Harman Patil (Editor)

Gauss–Laguerre quadrature

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In numerical analysis Gauss–Laguerre quadrature is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind:

Contents

∫ 0 + ∞ e − x f ( x ) d x .

In this case

∫ 0 + ∞ e − x f ( x ) d x ≈ ∑ 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

w i = 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

∫ 0 ∞ f ( x ) d x = ∫ 0 ∞ f ( x ) e x e − x d x = ∫ 0 ∞ g ( x ) e − x d x

where g ( x ) := e x f ( x ) . 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 α power-law singularity at x=0, for some real number α > − 1 , leading to integrals of the form:

∫ 0 + ∞ x α e − x f ( x ) d x .

This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.

References

Gauss–Laguerre quadrature Wikipedia
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