Gauss–Laguerre quadrature

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}^{%2B\infty} e^{-x} f(x)\,dx.

In this case

\int_{0}^{%2B\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%2B1)^2[L_{n%2B1}(x_i)]^2}.

Contents

For more general functions

To integrate the function f

\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.

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}^{%2B\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]

Program to Calculate Modified Gauss-Laguerre Abscissa And Weights

In Mathematica 7

Suppose want to calculate for n=5

  n=5;
  t = Array[N[Root[LaguerreL[n, x], #]] &, n]
  w = t/((n + 1)*LaguerreL[(n + 1), t])^2
  modW = Array[(Exp[t#]*t#)/(((n + 1)*LaguerreL[(n + 1), t#])^2) &, n]

t is an Abscissa array. w is the corresponding array of weights and modW is the corresponding array of modified weights.

References

  1. ^ Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions, 10th printing with corrections (1972), Dover, ISBN 978-0-486-61272-0. Equation 25.4.45.access online
  2. ^ Philip Rabinowitz and George Weiss, "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, vol. 13, pp. 285–294 (1959).

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