def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
**Examples**
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))
>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
return exc.expr
def viete(f, roots=None, *gens, **args):
"""
Generate Viete's formulas for ``f``.
Examples
========
>>> from sympy.polys.polyfuncs import viete
>>> from sympy import symbols
>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')
>>> viete(a*x**2 + b*x + c, [r1, r2], x)
[(r1 + r2, -b/a), (r1*r2, c/a)]
"""
allowed_flags(args, [])
if isinstance(roots, Basic):
gens, roots = (roots,) + gens, None
try:
f, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
raise ComputationFailed('viete', 1, exc)
def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
Examples
========
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))
>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
return exc.expr
def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
**Examples**
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
5 + x*(6 + x*(7 + x*(8 + 9*x)))
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(b + a*x)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(y*(1 + 2*y) + x*y*(2 + 4*y))
>>> horner(f, wrt=y)
y*(x*(1 + 2*x) + x*y*(2 + 4*x))
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
return exc.expr
def viete(f, roots=None, *gens, **args):
"""
Generate Viete's formulas for ``f``.
Examples
========
>>> from sympy.polys.polyfuncs import viete
>>> from sympy import symbols
>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')
>>> viete(a*x**2 + b*x + c, [r1, r2], x)
[(r1 + r2, -b/a), (r1*r2, c/a)]
"""
allowed_flags(args, [])
if isinstance(roots, Basic):
gens, roots = (roots,) + gens, None
try:
f, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed, exc:
raise ComputationFailed("viete", 1, exc)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f`` compute partial fraction decomposition
of ``f``. Two algorithms are available: one is based on undetermined
coefficients method and the other is Bronstein's full partial fraction
decomposition algorithm.
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
options = set_defaults(options, extension=True)
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
if P.is_multivariate:
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P/Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common*(poly.as_expr() + terms)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f`` compute partial fraction decomposition
of ``f``. Two algorithms are available: one is based on undetermined
coefficients method and the other is Bronstein's full partial fraction
decomposition algorithm.
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
options = set_defaults(options, extension=True)
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
if P.is_multivariate:
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P / Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common * (poly.as_expr() + terms)
def viete(f, roots=None, *gens, **args):
"""
Generate Viete's formulas for ``f``.
Examples
========
>>> from sympy.polys.polyfuncs import viete
>>> from sympy import symbols
>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')
>>> viete(a*x**2 + b*x + c, [r1, r2], x)
[(r1 + r2, -b/a), (r1*r2, c/a)]
"""
allowed_flags(args, [])
if isinstance(roots, Basic):
gens, roots = (roots, ) + gens, None
try:
f, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('viete', 1, exc)
if f.is_multivariate:
raise MultivariatePolynomialError(
"multivariate polynomials are not allowed")
n = f.degree()
if n < 1:
raise ValueError(
"can't derive Viete's formulas for a constant polynomial")
if roots is None:
roots = numbered_symbols('r', start=1)
roots = take(roots, n)
if n != len(roots):
raise ValueError("required %s roots, got %s" % (n, len(roots)))
lc, coeffs = f.LC(), f.all_coeffs()
result, sign = [], -1
for i, coeff in enumerate(coeffs[1:]):
poly = symmetric_poly(i + 1, roots)
coeff = sign * (coeff / lc)
result.append((poly, coeff))
sign = -sign
return result
def viete(f, roots=None, *gens, **args):
"""
Generate Viete's formulas for ``f``.
Examples
========
>>> from sympy.polys.polyfuncs import viete
>>> from sympy import symbols
>>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')
>>> viete(a*x**2 + b*x + c, [r1, r2], x)
[(r1 + r2, -b/a), (r1*r2, c/a)]
"""
allowed_flags(args, [])
if isinstance(roots, Basic):
gens, roots = (roots,) + gens, None
try:
f, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('viete', 1, exc)
if f.is_multivariate:
raise MultivariatePolynomialError(
"multivariate polynomials are not allowed")
n = f.degree()
if n < 1:
raise ValueError(
"can't derive Viete's formulas for a constant polynomial")
if roots is None:
roots = numbered_symbols('r', start=1)
roots = take(roots, n)
if n != len(roots):
raise ValueError("required %s roots, got %s" % (n, len(roots)))
lc, coeffs = f.LC(), f.all_coeffs()
result, sign = [], -1
for i, coeff in enumerate(coeffs[1:]):
poly = symmetric_poly(i + 1, roots)
coeff = sign*(coeff/lc)
result.append((poly, coeff))
sign = -sign
return result
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
Examples
========
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ["formal", "symbols"])
iterable = True
if not hasattr(F, "__iter__"):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed, exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed("symmetrize", len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([],)
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
Examples
========
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ['formal', 'symbols'])
iterable = True
if not hasattr(F, '__iter__'):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed, exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed('symmetrize', len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([],)
def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
Among other applications, evaluation of a polynomial at a point is optimal
when it is applied using the Horner scheme ([1]).
Examples
========
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))
>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))
References
==========
[1] - http://en.wikipedia.org/wiki/Horner_scheme
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
return exc.expr
form, gen = S.Zero, F.gen
if F.is_univariate:
for coeff in F.all_coeffs():
form = form*gen + coeff
else:
F, gens = Poly(F, gen), gens[1:]
for coeff in F.all_coeffs():
form = form*gen + horner(coeff, *gens, **args)
return form
def horner(f, *gens, **args):
"""
Rewrite a polynomial in Horner form.
Among other applications, evaluation of a polynomial at a point is optimal
when it is applied using the Horner scheme ([1]).
Examples
========
>>> from sympy.polys.polyfuncs import horner
>>> from sympy.abc import x, y, a, b, c, d, e
>>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
x*(x*(x*(9*x + 8) + 7) + 6) + 5
>>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
e + x*(d + x*(c + x*(a*x + b)))
>>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y
>>> horner(f, wrt=x)
x*(x*y*(4*y + 2) + y*(2*y + 1))
>>> horner(f, wrt=y)
y*(x*y*(4*x + 2) + x*(2*x + 1))
References
==========
[1] - http://en.wikipedia.org/wiki/Horner_scheme
"""
allowed_flags(args, [])
try:
F, opt = poly_from_expr(f, *gens, **args)
except PolificationFailed as exc:
return exc.expr
form, gen = S.Zero, F.gen
if F.is_univariate:
for coeff in F.all_coeffs():
form = form * gen + coeff
else:
F, gens = Poly(F, gen), gens[1:]
for coeff in F.all_coeffs():
form = form * gen + horner(coeff, *gens, **args)
return form
def __new__(cls, iterative, F, n, *gens, **args):
"""Compute a reduced Groebner basis for a system of polynomials. """
options.allowed_flags(args, ['polys', 'method'])
try:
polys, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('groebner', len(F), exc)
from sympy.polys.rings import PolyRing
ring = PolyRing(opt.gens, opt.domain, opt.order)
polys = [ring.from_dict(poly.rep.to_dict()) for poly in polys if poly]
if iterative:
G = iter_groebner(polys, n, ring, method=opt.method
) #Assumes last element is the "new" polynomial
else:
G = _groebner(polys, ring, method=opt.method)
G = [Poly._from_dict(g, opt) for g in G]
return cls._new(G, opt)
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
A symmetric polynomial is a multivariate polynomial that remains invariant
under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``,
then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where
``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an
element of the group ``S_n``).
Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that
``f = f1 + f2 + ... + fn``.
