Python polys.factor函数代码示例

def _separatevars(expr, force):
    if len(expr.free_symbols) == 1:
        return expr
    # don't destroy a Mul since much of the work may already be done
    if expr.is_Mul:
        args = list(expr.args)
        changed = False
        for i, a in enumerate(args):
            args[i] = separatevars(a, force)
            changed = changed or args[i] != a
        if changed:
            expr = expr.func(*args)
        return expr

    # get a Pow ready for expansion
    if expr.is_Pow:
        expr = Pow(separatevars(expr.base, force=force), expr.exp)

    # First try other expansion methods
    expr = expr.expand(mul=False, multinomial=False, force=force)

    _expr, reps = posify(expr) if force else (expr, {})
    expr = factor(_expr).subs(reps)

    if not expr.is_Add:
        return expr

    # Find any common coefficients to pull out
    args = list(expr.args)
    commonc = args[0].args_cnc(cset=True, warn=False)[0]
    for i in args[1:]:
        commonc &= i.args_cnc(cset=True, warn=False)[0]
    commonc = Mul(*commonc)
    commonc = commonc.as_coeff_Mul()[1]  # ignore constants
    commonc_set = commonc.args_cnc(cset=True, warn=False)[0]

    # remove them
    for i, a in enumerate(args):
        c, nc = a.args_cnc(cset=True, warn=False)
        c = c - commonc_set
        args[i] = Mul(*c)*Mul(*nc)
    nonsepar = Add(*args)

    if len(nonsepar.free_symbols) > 1:
        _expr = nonsepar
        _expr, reps = posify(_expr) if force else (_expr, {})
        _expr = (factor(_expr)).subs(reps)

        if not _expr.is_Add:
            nonsepar = _expr

    return commonc*nonsepar

def heuristics(e, z, z0, dir):
    """Computes the limit of an expression term-wise.
    Parameters are the same as for the ``limit`` function.
    Works with the arguments of expression ``e`` one by one, computing
    the limit of each and then combining the results. This approach
    works only for simple limits, but it is fast.
    """

    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, dir)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul): # try together
                        m = together(m)
                    if not isinstance(m, Mul): # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, dir)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])

                if len(e2) > 0:
                    e3 = Mul(*e2).simplify()
                    l = limit(e3, z, z0, dir)
                    rv = l * Mul(*r2)

            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, dir)
    return rv

def _futrig(e, **kwargs):
    """Helper for futrig."""
    from sympy.simplify.fu import (
        TR1, TR2, TR3, TR2i, TR10, L, TR10i,
        TR8, TR6, TR15, TR16, TR111, TR5, TRmorrie, TR11, TR14, TR22,
        TR12)
    from sympy.core.compatibility import _nodes

    if not e.has(TrigonometricFunction):
        return e

    if e.is_Mul:
        coeff, e = e.as_independent(TrigonometricFunction)
    else:
        coeff = S.One

    Lops = lambda x: (L(x), x.count_ops(), _nodes(x), len(x.args), x.is_Add)
    trigs = lambda x: x.has(TrigonometricFunction)

    tree = [identity,
        (
        TR3,  # canonical angles
        TR1,  # sec-csc -> cos-sin
        TR12,  # expand tan of sum
        lambda x: _eapply(factor, x, trigs),
        TR2,  # tan-cot -> sin-cos
        [identity, lambda x: _eapply(_mexpand, x, trigs)],
        TR2i,  # sin-cos ratio -> tan
        lambda x: _eapply(lambda i: factor(i.normal()), x, trigs),
        TR14,  # factored identities
        TR5,  # sin-pow -> cos_pow
        TR10,  # sin-cos of sums -> sin-cos prod
        TR11, TR6, # reduce double angles and rewrite cos pows
        lambda x: _eapply(factor, x, trigs),
        TR14,  # factored powers of identities
        [identity, lambda x: _eapply(_mexpand, x, trigs)],
        TRmorrie,
        TR10i,  # sin-cos products > sin-cos of sums
        [identity, TR8],  # sin-cos products -> sin-cos of sums
        [identity, lambda x: TR2i(TR2(x))],  # tan -> sin-cos -> tan
        [
            lambda x: _eapply(expand_mul, TR5(x), trigs),
            lambda x: _eapply(
                expand_mul, TR15(x), trigs)], # pos/neg powers of sin
        [
            lambda x:  _eapply(expand_mul, TR6(x), trigs),
            lambda x:  _eapply(
                expand_mul, TR16(x), trigs)], # pos/neg powers of cos
        TR111,  # tan, sin, cos to neg power -> cot, csc, sec
        [identity, TR2i],  # sin-cos ratio to tan
        [identity, lambda x: _eapply(
            expand_mul, TR22(x), trigs)],  # tan-cot to sec-csc
        TR1, TR2, TR2i,
        [identity, lambda x: _eapply(
            factor_terms, TR12(x), trigs)],  # expand tan of sum
        )]
    e = greedy(tree, objective=Lops)(e)

    return coeff*e

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 gosper_sum(f, k):
    r"""
    Gosper's hypergeometric summation algorithm.

