In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind:

${\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx.}$

In this case

${\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})}$

where x_i is the i-th root of Laguerre polynomial L_n(x) and the weight w_i is given by^[1]

${\displaystyle w_{i}={\frac {x_{i}}{\left(n+1\right)^{2}\left[L_{n+1}\left(x_{i}\right)\right]^{2}}}.}$

The following Python code with the SymPy library will allow for calculation of the values of ${\displaystyle x_{i}}$ and ${\displaystyle w_{i}}$ to 20 digits of precision:

from sympy import *

def lag_weights_roots(n):
    x = Symbol("x")
    roots = Poly(laguerre(n, x)).all_roots()
    x_i = [rt.evalf(20) for rt in roots]
    w_i = [(rt / ((n + 1) * laguerre(n + 1, rt)) ** 2).evalf(20) for rt in roots]
    return x_i, w_i

print(lag_weights_roots(5))

To integrate the function ${\displaystyle f}$ we apply the following transformation

${\displaystyle \int _{0}^{\infty }f(x)\,dx=\int _{0}^{\infty }f(x)e^{x}e^{-x}\,dx=\int _{0}^{\infty }g(x)e^{-x}\,dx}$

where ${\displaystyle g\left(x\right):=e^{x}f\left(x\right)}$. For the last integral one then uses Gauss-Laguerre quadrature. Note, that while this approach works from an analytical perspective, it is not always numerically stable.

More generally, one can also consider integrands that have a known ${\displaystyle x^{\alpha }}$ power-law singularity at x=0, for some real number ${\displaystyle \alpha >-1}$, leading to integrals of the form:

${\displaystyle \int _{0}^{+\infty }x^{\alpha }e^{-x}f(x)\,dx.}$

In this case, the weights are given^[2] in terms of the generalized Laguerre polynomials:

${\displaystyle w_{i}={\frac {\Gamma (n+\alpha +1)x_{i}}{n!(n+1)^{2}[L_{n+1}^{(\alpha )}(x_{i})]^{2}}}\,,}$

where ${\displaystyle x_{i}}$ are the roots of ${\displaystyle L_{n}^{(\alpha )}}$.

This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.^[3]