Examples
========
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ['formal', 'symbols'])
iterable = True
if not hasattr(F, '__iter__'):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed as exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed('symmetrize', len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([], )
polys, symbols = [], opt.symbols
gens, dom = opt.gens, opt.domain
for i in xrange(0, len(gens)):
poly = symmetric_poly(i + 1, gens, polys=True)
polys.append((symbols.next(), poly.set_domain(dom)))
indices = range(0, len(gens) - 1)
weights = range(len(gens), 0, -1)
result = []
for f in F:
symmetric = []
if not f.is_homogeneous:
symmetric.append(f.TC())
f -= f.TC()
while f:
_height, _monom, _coeff = -1, None, None
for i, (monom, coeff) in enumerate(f.terms()):
if all(monom[i] >= monom[i + 1] for i in indices):
height = max([n * m for n, m in zip(weights, monom)])
if height > _height:
_height, _monom, _coeff = height, monom, coeff
if _height != -1:
monom, coeff = _monom, _coeff
else:
break
exponents = []
for m1, m2 in zip(monom, monom[1:] + (0, )):
exponents.append(m1 - m2)
term = [s**n for (s, _), n in zip(polys, exponents)]
poly = [p**n for (_, p), n in zip(polys, exponents)]
symmetric.append(Mul(coeff, *term))
product = poly[0].mul(coeff)
for p in poly[1:]:
product = product.mul(p)
f -= product
result.append((Add(*symmetric), f.as_expr()))
polys = [(s, p.as_expr()) for s, p in polys]
if not opt.formal:
for i, (sym, non_sym) in enumerate(result):
result[i] = (sym.subs(polys), non_sym)
if not iterable:
result, = result
if not opt.formal:
return result
else:
if iterable:
return result, polys
else:
return result + (polys, )
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
A symmetric polynomial is a multivariate polynomial that remains invariant
under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``,
then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where
``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an
element of the group ``S_n``).
Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that
``f = f1 + f2 + ... + fn``.
Examples
========
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ["formal", "symbols"])
iterable = True
if not hasattr(F, "__iter__"):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed, exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed("symmetrize", len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([],)
def apart_list(f, x=None, dummies=None, **options):
"""
Compute partial fraction decomposition of a rational function
and return the result in structured form.
Given a rational function ``f`` compute the partial fraction decomposition
of ``f``. Only Bronstein's full partial fraction decomposition algorithm
is supported by this method. The return value is highly structured and
perfectly suited for further algorithmic treatment rather than being
human-readable. The function returns a tuple holding three elements:
* The first item is the common coefficient, free of the variable `x` used
for decomposition. (It is an element of the base field `K`.)
* The second item is the polynomial part of the decomposition. This can be
the zero polynomial. (It is an element of `K[x]`.)
* The third part itself is a list of quadruples. Each quadruple
has the following elements in this order:
- The (not necessarily irreducible) polynomial `D` whose roots `w_i` appear
in the linear denominator of a bunch of related fraction terms. (This item
can also be a list of explicit roots. However, at the moment ``apart_list``
never returns a result this way, but the related ``assemble_partfrac_list``
function accepts this format as input.)
- The numerator of the fraction, written as a function of the root `w`
- The linear denominator of the fraction *excluding its power exponent*,
written as a function of the root `w`.
- The power to which the denominator has to be raised.
On can always rebuild a plain expression by using the function ``assemble_partfrac_list``.
Examples
========
A first example:
>>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list
>>> from sympy.abc import x, t
>>> f = (2*x**3 - 2*x) / (x**2 - 2*x + 1)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(2*x + 4, x, domain='ZZ'),
[(Poly(_w - 1, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
2*x + 4 + 4/(x - 1)
Second example:
>>> f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
>>> pfd = apart_list(f)
>>> pfd
(-1,
Poly(2/3, x, domain='QQ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-2/3 - 2/(x - 2)
Another example, showing symbolic parameters:
>>> pfd = apart_list(t/(x**2 + x + t), x)
>>> pfd
(1,
Poly(0, x, domain='ZZ[t]'),
[(Poly(_w**2 + _w + t, _w, domain='ZZ[t]'),
Lambda(_a, -2*_a*t/(4*t - 1) - t/(4*t - 1)),
Lambda(_a, -_a + x),
1)])
>>> assemble_partfrac_list(pfd)
RootSum(_w**2 + _w + t, Lambda(_a, (-2*_a*t/(4*t - 1) - t/(4*t - 1))/(-_a + x)))
This example is taken from Bronstein's original paper:
>>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
See also
========
apart, assemble_partfrac_list
References
==========
1. [Bronstein93]_
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
options = set_defaults(options, extension=True)
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
if P.is_multivariate:
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
polypart = poly
if dummies is None:
def dummies(name):
d = Dummy(name)
while True:
yield d
dummies = dummies("w")
rationalpart = apart_list_full_decomposition(P, Q, dummies)
return (common, polypart, rationalpart)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f`` compute partial fraction decomposition
of ``f``. Two algorithms are available: one is based on undetermined
coefficients method and the other is Bronstein's full partial fraction
decomposition algorithm.
