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

↑ 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.
↑ 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.

External links

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

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Gauss–Laguerre quadrature

For more general functions

Generalized Gauss–Laguerre quadrature

References

Further reading

External links