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 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 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

#.........这里部分代码省略.........
    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:
#.........这里部分代码省略.........

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 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)
#.........这里部分代码省略.........