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
By default, using the undetermined coefficients method:
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
You can choose Bronstein's algorithm by setting ``full=True``:
>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)
>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))
See Also
========
apart_list, assemble_partfrac_list
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
_options = options.copy()
options = set_defaults(options, extension=True)
try:
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
except PolynomialError as msg:
if f.is_commutative:
raise PolynomialError(msg)
# non-commutative
if f.is_Mul:
c, nc = f.args_cnc(split_1=False)
nc = Mul(*[apart(i, x=x, full=full, **_options) for i in nc])
if c:
c = apart(Mul._from_args(c), x=x, full=full, **_options)
return c*nc
else:
return nc
elif f.is_Add:
c = []
nc = []
for i in f.args:
if i.is_commutative:
c.append(i)
else:
try:
nc.append(apart(i, x=x, full=full, **_options))
except NotImplementedError:
nc.append(i)
return apart(Add(*c), x=x, full=full, **_options) + Add(*nc)
else:
reps = []
pot = preorder_traversal(f)
pot.next()
for e in pot:
try:
reps.append((e, apart(e, x=x, full=full, **_options)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
if P.is_multivariate:
fc = f.cancel()
if fc != f:
return apart(fc, x=x, full=full, **_options)
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P/Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common*(poly.as_expr() + terms)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f``, computes the partial fraction
decomposition of ``f``. Two algorithms are available: One is based on the
undertermined coefficients method, the other is Bronstein's full partial
fraction decomposition algorithm.
The undetermined coefficients method (selected by ``full=False``) uses
polynomial factorization (and therefore accepts the same options as
factor) for the denominator. Per default it works over the rational
numbers, therefore decomposition of denominators with non-rational roots
(e.g. irrational, complex roots) is not supported by default (see options
of factor).
Bronstein's algorithm can be selected by using ``full=True`` and allows a
decomposition of denominators with non-rational roots. A human-readable
result can be obtained via ``doit()`` (see examples below).
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
By default, using the undetermined coefficients method:
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
The undetermined coefficients method does not provide a result when the
denominators roots are not rational:
>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)
You can choose Bronstein's algorithm by setting ``full=True``:
>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))
Calling ``doit()`` yields a human-readable result:
>>> apart(y/(x**2 + x + 1), x, full=True).doit()
(-y/3 - 2*y*(-1/2 - sqrt(3)*I/2)/3)/(x + 1/2 + sqrt(3)*I/2) + (-y/3 -
2*y*(-1/2 + sqrt(3)*I/2)/3)/(x + 1/2 - sqrt(3)*I/2)
See Also
========
apart_list, assemble_partfrac_list
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
_options = options.copy()
options = set_defaults(options, extension=True)
try:
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
except PolynomialError as msg:
if f.is_commutative:
raise PolynomialError(msg)
# non-commutative
if f.is_Mul:
c, nc = f.args_cnc(split_1=False)
nc = f.func(*nc)
if c:
c = apart(f.func._from_args(c), x=x, full=full, **_options)
return c*nc
else:
return nc
elif f.is_Add:
c = []
nc = []
for i in f.args:
if i.is_commutative:
c.append(i)
else:
try:
nc.append(apart(i, x=x, full=full, **_options))
except NotImplementedError:
nc.append(i)
return apart(f.func(*c), x=x, full=full, **_options) + f.func(*nc)
else:
reps = []
pot = preorder_traversal(f)
next(pot)
for e in pot:
try:
reps.append((e, apart(e, x=x, full=full, **_options)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
if P.is_multivariate:
fc = f.cancel()
if fc != f:
return apart(fc, x=x, full=full, **_options)
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P/Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common*(poly.as_expr() + terms)
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
A symmetric polynomial is a multivariate polynomial that remains invariant
under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``,
then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where
``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an
element of the group ``S_n``).
Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that
``f = f1 + f2 + ... + fn``.