    Given a hypergeometric term ``f`` such that:

    .. math ::
        s_n = \sum_{k=0}^{n-1} f_k

    and `f(n)` doesn't depend on `n`, returns `g_{n} - g(0)` where
    `g_{n+1} - g_n = f_n`, or ``None`` if `s_n` can not be expressed
    in closed form as a sum of hypergeometric terms.

    Examples
    ========

    >>> from sympy.concrete.gosper import gosper_sum
    >>> from sympy.functions import factorial
    >>> from sympy.abc import i, n, k

    >>> f = (4*k + 1)*factorial(k)/factorial(2*k + 1)
    >>> gosper_sum(f, (k, 0, n))
    (-n! + 2*(2*n + 1)!)/(2*n + 1)!
    >>> _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2])
    True
    >>> gosper_sum(f, (k, 3, n))
    (-60*n! + (2*n + 1)!)/(60*(2*n + 1)!)
    >>> _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5])
    True

    References
    ==========

    .. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B,
           AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100

    """
    indefinite = False

    if is_sequence(k):
        k, a, b = k
    else:
        indefinite = True

    g = gosper_term(f, k)

    if g is None:
        return None

    if indefinite:
        result = f*g
    else:
        result = (f*(g+1)).subs(k, b) - (f*g).subs(k, a)

    return factor(result)

def heuristics(e, z, z0, dir):
    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, dir)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul): # try together
                        m = together(m)
                    if not isinstance(m, Mul): # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, dir)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])

                if len(e2) > 0:
                    e3 = Mul(*e2).simplify()
                    l = limit(e3, z, z0, dir)
                    rv = l * Mul(*r2)

            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, dir)
    return rv

def gosper_sum(f, k):
    """
    Gosper's hypergeometric summation algorithm.

    Given a hypergeometric term ``f`` such that::

    .. math::

        s_n = \sum_{k=0}^{n-1} f_k

    and $f(n)$ doesn't depend on $n$, returns $g_{n} - g(0)$ where
    $g_{n+1} - g_n = f_n$, or ``None`` if $s_n$ can not be expressed
    in closed form as a sum of hypergeometric terms.

    **Examples**

    >>> from sympy.concrete.gosper import gosper_sum
    >>> from sympy.functions import factorial
    >>> from sympy.abc import n, k

    >>> gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n))
    (-n! + 2*(2*n + 1)!)/(2*n + 1)!

    **References**

    .. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B,
           AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100

    """
    indefinite = False

    if is_sequence(k):
        k, a, b = k
    else:
        indefinite = True

    g = gosper_term(f, k)

    if g is None:
        return None

    if indefinite:
        result = f*g
    else:
        result = (f*(g+1)).subs(k, b) - (f*g).subs(k, a)

    return factor(result)

def _separatevars(expr):
    # First try other expansion methods
    expr = expr.expand(mul=False, multinomial=False)
    try:
        expr = factor(expr)
    except PolynomialError:
        pass

    _coeff = Symbol('_coeff', dummy=True)

    if expr.is_Add:

        nonsepar = sympify(0)
        # Find any common coeficients to pull out
        commoncsetlist = []
        for i in expr.args:
            if i.is_Mul:
                commoncsetlist.append(set(i.args))
            else:
                commoncsetlist.append(set((i,)))
        commoncset = set(flatten(commoncsetlist))
        commonc = sympify(1)

        for i in commoncsetlist:
            commoncset = commoncset.intersection(i)
        commonc = Mul(*commoncset)

        for i in expr.args:
            coe = i.extract_multiplicatively(commonc)
            if coe == None:
                nonsepar += sympify(1)
            else:
                nonsepar += coe
        if nonsepar == 0:
            return commonc
        else:
            return commonc*nonsepar

    else:
        return expr

def heuristics(e, z, z0, dir):
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, dir)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul): # try together
                        m = together(m)
                    if not isinstance(m, Mul): # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, dir)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, dir)
    return rv

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:
#.........这里部分代码省略.........

def factor_nc(expr):
    """Return the factored form of ``expr`` while handling non-commutative
    expressions.

