
import numpy as np
import sympy as sp
from sympy.core import S, Dummy
from sympy.functions.special.gamma_functions import gamma
from sympy.polys.orthopolys import (legendre_poly, laguerre_poly,
                                    hermite_poly, jacobi_poly)
from sympy.polys.rootoftools import RootOf



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, gauss_lobatto

    References
    ==========

    .. [1] https://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



alpha = 0
digits = 36
npoints = list(range(1,17))

for n in npoints:
    xi, wi = gauss_gen_laguerre(n, alpha, digits)
    
    print('\nN = %3d'%(n))
    for i in range(0,n):
        print('%3d %40.36f %40.36f'%(i+1,xi[i],wi[i]))
    
    