예제 #1
0
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)
예제 #2
0
파일: partfrac.py 프로젝트: FireJade/sympy
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)
예제 #3
0
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)
예제 #4
0
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)
예제 #5
0
파일: partfrac.py 프로젝트: vidyar/sympy
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)
예제 #6
0
파일: partfrac.py 프로젝트: vidyar/sympy
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)
예제 #7
0
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)
예제 #8
0
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)
예제 #9
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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))