Examples
========
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ['formal', 'symbols'])
iterable = True
if not hasattr(F, '__iter__'):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed as exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed('symmetrize', len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([],)
polys, symbols = [], opt.symbols
gens, dom = opt.gens, opt.domain
for i in xrange(0, len(gens)):
poly = symmetric_poly(i + 1, gens, polys=True)
polys.append((next(symbols), poly.set_domain(dom)))
indices = list(range(0, len(gens) - 1))
weights = list(range(len(gens), 0, -1))
result = []
for f in F:
symmetric = []
if not f.is_homogeneous:
symmetric.append(f.TC())
f -= f.TC()
while f:
_height, _monom, _coeff = -1, None, None
for i, (monom, coeff) in enumerate(f.terms()):
if all(monom[i] >= monom[i + 1] for i in indices):
height = max([ n*m for n, m in zip(weights, monom) ])
if height > _height:
_height, _monom, _coeff = height, monom, coeff
if _height != -1:
monom, coeff = _monom, _coeff
else:
break
exponents = []
for m1, m2 in zip(monom, monom[1:] + (0,)):
exponents.append(m1 - m2)
term = [ s**n for (s, _), n in zip(polys, exponents) ]
poly = [ p**n for (_, p), n in zip(polys, exponents) ]
symmetric.append(Mul(coeff, *term))
product = poly[0].mul(coeff)
for p in poly[1:]:
product = product.mul(p)
f -= product
result.append((Add(*symmetric), f.as_expr()))
polys = [ (s, p.as_expr()) for s, p in polys ]
if not opt.formal:
for i, (sym, non_sym) in enumerate(result):
result[i] = (sym.subs(polys), non_sym)
if not iterable:
result, = result
if not opt.formal:
return result
else:
if iterable:
return result, polys
else:
return result + (polys,)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f``, computes the partial fraction
decomposition of ``f``. Two algorithms are available: One is based on the
undertermined coefficients method, the other is Bronstein's full partial
fraction decomposition algorithm.
The undetermined coefficients method (selected by ``full=False``) uses
polynomial factorization (and therefore accepts the same options as
factor) for the denominator. Per default it works over the rational
numbers, therefore decomposition of denominators with non-rational roots
(e.g. irrational, complex roots) is not supported by default (see options
of factor).
Bronstein's algorithm can be selected by using ``full=True`` and allows a
decomposition of denominators with non-rational roots. A human-readable
result can be obtained via ``doit()`` (see examples below).
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
By default, using the undetermined coefficients method:
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
The undetermined coefficients method does not provide a result when the
denominators roots are not rational:
>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)
You can choose Bronstein's algorithm by setting ``full=True``:
>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))
Calling ``doit()`` yields a human-readable result:
>>> apart(y/(x**2 + x + 1), x, full=True).doit()
(-y/3 - 2*y*(-1/2 - sqrt(3)*I/2)/3)/(x + 1/2 + sqrt(3)*I/2) + (-y/3 -
2*y*(-1/2 + sqrt(3)*I/2)/3)/(x + 1/2 - sqrt(3)*I/2)
See Also
========
apart_list, assemble_partfrac_list
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
_options = options.copy()
options = set_defaults(options, extension=True)
try:
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
except PolynomialError as msg:
if f.is_commutative:
raise PolynomialError(msg)
# non-commutative
if f.is_Mul:
c, nc = f.args_cnc(split_1=False)
nc = f.func(*nc)
if c:
c = apart(f.func._from_args(c), x=x, full=full, **_options)
return c*nc
else:
return nc
elif f.is_Add:
c = []
nc = []
for i in f.args:
if i.is_commutative:
c.append(i)
else:
try:
nc.append(apart(i, x=x, full=full, **_options))
except NotImplementedError:
nc.append(i)
return apart(f.func(*c), x=x, full=full, **_options) + f.func(*nc)
else:
reps = []
pot = preorder_traversal(f)
next(pot)
for e in pot:
try:
reps.append((e, apart(e, x=x, full=full, **_options)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
if P.is_multivariate:
fc = f.cancel()
if fc != f:
return apart(fc, x=x, full=full, **_options)
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P/Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common*(poly.as_expr() + terms)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f`` compute partial fraction decomposition
of ``f``. Two algorithms are available: one is based on undetermined
coefficients method and the other is Bronstein's full partial fraction
decomposition algorithm.
