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

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

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 Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions. Dover. ISBN 978-0-486-61272-0. 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.

Further reading

• Salzer, H. E.; Zucker, R. (1949). "Table of zeros and weight factors of the first fifteen Laguerre polynomials". Bulletin of the American Mathematical Society 55: 1004–1012. doi:10.1090/S0002-9904-1949-09327-8.
• Concus, P.; Cassatt, D.; Jaehnig, G.; Melby, E. (1963). "Tables for the evaluation of $\int_0^\infty x^\beta \exp(-x) f(x) dx$ by Gauss-Laguerre quadrature". Mathematics of Computation 17: 245–256. doi:10.1090/S0025-5718-1963-0158534-9.
• Shao, T. S.; Chen, T. C.; Frank, R. M. (1964). "Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials". Mathematics of Computation 18 (88): 598–616. doi:10.1090/S0025-5718-1964-0166397-1. JSTOR 2002946. MR 0166397.
• Ehrich, S. (2002). "On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas". Journal of Computational and Applied Mathematics 140 (1-2): 291–299. doi:10.1016/S0377-0427(01)00407-1.

External links

• Matlab routine for Gauss–Laguerre quadrature
• Generalized Gauss–Laguerre quadrature, free software in Matlab, C++, and Fortran.