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[1]
The following Python code with the SymPy library will allow for calculation of the values of
to 20 digits of precision:To integrate the function
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.
power-law singularity at x=0, for some real number
, leading to integrals of the form: In this case, the weights are given[2] in terms of the generalized Laguerre polynomials: where
This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.