    **examples**
    >>> from sympy.core.exprtools import factor_nc
    >>> from sympy import Symbol
    >>> from sympy.abc import x
    >>> A = Symbol('A', commutative=False)
    >>> B = Symbol('B', commutative=False)
    >>> factor_nc((x**2 + 2*A*x + A**2).expand())
    (x + A)**2
    >>> factor_nc(((x + A)*(x + B)).expand())
    (x + A)*(x + B)
    """
    from sympy.simplify.simplify import _mexpand
    from sympy.polys import gcd, factor

    expr = sympify(expr)
    if not isinstance(expr, Expr) or not expr.args:
        return expr
    if not expr.is_Add:
        return expr.func(*[factor_nc(a) for a in expr.args])

    expr, rep, nc_symbols = _mask_nc(expr)
    if rep:
        return factor(expr).subs(rep)
    else:
        args = [a.args_cnc() for a in Add.make_args(expr)]
        c = g = l = r = S.One
        hit = False
        # find any commutative gcd term
        for i, a in enumerate(args):
            if i == 0:
                c = Mul._from_args(a[0])
            elif a[0]:
                c = gcd(c, Mul._from_args(a[0]))
            else:
                c = S.One
        if c is not S.One:
            hit = True
            c, g = c.as_coeff_Mul()
            if g is not S.One:
                for i, (cc, _) in enumerate(args):
                    cc = list(Mul.make_args(Mul._from_args(list(cc))/g))
                    args[i][0] = cc
            else:
                for i, (cc, _) in enumerate(args):
                    cc[0] = cc[0]/c
                    args[i][0] = cc
        # find any noncommutative common prefix
        for i, a in enumerate(args):
            if i == 0:
                n = a[1][:]
            else:
                n = common_prefix(n, a[1])
            if not n:
                # is there a power that can be extracted?
                if not args[0][1]:
                    break
                b, e = args[0][1][0].as_base_exp()
                ok = False
                if e.is_Integer:
                    for t in args:
                        if not t[1]:
                            break
                        bt, et = t[1][0].as_base_exp()
                        if et.is_Integer and bt == b:
                            e = min(e, et)
                        else:
                            break
                    else:
                        ok = hit = True
                        l = b**e
                        il = b**-e
                        for i, a in enumerate(args):
                            args[i][1][0] = il*args[i][1][0]
                        break
                if not ok:
                    break
        else:
            hit = True
            lenn = len(n)
            l = Mul(*n)
            for i, a in enumerate(args):
                args[i][1] = args[i][1][lenn:]
        # find any noncommutative common suffix
        for i, a in enumerate(args):
            if i == 0:
                n = a[1][:]
            else:
                n = common_suffix(n, a[1])
            if not n:
                # is there a power that can be extracted?
                if not args[0][1]:
                    break
                b, e = args[0][1][-1].as_base_exp()
                ok = False
                if e.is_Integer:
                    for t in args:
#.........这里部分代码省略.........

 def factor(self, *gens, **args):
     """See the factor function in sympy.simplify"""
     from sympy.polys import factor
     return factor(self, *gens, **args)

def expand_polyeq(eq):
    if not isinstance(eq, Rel):
        return eq
    rel = eq.func
    expr = factor(eq.args[0]-eq.args[1])
    if expr.is_number:
        return rel(expr, 0, evaluate=True)
    elif isinstance(expr, Mul):
        terms = expr.args
    else:
        terms = (expr,)

    # eliminate positive/negative terms
    terms = tuple(t for t in terms if not _is_positive(t))
    sign = len(terms)
    terms = tuple(t for t in terms if not _is_negative(t))
    sign -= len(terms)

    # reduce power terms
    terms1 = (t for t in terms if not isinstance(t, Pow))
    termspow = tuple(t for t in terms if isinstance(t, Pow))
    terms2 = (t.args[0] for t in termspow if t.args[1]%2==0)
    terms3 = (t.args[0] for t in termspow if t.args[1]%2==1)

    # find positive-or-zero terms
    termsp_ = list(terms2)
    termsn = []
    for term in chain(terms1, terms3):
        if _is_positive_or_zero(term):
            termsp_.append(term)
        elif _is_negative_or_zero(term):
            termsp_.append(-term)
            sign += 1
        else:
            termsn.append(term)

    # reduce polynomial positive-or-zero terms
    termsp = []
    for term in termsp_:
        if is_polynomial(term):
            termsp.extend(term.free_symbols)
        else:
            termsp.append(term)

    # normalize leading coeff
    for i in range(len(termsn)):
        lc = LC(termsn[i], sorted(termsn[i].free_symbols, key=lambda x: x.name))
        if lc < 0:
            sign += 1
        termsn[i] = termsn[i]/lc
    for i in range(len(termsp)):
        lc = LC(termsp[i], sorted(termsp[i].free_symbols, key=lambda x: x.name))
        termsp[i] = termsp[i]/lc

