(n, alpha, n_digits)
return xi, w
r"""
    Computes the generalized Gauss-Laguerre quadrature [1]_ points and weights.

    The generalized Gauss-Laguerre quadrature approximates the integral:

    .. math::
        \int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)

    The nodes `x_i` of an order `n` quadrature rule are the roots of `L^{\alpha}_n`
    and the weights `w_i` are given by:

    .. math::
        w_i = \frac{\Gamma(\alpha+n)}{n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)}

    Parameters
    ==========

    n : the order of quadrature

    alpha : the exponent of the singularity, `\alpha > -1`

    n_digits : number of significant digits of the points and weights to return

    Returns
    =======

    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
             The points `x_i` and weights `w_i` are returned as ``(x, w)``
             tuple of lists.

    Examples
    ========

    >>> from sympy import S
    >>> from sympy.integrals.quadrature import gauss_gen_laguerre
    >>> x, w = gauss_gen_laguerre(3, -S.Half, 5)
    >>> x
    [0.19016, 1.7845, 5.5253]
    >>> w
    [1.4493, 0.31413, 0.00906]

    >>> x, w = gauss_gen_laguerre(4, 3*S.Half, 5)
    >>> x
    [0.97851, 2.9904, 6.3193, 11.712]
    >>> w
    [0.53087, 0.67721, 0.11895, 0.0023152]

    See Also
    ========

    gauss_legendre, gauss_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
    .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html
r"""
    Computes the generalized Gauss-Laguerre quadrature [1]_ points and weights.
r Computes the generalized Gauss - Laguerre quadrature [ 1 ] _ points and weights .
def gauss_gen_laguerre(n, alpha, n_digits):
    r"""
    Computes the generalized Gauss-Laguerre quadrature [1]_ points and weights.

    The generalized Gauss-Laguerre quadrature approximates the integral:

    .. math::
        \int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)

    The nodes `x_i` of an order `n` quadrature rule are the roots of `L^{\alpha}_n`
    and the weights `w_i` are given by:

    .. math::
        w_i = \frac{\Gamma(\alpha+n)}{n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)}

    Parameters
    ==========

    n : the order of quadrature

    alpha : the exponent of the singularity, `\alpha > -1`

    n_digits : number of significant digits of the points and weights to return

    Returns
    =======

    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
             The points `x_i` and weights `w_i` are returned as ``(x, w)``
             tuple of lists.

    Examples
    ========

    >>> from sympy import S
    >>> from sympy.integrals.quadrature import gauss_gen_laguerre
    >>> x, w = gauss_gen_laguerre(3, -S.Half, 5)
    >>> x
    [0.19016, 1.7845, 5.5253]
    >>> w
    [1.4493, 0.31413, 0.00906]

    >>> x, w = gauss_gen_laguerre(4, 3*S.Half, 5)
    >>> x
    [0.97851, 2.9904, 6.3193, 11.712]
    >>> w
    [0.53087, 0.67721, 0.11895, 0.0023152]

    See Also
    ========

    gauss_legendre, gauss_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
    .. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html
    """
    x = Dummy("x")
    p = laguerre_poly(n, x, alpha=alpha, polys=True)
    p1 = laguerre_poly(n-1, x, alpha=alpha, polys=True)
    p2 = laguerre_poly(n-1, x, alpha=alpha+1, polys=True)
    xi = []
    w  = []
    for r in p.real_roots():
        if isinstance(r, RootOf):
            r = r.eval_rational(S(1)/10**(n_digits+2))
        xi.append(r.n(n_digits))
        w.append((gamma(alpha+n)/(n*gamma(n)*p1.subs(x, r)*p2.subs(x, r))).n(n_digits))
    return xi, w
def gauss_gen_laguerre ( n , alpha , n_digits ) : x = Dummy ( "x" ) p = laguerre_poly ( n , x , alpha = alpha , polys = True ) p1 = laguerre_poly ( n - 1 , x , alpha = alpha , polys = True ) p2 = laguerre_poly ( n - 1 , x , alpha = alpha + 1 , polys = True ) xi = [ ] w = [ ] for r in p . real_roots ( ) : if isinstance ( r , RootOf ) : r = r . eval_rational ( S ( 1 ) / 10 ** ( n_digits + 2 ) ) xi . append ( r . n ( n_digits ) ) w . append ( ( gamma ( alpha + n ) / ( n * gamma ( n ) * p1 . subs ( x , r ) * p2 . subs ( x , r ) ) ) . n ( n_digits ) ) return xi , w