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
By default, using the undetermined coefficients method:
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
You can choose Bronstein's algorithm by setting ``full=True``:
>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)
>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))
See Also
========
apart_list, assemble_partfrac_list
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
_options = options.copy()
options = set_defaults(options, extension=True)
try:
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
except PolynomialError as msg:
if f.is_commutative:
raise PolynomialError(msg)
# non-commutative
if f.is_Mul:
c, nc = f.args_cnc(split_1=False)
nc = f.func(*[apart(i, x=x, full=full, **_options) for i in nc])
if c:
c = apart(f.func._from_args(c), x=x, full=full, **_options)
return c*nc
else:
return nc
elif f.is_Add:
c = []
nc = []
for i in f.args:
if i.is_commutative:
c.append(i)
else:
try:
nc.append(apart(i, x=x, full=full, **_options))
except NotImplementedError:
nc.append(i)
return apart(f.func(*c), x=x, full=full, **_options) + f.func(*nc)
else:
reps = []
pot = preorder_traversal(f)
next(pot)
for e in pot:
try:
reps.append((e, apart(e, x=x, full=full, **_options)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
if P.is_multivariate:
fc = f.cancel()
if fc != f:
return apart(fc, x=x, full=full, **_options)
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
if Q.degree() <= 1:
partial = P/Q
else:
if not full:
partial = apart_undetermined_coeffs(P, Q)
else:
partial = apart_full_decomposition(P, Q)
terms = S.Zero
for term in Add.make_args(partial):
if term.has(RootSum):
terms += term
else:
terms += factor(term)
return common*(poly.as_expr() + terms)
def apart(f, x=None, full=False, **options):
"""
Compute partial fraction decomposition of a rational function.
Given a rational function ``f`` compute partial fraction decomposition
of ``f``. Two algorithms are available: one is based on undetermined
coefficients method and the other is Bronstein's full partial fraction
decomposition algorithm.
Examples
========
>>> from sympy.polys.partfrac import apart
>>> from sympy.abc import x, y
By default, using the undetermined coefficients method:
>>> apart(y/(x + 2)/(x + 1), x)
-y/(x + 2) + y/(x + 1)
You can choose Bronstein's algorithm by setting ``full=True``:
>>> apart(y/(x**2 + x + 1), x)
y/(x**2 + x + 1)
>>> apart(y/(x**2 + x + 1), x, full=True)
RootSum(_w**2 + _w + 1, Lambda(_a, (-2*_a*y/3 - y/3)/(-_a + x)))
See Also
========
apart_list, assemble_partfrac_list
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
_options = options.copy()
options = set_defaults(options, extension=True)
try:
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
except PolynomialError, msg:
if f.is_commutative:
raise PolynomialError(msg)
# non-commutative
if f.is_Mul:
c, nc = f.args_cnc(split_1=False)
nc = Mul(*[apart(i, x=x, full=full, **_options) for i in nc])
if c:
c = apart(Mul._from_args(c), x=x, full=full, **_options)
return c*nc
else:
return nc
elif f.is_Add:
c = []
nc = []
for i in f.args:
if i.is_commutative:
c.append(i)
else:
try:
nc.append(apart(i, x=x, full=full, **_options))
except NotImplementedError:
nc.append(i)
return apart(Add(*c), x=x, full=full, **_options) + Add(*nc)
else:
reps = []
pot = preorder_traversal(f)
pot.next()
for e in pot:
try:
reps.append((e, apart(e, x=x, full=full, **_options)))
pot.skip() # this was handled successfully
except NotImplementedError:
pass
return f.xreplace(dict(reps))
def apart_list(f, x=None, dummies=None, **options):
"""
Compute partial fraction decomposition of a rational function
and return the result in structured form.
Given a rational function ``f`` compute the partial fraction decomposition
of ``f``. Only Bronstein's full partial fraction decomposition algorithm
is supported by this method. The return value is highly structured and
perfectly suited for further algorithmic treatment rather than being
human-readable. The function returns a tuple holding three elements:
* The first item is the common coefficient, free of the variable `x` used
for decomposition. (It is an element of the base field `K`.)
* The second item is the polynomial part of the decomposition. This can be
the zero polynomial. (It is an element of `K[x]`.)
