Gauss–Laguerre quadrature

In numerical analysis Gauss–Laguerre quadrature is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind:

\int _{0}^{+\infty }e^{-x}f(x)\,dx.

In this case

\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]

w_{i}={\frac {x_{i}}{(n+1)^{2}[L_{n+1}(x_{i})]^{2}}}.

For more general functions

To integrate the function $f$ we apply the following transformation

\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 $g\left(x\right):=e^{x}f\left(x\right)$ . 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 $x^{\alpha }$ power-law singularity at x=0, for some real number $\alpha >-1$ , leading to integrals of the form: