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 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``, 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_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 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))
