
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_laguerre(n, n_digits):
    r"""
    Computes the Gauss-Laguerre quadrature [1]_ points and weights.

    The Gauss-Laguerre quadrature approximates the integral:

    .. math::
        \int_0^{\infty} 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_n`
    and the weights `w_i` are given by:

    .. math::
        w_i = \frac{x_i}{(n+1)^2 \left(L_{n+1}(x_i)\right)^2}

    Parameters
    ==========

    n : the order of quadrature

    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.integrals.quadrature import gauss_laguerre
    >>> x, w = gauss_laguerre(3, 5)
    >>> x
    [0.41577, 2.2943, 6.2899]
    >>> w
    [0.71109, 0.27852, 0.010389]
    >>> x, w = gauss_laguerre(6, 5)
    >>> x
    [0.22285, 1.1889, 2.9927, 5.7751, 9.8375, 15.983]
    >>> w
    [0.45896, 0.417, 0.11337, 0.010399, 0.00026102, 8.9855e-7]

    See Also
    ========

    gauss_legendre, gauss_gen_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/laguerre_rule/laguerre_rule.html
    """
    x = Dummy("x")
    p = laguerre_poly(n, x, polys=True)
    p1 = laguerre_poly(n+1, x, 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((r/((n+1)**2 * p1.subs(x, r)**2)).n(n_digits))
    return xi, w


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

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