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:

In this case 

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

The following Python code with the SymPy library will allow for calculation of the values of  and  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  we apply the following transformation

where . 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  power-law singularity at x=0, for some real number , leading to integrals of the form:

In this case, the weights are given in terms of the generalized Laguerre polynomials:

where  are the roots of .

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

References

Further reading

External links
 Matlab routine for Gauss–Laguerre quadrature
 Generalized Gauss–Laguerre quadrature, free software in Matlab, C++, and Fortran.

Numerical integration (quadrature)