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:

In this case

where xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by [1]

For more general functions

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.

Generalized Gauss–Laguerre quadrature

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:

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 ". 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 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.
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