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 x_i and w_i to 20 digits of precision:

For more general functions

To integrate the function f 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 x^\alpha power-law singularity at x=0, for some real number \alpha > -1, leading to integrals of the form: In this case, the weights are given in terms of the generalized Laguerre polynomials: where x_i 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.