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In numerical analysis Gauss–Laguerre quadrature is an extension of 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 xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi 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

$\int_{0}^{\infty}f\left(x\right)dx=\int_{0}^{\infty}f\left(x\right)e^{x}e^{-x}dx=\int_{0}^{\infty}g\left(x\right)e^{-x}dx$

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:

$\int_{0}^{+\infty} x^\alpha e^{-x} f(x)\,dx.$

This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.[2]

## References

1. ^ Equation 25.4.45 in 10th reprint with corrections.
2. ^ Rabinowitz, P.; Weiss, G. (1959). "Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form $\int_0^{\infty} \exp(-x) x^n f(x) dx$". Mathematical Tables and Other Aids to Computation 13: 285–294. doi:10.1090/S0025-5718-1959-0107992-3.