Gauss–Laguerre quadrature

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))

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.

Generalized Gauss–Laguerre quadrature

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]