* The third part itself is a list of quadruples. Each quadruple
has the following elements in this order:
- The (not necessarily irreducible) polynomial `D` whose roots `w_i` appear
in the linear denominator of a bunch of related fraction terms. (This item
can also be a list of explicit roots. However, at the moment ``apart_list``
never returns a result this way, but the related ``assemble_partfrac_list``
function accepts this format as input.)
- The numerator of the fraction, written as a function of the root `w`
- The linear denominator of the fraction *excluding its power exponent*,
written as a function of the root `w`.
- The power to which the denominator has to be raised.
On can always rebuild a plain expression by using the function ``assemble_partfrac_list``.
Examples
========
A first example:
>>> from sympy.polys.partfrac import apart_list, assemble_partfrac_list
>>> from sympy.abc import x, t
>>> f = (2*x**3 - 2*x) / (x**2 - 2*x + 1)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(2*x + 4, x, domain='ZZ'),
[(Poly(_w - 1, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
2*x + 4 + 4/(x - 1)
Second example:
>>> f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
>>> pfd = apart_list(f)
>>> pfd
(-1,
Poly(2/3, x, domain='QQ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-2/3 - 2/(x - 2)
Another example, showing symbolic parameters:
>>> pfd = apart_list(t/(x**2 + x + t), x)
>>> pfd
(1,
Poly(0, x, domain='ZZ[t]'),
[(Poly(_w**2 + _w + t, _w, domain='ZZ[t]'),
Lambda(_a, -2*_a*t/(4*t - 1) - t/(4*t - 1)),
Lambda(_a, -_a + x),
1)])
>>> assemble_partfrac_list(pfd)
RootSum(_w**2 + _w + t, Lambda(_a, (-2*_a*t/(4*t - 1) - t/(4*t - 1))/(-_a + x)))
This example is taken from Bronstein's original paper:
>>> f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
>>> pfd = apart_list(f)
>>> pfd
(1,
Poly(0, x, domain='ZZ'),
[(Poly(_w - 2, _w, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(_w**2 - 1, _w, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(_w + 1, _w, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
>>> assemble_partfrac_list(pfd)
-4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
See also
========
apart, assemble_partfrac_list
References
==========
.. [1] [Bronstein93]_
"""
allowed_flags(options, [])
f = sympify(f)
if f.is_Atom:
return f
else:
P, Q = f.as_numer_denom()
options = set_defaults(options, extension=True)
(P, Q), opt = parallel_poly_from_expr((P, Q), x, **options)
if P.is_multivariate:
raise NotImplementedError(
"multivariate partial fraction decomposition")
common, P, Q = P.cancel(Q)
poly, P = P.div(Q, auto=True)
P, Q = P.rat_clear_denoms(Q)
polypart = poly
if dummies is None:
def dummies(name):
d = Dummy(name)
while True:
yield d
dummies = dummies("w")
rationalpart = apart_list_full_decomposition(P, Q, dummies)
return (common, polypart, rationalpart)
def symmetrize(F, *gens, **args):
"""
Rewrite a polynomial in terms of elementary symmetric polynomials.
A symmetric polynomial is a multivariate polynomial that remains invariant
under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``,
then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where
``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an
element of the group ``S_n``).
Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that
``f = f1 + f2 + ... + fn``.
Examples
========
>>> from sympy.polys.polyfuncs import symmetrize
>>> from sympy.abc import x, y
>>> symmetrize(x**2 + y**2)
(-2*x*y + (x + y)**2, 0)
>>> symmetrize(x**2 + y**2, formal=True)
(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])
>>> symmetrize(x**2 - y**2)
(-2*x*y + (x + y)**2, -2*y**2)
>>> symmetrize(x**2 - y**2, formal=True)
(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])
"""
allowed_flags(args, ['formal', 'symbols'])
iterable = True
if not hasattr(F, '__iter__'):
iterable = False
F = [F]
try:
F, opt = parallel_poly_from_expr(F, *gens, **args)
except PolificationFailed, exc:
result = []
for expr in exc.exprs:
if expr.is_Number:
result.append((expr, S.Zero))
else:
raise ComputationFailed('symmetrize', len(F), exc)
else:
if not iterable:
result, = result
if not exc.opt.formal:
return result
else:
if iterable:
return result, []
else:
return result + ([], )
