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 xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi 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))

## For more general functions

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 $\displaystyle{ \alpha \gt -1 }$, 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]

## References

1. Equation 25.4.45 in Handbook of Mathematical Functions. Dover. ISBN 978-0-486-61272-0.  10th reprint with corrections.
2. Weisstein, Eric W., "Laguerre-Gauss Quadrature" From MathWorld--A Wolfram Web Resource, Accessed March 9, 2020
3. "Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form $\displaystyle{ \int_0^{\infty} \exp(-x) x^n f(x)\,dx }$". Mathematical Tables and Other Aids to Computation 13: 285–294. 1959. doi:10.1090/S0025-5718-1959-0107992-3.