Gauss–Laguerre quadrature

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In numerical analysis Gauss–Laguerre 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 xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by [1]

For more general functions[edit]

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[edit]

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]


  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[edit]

External links[edit]