    # swap relation
    if sign % 2 == 1:
        swap = {Eq: Eq, Ne: Ne,
                Lt: Gt, Le: Ge,
                Gt: Lt, Ge: Le}
        rel = swap[rel]

    # construct expression
    eqs = []
    if rel == Ne:
        eqs.append(And(*[Ne(term,0) for term in termsn+termsp]))
    elif rel == Gt:
        eq0 = And(*[Ne(term,0) for term in termsp])
        for i in range(0, len(termsn)+1, 2):
            for neg in combinations(termsn, i):
                pos = tuple(set(termsn)-set(neg))
                eqp = And(*[Gt(p,0) for p in pos])
                eqn = And(*[Lt(n,0) for n in neg])
                eqs.append(And(eq0, eqp, eqn))
    elif rel == Ge:
        for i in range(0, len(termsn)+1, 2):
            for neg in combinations(termsn, i):
                pos = tuple(set(termsn)-set(neg))
                eqp = And(*[Gt(p,0) for p in pos])
                eqn = And(*[Lt(n,0) for n in neg])
                eqs.append(And(eqp, eqn))
    elif rel == Lt:
        eq0 = And(*[Ne(term,0) for term in termsp])
        for i in range(1, len(termsn)+1, 2):
            for neg in combinations(termsn, i):
                pos = tuple(set(termsn)-set(neg))
                eqp = And(*[Gt(p,0) for p in pos])
                eqn = And(*[Lt(n,0) for n in neg])
                eqs.append(And(eq0, eqp, eqn))
    elif rel == Le:
        for i in range(1, len(termsn)+1, 2):
            for neg in combinations(termsn, i):
                pos = tuple(set(termsn)-set(neg))
                eqp = And(*[Gt(p,0) for p in pos])
                eqn = And(*[Lt(n,0) for n in neg])
                eqs.append(And(eqp, eqn))
    if rel in (Eq, Ge, Le):
        for term in termsn+termsp:
            eqs.append(Eq(term,0))

    return Or(*eqs)

#.........这里部分代码省略.........

    >>> 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 _gammasimp(expr, as_comb):
    """
    Helper function for gammasimp and combsimp.

    Simplifies expressions written in terms of gamma function. If
    as_comb is True, it tries to preserve integer arguments. See
    docstring of gammasimp for more information. This was part of
    combsimp() in combsimp.py.
    """

    expr = expr.replace(gamma,
        lambda n: _rf(1, (n - 1).expand()))

    if as_comb:
        expr = expr.replace(_rf,
            lambda a, b: gamma(b + 1))
    else:
        expr = expr.replace(_rf,
            lambda a, b: gamma(a + b)/gamma(a))

    def rule(n, k):
        coeff, rewrite = S.One, False

        cn, _n = n.as_coeff_Add()

        if _n and cn.is_Integer and cn:
            coeff *= _rf(_n + 1, cn)/_rf(_n - k + 1, cn)
            rewrite = True
            n = _n

        # this sort of binomial has already been removed by
        # rising factorials but is left here in case the order
        # of rule application is changed
        if k.is_Add:
            ck, _k = k.as_coeff_Add()
            if _k and ck.is_Integer and ck:
                coeff *= _rf(n - ck - _k + 1, ck)/_rf(_k + 1, ck)
                rewrite = True
                k = _k

        if count_ops(k) > count_ops(n - k):
            rewrite = True
            k = n - k

        if rewrite:
            return coeff*binomial(n, k)

    expr = expr.replace(binomial, rule)

    def rule_gamma(expr, level=0):
        """ Simplify products of gamma functions further. """

        if expr.is_Atom:
            return expr

        def gamma_rat(x):
            # helper to simplify ratios of gammas
            was = x.count(gamma)
            xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand()
                ).replace(_rf, lambda a, b: gamma(a + b)/gamma(a)))
            if xx.count(gamma) < was:
                x = xx
            return x

        def gamma_factor(x):
            # return True if there is a gamma factor in shallow args
            if isinstance(x, gamma):
                return True
            if x.is_Add or x.is_Mul:
                return any(gamma_factor(xi) for xi in x.args)
            if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
                return gamma_factor(x.base)
            return False

        # recursion step
        if level == 0:
            expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
            level += 1

        if not expr.is_Mul:
            return expr

        # non-commutative step
        if level == 1:
            args, nc = expr.args_cnc()
            if not args:
                return expr
            if nc:
                return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc)
            level += 1

        # pure gamma handling, not factor absorption
        if level == 2:
            T, F = sift(expr.args, gamma_factor, binary=True)
            gamma_ind = Mul(*F)
            d = Mul(*T)

            nd, dd = d.as_numer_denom()
            for ipass in range(2):
                args = list(ordered(Mul.make_args(nd)))
#.........这里部分代码省略.........