In numerical analysis GaussâÂÂLaguerre 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 x<sub>i</sub> is the i-th root of Laguerre polynomial L<sub>n</sub>(x) and the weight w<sub>i</sub> 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:
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.
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 ñ is not an integer.