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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:

∫ 0 + ∞ e − x f ( x ) d x . {\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx.}

In this case

∫ 0 + ∞ e − x f ( x ) d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \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

w i = x i ( n + 1 ) 2 [ L n + 1 ( x i ) ] 2 . {\displaystyle w_{i}={\frac {x_{i}}{\left(n+1\right)^{2}\left^{2}}}.}

The following Python code with the SymPy library will allow for calculation of the values of x i {\displaystyle x_{i}} and w i {\displaystyle w_{i}} to 20 digits of precision:

from sympy import *
def lag_weights_roots(n):
    x = Symbol("x")
    roots = Poly(laguerre(n, x)).all_roots()
    x_i = 
    w_i = 
    return x_i, w_i
print(lag_weights_roots(5))

For more general functions

To integrate the function f {\displaystyle 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 {\displaystyle \int _{0}^{\infty }f(x)\,dx=\int _{0}^{\infty }f(x)e^{x}e^{-x}\,dx=\int _{0}^{\infty }g(x)e^{-x}\,dx}

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

∫ 0 + ∞ x α e − x f ( x ) d x . {\displaystyle \int _{0}^{+\infty }x^{\alpha }e^{-x}f(x)\,dx.}

In this case, the weights are given in terms of the generalized Laguerre polynomials:

w i = Γ ( n + α + 1 ) x i n ! ( n + 1 ) 2 [ L n + 1 ( α ) ( x i ) ] 2 , {\displaystyle w_{i}={\frac {\Gamma (n+\alpha +1)x_{i}}{n!(n+1)^{2}^{2}}}\,,}

where x i {\displaystyle x_{i}} are the roots of L n ( α ) {\displaystyle L_{n}^{(\alpha )}} .

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

References

  1. Equation 25.4.45 in Abramowitz, M.; Stegun, I. A. (1964). Handbook of Mathematical Functions. Dover. ISBN 978-0-486-61272-0. 10th reprint with corrections.
  2. Weisstein, Eric W., "Laguerre-Gauss Quadrature" From MathWorld--A Wolfram Web Resource, Accessed March 9, 2020
  3. Rabinowitz, P.; