sqrt(2) + sqrt(3)
>>> _minimal_polynomial_sq(q, 3, x)
x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8
"""
from sympy.simplify.simplify import _is_sum_surds
p = sympify(p)
n = sympify(n)
r = _is_sum_surds(p)
if not n.is_Integer or not n > 0 or not _is_sum_surds(p):
return None
pn = p**Rational(1, n)
# eliminate the square roots
p -= x
while 1:
p1 = _separate_sq(p)
if p1 is p:
p = p1.subs({x:x**n})
break
else:
p = p1
# _separate_sq eliminates field extensions in a minimal way, so that
# if n = 1 then `p = constant*(minimal_polynomial(p))`
# if n > 1 it contains the minimal polynomial as a factor.
if n == 1:
p1 = Poly(p)
if p.coeff(x**p1.degree(x)) < 0:
p = -p
p = p.primitive()[1]
return p
# by construction `p` has root `pn`
# the minimal polynomial is the factor vanishing in x = pn
factors = factor_list(p)[1]
result = _choose_factor(factors, x, pn)
return result
def _minimal_polynomial_sq(p, n, x):
"""
Returns the minimal polynomial for the ``nth-root`` of a sum of surds
or ``None`` if it fails.
Parameters
==========
p : sum of surds
n : positive integer
x : variable of the returned polynomial
Examples
========
>>> from sympy.polys.numberfields import _minimal_polynomial_sq
>>> from sympy import sqrt
>>> from sympy.abc import x
>>> q = 1 + sqrt(2) + sqrt(3)
>>> _minimal_polynomial_sq(q, 3, x)
x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8
"""
from sympy.simplify.simplify import _is_sum_surds
p = sympify(p)
n = sympify(n)
r = _is_sum_surds(p)
if not n.is_Integer or not n > 0 or not _is_sum_surds(p):
return None
pn = p**Rational(1, n)
# eliminate the square roots
p -= x
while 1:
p1 = _separate_sq(p)
if p1 is p:
p = p1.subs({x: x**n})
break
else:
p = p1
# _separate_sq eliminates field extensions in a minimal way, so that
# if n = 1 then `p = constant*(minimal_polynomial(p))`
# if n > 1 it contains the minimal polynomial as a factor.
if n == 1:
p1 = Poly(p)
if p.coeff(x**p1.degree(x)) < 0:
p = -p
p = p.primitive()[1]
return p
# by construction `p` has root `pn`
# the minimal polynomial is the factor vanishing in x = pn
factors = factor_list(p)[1]
result = _choose_factor(factors, x, pn)
return result
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
from sympy.functions import hyper
from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
from sympy.polys.polytools import Poly, factor
from sympy.core.numbers import Float
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
if simplify(f.subs(i, Dummy("i", integer=True, positive=True))) == 0:
return S.Zero, True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
from sympy.simplify.simplify import nsimplify
hs = nsimplify(hs)
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m ** mul
params[k] += [n / m] * mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0] / ab[1]
h = hyper(ap, bq, x)
f = combsimp(f)
return f.subs(i, 0) * hyperexpand(h), h.convergence_statement
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
from sympy.functions import hyper
from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
from sympy.polys.polytools import Poly, factor
from sympy.core.numbers import Float
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
return S(0), True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
from sympy.simplify.simplify import nsimplify
hs = nsimplify(hs)
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m**mul
params[k] += [n/m]*mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0]/ab[1]
h = hyper(ap, bq, x)
return f.subs(i, 0)*hyperexpand(h), h.convergence_statement
def __new__(cls, expr, coeffs=Tuple(), alias=None, **args):
"""Construct a new algebraic number. """
expr = sympify(expr)
if isinstance(expr, (tuple, Tuple)):
minpoly, root = expr
if not minpoly.is_Poly:
minpoly = Poly(minpoly)
elif expr.is_AlgebraicNumber:
minpoly, root = expr.minpoly, expr.root
else:
minpoly, root = minimal_polynomial(expr,
args.get('gen'),
polys=True), expr
dom = minpoly.get_domain()
if coeffs != Tuple():
if not isinstance(coeffs, ANP):
rep = DMP.from_sympy_list(sympify(coeffs), 0, dom)
scoeffs = Tuple(*coeffs)
else:
rep = DMP.from_list(coeffs.to_list(), 0, dom)
scoeffs = Tuple(*coeffs.to_list())
if rep.degree() >= minpoly.degree():
rep = rep.rem(minpoly.rep)
sargs = (root, scoeffs)
else:
rep = DMP.from_list([1, 0], 0, dom)
if ask(Q.negative(root)):
rep = -rep
sargs = (root, coeffs)
if alias is not None:
if not isinstance(alias, Symbol):
alias = Symbol(alias)
sargs = sargs + (alias, )
obj = Expr.__new__(cls, *sargs)
obj.rep = rep
obj.root = root
obj.alias = alias
obj.minpoly = minpoly
return obj
def __new__(cls, expr, coeffs=Tuple(), alias=None, **args):
"""Construct a new algebraic number. """
expr = sympify(expr)
if isinstance(expr, (tuple, Tuple)):
minpoly, root = expr
if not minpoly.is_Poly:
minpoly = Poly(minpoly)
elif expr.is_AlgebraicNumber:
minpoly, root = expr.minpoly, expr.root
else:
minpoly, root = minimal_polynomial(
expr, args.get('gen'), polys=True), expr
dom = minpoly.get_domain()
if coeffs != Tuple():
if not isinstance(coeffs, ANP):
rep = DMP.from_sympy_list(sympify(coeffs), 0, dom)
scoeffs = Tuple(*coeffs)
else:
rep = DMP.from_list(coeffs.to_list(), 0, dom)
scoeffs = Tuple(*coeffs.to_list())
if rep.degree() >= minpoly.degree():
rep = rep.rem(minpoly.rep)
sargs = (root, scoeffs)
else:
rep = DMP.from_list([1, 0], 0, dom)
if ask(Q.negative(root)):
rep = -rep
sargs = (root, coeffs)
if alias is not None:
if not isinstance(alias, Symbol):
alias = Symbol(alias)
sargs = sargs + (alias,)
obj = Expr.__new__(cls, *sargs)
obj.rep = rep
obj.root = root
obj.alias = alias
obj.minpoly = minpoly
return obj
def root_factors(f, *gens, **args):
"""
Returns all factors of a univariate polynomial.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.polys.polyroots import root_factors
>>> root_factors(x**2 - y, x)
[x - sqrt(y), x + sqrt(y)]
"""
args = dict(args)
filter = args.pop('filter', None)
F = Poly(f, *gens, **args)
if not F.is_Poly:
return [f]
if F.is_multivariate:
raise ValueError('multivariate polynomials are not supported')
x = F.gens[0]
zeros = roots(F, filter=filter)
if not zeros:
factors = [F]
else:
factors, N = [], 0
for r, n in ordered(zeros.items()):
factors, N = factors + [Poly(x - r, x)]*n, N + n
if N < F.degree():
G = reduce(lambda p, q: p*q, factors)
factors.append(F.quo(G))
if not isinstance(f, Poly):
factors = [ f.as_expr() for f in factors ]
return factors
def bivariate_type(f, x, y, **kwargs):
"""Given an expression, f, 3 tests will be done to see what type
of composite bivariate it might be, options for u(x, y) are::
x*y
x+y
x*y+x
x*y+y
If it matches one of these types, ``u(x, y)``, ``P(u)`` and dummy
variable ``u`` will be returned. Solving ``P(u)`` for ``u`` and
equating the solutions to ``u(x, y)`` and then solving for ``x`` or
``y`` is equivalent to solving the original expression for ``x`` or
``y``. If ``x`` and ``y`` represent two functions in the same
variable, e.g. ``x = g(t)`` and ``y = h(t)``, then if ``u(x, y) - p``
can be solved for ``t`` then these represent the solutions to
``P(u) = 0`` when ``p`` are the solutions of ``P(u) = 0``.
Only positive values of ``u`` are considered.
Examples
========
>>> from sympy.solvers.solvers import solve
>>> from sympy.solvers.bivariate import bivariate_type
>>> from sympy.abc import x, y
>>> eq = (x**2 - 3).subs(x, x + y)
>>> bivariate_type(eq, x, y)
(x + y, _u**2 - 3, _u)
>>> uxy, pu, u = _
>>> usol = solve(pu, u); usol
[sqrt(3)]
>>> [solve(uxy - s) for s in solve(pu, u)]
[[{x: -y + sqrt(3)}]]
>>> all(eq.subs(s).equals(0) for sol in _ for s in sol)
True
"""
u = Dummy('u', positive=True)
if kwargs.pop('first', True):
p = Poly(f, x, y)
f = p.as_expr()
_x = Dummy()
_y = Dummy()
rv = bivariate_type(Poly(f.subs({x: _x, y: _y}), _x, _y), _x, _y, first=False)
if rv:
reps = {_x: x, _y: y}
return rv[0].xreplace(reps), rv[1].xreplace(reps), rv[2]
return
p = f
f = p.as_expr()
# f(x*y)
args = Add.make_args(p.as_expr())
new = []
for a in args:
a = _mexpand(a.subs(x, u/y))
free = a.free_symbols
if x in free or y in free:
break
new.append(a)
else:
return x*y, Add(*new), u
def ok(f, v, c):
new = _mexpand(f.subs(v, c))
free = new.free_symbols
return None if (x in free or y in free) else new
# f(a*x + b*y)
new = []
d = p.degree(x)
if p.degree(y) == d:
a = root(p.coeff_monomial(x**d), d)
b = root(p.coeff_monomial(y**d), d)
new = ok(f, x, (u - b*y)/a)
if new is not None:
return a*x + b*y, new, u
# f(a*x*y + b*y)
new = []
d = p.degree(x)
if p.degree(y) == d:
for itry in range(2):
a = root(p.coeff_monomial(x**d*y**d), d)
b = root(p.coeff_monomial(y**d), d)
new = ok(f, x, (u - b*y)/a/y)
if new is not None:
return a*x*y + b*y, new, u
x, y = y, x
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.
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 tuple containing all
those roots set the ``multiple`` flag to True.
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}
"""
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
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)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for root in _try_heuristics(factors[0]):
roots.append(root)
for factor in factors[1:]:
previous, roots = list(roots), []
for root in previous:
g = factor - Poly(root, f.gen)
for root in _try_heuristics(g):
roots.append(root)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S(0)]*f.degree()
if f.length() == 2:
if f.degree() == 1:
return 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 += map(cancel, roots_linear(f))
elif n == 2:
result += map(cancel, roots_quadratic(f))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
(k,), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S(0): k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().has_Ring:
f = f.to_field()
result = {}
if not f.is_ground:
if not f.get_domain().is_Exact:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
elif f.length() == 2:
for r in roots_binomial(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and factors[0][1] == 1:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)
if coeff is not S.One:
_result, result, = result, {}
for root, k in _result.iteritems():
result[coeff*root] = k
result.update(zeros)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).iterkeys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).iterkeys():
if not predicate(zero):
del result[zero]
if not multiple:
return result
else:
zeros = []
for zero, k in result.iteritems():
zeros.extend([zero]*k)
return sorted(zeros, key=default_sort_key)
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.
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 tuple containing all
those roots set the ``multiple`` flag to True.
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}
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
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)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for root in _try_heuristics(factors[0]):
roots.append(root)
for factor in factors[1:]:
previous, roots = list(roots), []
for root in previous:
g = factor - Poly(root, f.gen)
for root in _try_heuristics(g):
roots.append(root)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S(0)] * 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)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
(k, ), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S(0): k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().has_Ring:
f = f.to_field()
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
if not f.get_domain().is_Exact:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
elif f.length() == 2:
for r in roots_binomial(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
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 root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)
if coeff is not S.One:
_result, result, = result, {}
for root, k in _result.items():
result[coeff * root] = k
result.update(zeros)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'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
if not multiple:
return result
else:
zeros = []
for zero, k in result.items():
zeros.extend([zero] * k)
return sorted(zeros, key=default_sort_key)
def __new__(cls, f, x=None, indices=None, radicals=True, expand=True):
"""Construct a new ``RootOf`` object for ``k``-th root of ``f``. """
if indices is None and (not isinstance(x, Basic) or x.is_Integer):
x, indices = None, x
poly = Poly(f, x, greedy=False, expand=expand)
if not poly.is_univariate:
raise PolynomialError("only univariate polynomials are allowed")
degree = poly.degree()
if degree <= 0:
raise PolynomialError("can't construct RootOf object for %s" % f)
if indices is not None and indices is not True:
if hasattr(indices, '__iter__'):
indices, iterable = list(indices), True
else:
indices, iterable = [indices], False
indices = map(int, indices)
for i, index in enumerate(indices):
if index < -degree or index >= degree:
raise IndexError("root index out of [%d, %d] range, got %d" % (-degree, degree-1, index))
elif index < 0:
indices[i] += degree
else:
iterable = True
if indices is True:
indices = range(degree)
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = roots_trivial(poly, radicals)
if roots is not None:
if indices is not None:
result = [ roots[index] for index in indices ]
else:
result = [ root for root in roots if root.is_real ]
else:
coeff, poly = _rootof_preprocess(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("RootOf is not supported over %s" % dom)
result = []
for data in _rootof_data(poly, indices):
poly, index, pointer, conjugate = data
roots = roots_trivial(poly, radicals)
if roots is not None:
result.append(coeff*roots[index])
else:
result.append(coeff*cls._new(poly, index, pointer, conjugate))
if not iterable:
return result[0]
else:
return result
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.
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 tuple containing all
those roots set the ``multiple`` flag to True.
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}
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
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)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
def compose_(fs):
r = fs[0]
for f in fs[1:]:
r = compose(r, f)
return r
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
if len(factors) == 1:
for root in _try_heuristics(factors[0]):
roots.append(root)
return (roots, False)
add_comment("Use the substitution")
t = Dummy("t1")
add_eq(t, compose_(factors[1:]).as_expr())
g = Poly(factors[0].as_expr().subs(f.gen, t), t)
add_comment("We have")
add_eq(g.as_expr(), 0)
for root in _try_heuristics(g):
roots.append(root)
# f(x) = f1(f2(f3(x))) = 0
# f(x) = f1(t) = 0 --> f2(f3(x)) = t1, t2, ..., tn
i = 1
for factor in factors[1:]:
previous, roots = list(roots), []
h = compose_(factors[i:]).as_expr()
add_comment("Therefore")
for root in previous:
add_eq(h, root)
for root in previous:
if i < len(factors) - 1:
add_comment("Solve the equation")
add_eq(h, root)
add_comment("Use the substitution")
t = Dummy("t" + str(i+1))
add_eq(t, compose_(factors[(i+1):]).as_expr())
g = Poly(factor.as_expr().subs(f.gen, t), t) - Poly(root, t)
else:
g = factor - Poly(root, f.gen)
for root in _try_heuristics(g):
roots.append(root)
i += 1
return (roots, True)
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
add_comment("The polynomial is constant. Therefore there is no root.")
return []
if f.is_monomial:
add_comment("The root of the equation is zero")
rr = [S(0)]*f.degree()
return rr
if f.length() == 2:
if f.degree() == 1:
rr = list(map(cancel, roots_linear(f)))
return rr
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)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
tmp_ = f
(k,), f = f.terms_gcd()
if (tmp_ != f):
add_comment("Rewrite the equation as")
add_eq(Mul(f.gen**k, f.as_expr(), evaluate=False), 0)
if not k:
zeros = {}
else:
zeros = {S(0): k}
if not f.is_ground:
add_comment("Solve the equation")
add_eq(f.as_expr(), 0)
else:
add_comment("The roots are")
for z in zeros:
add_eq(f.gen, 0)
return {S(0): k}
coeff, fp = preprocess_roots(f)
if coeff.is_Rational or f.degree() <= 2:
coeff = S.One
else:
f = fp
if (coeff is not S.One):
add_comment("Use the substitution")
t = Dummy("t")
add_eq(f.gen, t*coeff)
add_comment("We have")
f = Poly(f.as_expr().subs(f.gen, t), t)
add_eq(f.as_expr(), 0)
if auto and f.get_domain().has_Ring:
f = f.to_field()
rescale_x = None
translate_x = None
result = {}
print_all_roots = False
if not f.is_ground:
if not f.get_domain().is_Exact:
add_comment("Use numerical methods")
for r in f.nroots():
add_eq(f.gen, r)
_update_dict(result, r, 1)
elif f.degree() == 1:
tmp = roots_linear(f)[0]
result[tmp] = 1
elif f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
elif f.length() == 2:
for r in roots_binomial(f):
_update_dict(result, r, 1)
else:
rr = find_rational_roots(f)
if len(rr) > 0:
print_all_roots = True
g = f
h = 1
for r in rr:
while g(r) == 0:
g = g.quo(Poly(f.gen - r, f.gen))
h = Mul(h, f.gen - r, evaluate=False)
_update_dict(result, r, 1)
add_comment("Rewrite the equation as")
add_eq(Mul(h, g.as_expr(), evaluate=False), 0)
if not g.is_ground:
add_comment("Solve the equation")
add_eq(g.as_expr(), 0)
f = g
if not f.is_ground:
_, factors = Poly(f.as_expr()).factor_list()
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:
td = _try_decompose(f)
print_all_roots = td[1]
for root in td[0]:
_update_dict(result, root, 1)
else:
td = _try_decompose(f)
print_all_roots = td[1]
for root in td[0]:
_update_dict(result, root, 1)
else:
print_all_roots = True
add_comment("The equation can be rewritten as")
eq1 = 1
for factor, k in factors:
eq1 = Mul(eq1, (factor.as_expr())**k, evaluate=False)
add_eq(eq1, 0)
for factor, k in factors:
add_comment("Solve the equation")
add_eq(factor.as_expr(), 0)
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)
else:
add_comment("Solve the equation")
add_eq(f.as_expr(), 0)
add_comment("Since the polynomial is constant, there are no roots")
if coeff is not S.One:
_result, result, = result, {}
add_comment("Therefore")
for root, k in _result.items():
result[coeff*root] = k
for i in range(k):
add_eq(tmp_.gen, coeff*root)
result.update(zeros)
if tmp_ != f or print_all_roots:
add_comment("Finally we have the following roots")
for r, k in result.items():
for i in range(k):
add_eq(f.gen, r)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'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
if not multiple:
return result
else:
zeros = []
for zero, k in result.items():
zeros.extend([zero]*k)
return sorted(zeros, key=default_sort_key)
def round_two(T, radicals=None):
r"""
Zassenhaus's "Round 2" algorithm.
Explanation
===========
Carry out Zassenhaus's "Round 2" algorithm on a monic irreducible
polynomial *T* over :ref:`ZZ`. This computes an integral basis and the
discriminant for the field $K = \mathbb{Q}[x]/(T(x))$.
Ordinarily this function need not be called directly, as one can instead
access the :py:meth:`~.AlgebraicField.maximal_order`,
:py:meth:`~.AlgebraicField.integral_basis`, and
:py:meth:`~.AlgebraicField.discriminant` methods of an
:py:class:`~.AlgebraicField`.
Examples
========
Working through an AlgebraicField:
>>> from sympy import Poly, QQ
>>> from sympy.abc import x
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> K = QQ.alg_field_from_poly(T, "theta")
>>> print(K.maximal_order())
Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2
>>> print(K.discriminant())
-503
>>> print(K.integral_basis(fmt='sympy'))
[1, theta, theta/2 + theta**2/2]
Calling directly:
>>> from sympy import Poly
>>> from sympy.abc import x
>>> from sympy.polys.numberfields.basis import round_two
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> print(round_two(T))
(Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2, -503)
The nilradicals mod $p$ that are sometimes computed during the Round Two
algorithm may be useful in further calculations. Pass a dictionary under
`radicals` to receive these:
>>> T = Poly(x**3 + 3*x**2 + 5)
>>> rad = {}
>>> ZK, dK = round_two(T, radicals=rad)
>>> print(rad)
{3: Submodule[[-1, 1, 0], [-1, 0, 1]]}
Parameters
==========
T : :py:class:`~.Poly`
The irreducible monic polynomial over :ref:`ZZ` defining the number
field.
radicals : dict, optional
This is a way for any $p$-radicals (if computed) to be returned by
reference. If desired, pass an empty dictionary. If the algorithm
reaches the point where it computes the nilradical mod $p$ of the ring
of integers $Z_K$, then an $\mathbb{F}_p$-basis for this ideal will be
stored in this dictionary under the key ``p``. This can be useful for
other algorithms, such as prime decomposition.
Returns
=======
Pair ``(ZK, dK)``, where:
``ZK`` is a :py:class:`~sympy.polys.numberfields.modules.Submodule`
representing the maximal order.
``dK`` is the discriminant of the field $K = \mathbb{Q}[x]/(T(x))$.
See Also
========
.AlgebraicField.maximal_order
.AlgebraicField.integral_basis
.AlgebraicField.discriminant
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
"""
if T.domain == QQ:
try:
T = Poly(T, domain=ZZ)
except CoercionFailed:
pass # Let the error be raised by the next clause.
if (not T.is_univariate or not T.is_irreducible or not T.is_monic
or not T.domain == ZZ):
raise ValueError(
'Round 2 requires a monic irreducible univariate polynomial over ZZ.'
)
n = T.degree()
D = T.discriminant()
D_modulus = ZZ.from_sympy(abs(D))
# D must be 0 or 1 mod 4 (see Cohen Sec 4.4), which ensures we can write
# it in the form D = D_0 * F**2, where D_0 is 1 or a fundamental discriminant.
_, F = extract_fundamental_discriminant(D)
Ztheta = PowerBasis(T)
H = Ztheta.whole_submodule()
nilrad = None
while F:
# Next prime:
p, e = F.popitem()
U_bar, m = _apply_Dedekind_criterion(T, p)
if m == 0:
continue
# For a given prime p, the first enlargement of the order spanned by
# the current basis can be done in a simple way:
U = Ztheta.element_from_poly(Poly(U_bar, domain=ZZ))
# TODO:
# Theory says only first m columns of the U//p*H term below are needed.
# Could be slightly more efficient to use only those. Maybe `Submodule`
# class should support a slice operator?
H = H.add(U // p * H, hnf_modulus=D_modulus)
if e <= m:
continue
# A second, and possibly more, enlargements for p will be needed.
# These enlargements require a more involved procedure.
q = p
while q < n:
q *= p
H1, nilrad = _second_enlargement(H, p, q)
while H1 != H:
H = H1
H1, nilrad = _second_enlargement(H, p, q)
# Note: We do not store all nilradicals mod p, only the very last. This is
# because, unless computed against the entire integral basis, it might not
# be accurate. (In other words, if H was not already equal to ZK when we
# passed it to `_second_enlargement`, then we can't trust the nilradical
# so computed.) Example: if T(x) = x ** 3 + 15 * x ** 2 - 9 * x + 13, then
# F is divisible by 2, 3, and 7, and the nilradical mod 2 as computed above
# will not be accurate for the full, maximal order ZK.
if nilrad is not None and isinstance(radicals, dict):
radicals[p] = nilrad
ZK = H
# Pre-set expensive boolean properties which we already know to be true:
ZK._starts_with_unity = True
ZK._is_sq_maxrank_HNF = True
dK = (D * ZK.matrix.det()**2) // ZK.denom**(2 * n)
return ZK, dK
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
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 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(0)]*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
(k,), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S(0): k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().is_Ring:
f = f.to_field()
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
result.update(zeros)
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'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
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero]*result[zero])
return zeros
def __new__(cls, f, x=None, indices=None, radicals=True, expand=True):
"""Construct a new ``RootOf`` object for ``k``-th root of ``f``. """
if indices is None and (not isinstance(x, Basic) or x.is_Integer):
x, indices = None, x
poly = Poly(f, x, greedy=False, expand=expand)
if not poly.is_univariate:
raise PolynomialError("only univariate polynomials are allowed")
degree = poly.degree()
if degree <= 0:
raise PolynomialError("can't construct RootOf object for %s" % f)
if indices is not None and indices is not True:
if hasattr(indices, '__iter__'):
indices, iterable = list(indices), True
else:
indices, iterable = [indices], False
indices = map(int, indices)
for i, index in enumerate(indices):
if index < -degree or index >= degree:
raise IndexError(
"root index out of [%d, %d] range, got %d" %
(-degree, degree - 1, index))
elif index < 0:
indices[i] += degree
else:
iterable = True
if indices is True:
indices = range(degree)
dom = poly.get_domain()
if not dom.is_Exact:
poly = poly.to_exact()
roots = roots_trivial(poly, radicals)
if roots is not None:
if indices is not None:
result = [roots[index] for index in indices]
else:
result = [root for root in roots if root.is_real]
else:
coeff, poly = _rootof_preprocess(poly)
dom = poly.get_domain()
if not dom.is_ZZ:
raise NotImplementedError("RootOf is not supported over %s" %
dom)
result = []
for data in _rootof_data(poly, indices):
poly, index, pointer, conjugate = data
roots = roots_trivial(poly, radicals)
if roots is not None:
result.append(coeff * roots[index])
else:
result.append(coeff *
cls._new(poly, index, pointer, conjugate))
if not iterable:
return result[0]
else:
return result
