Gauss–Laguerre quadrature

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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[edit]

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[edit]

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[edit]

  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[edit]

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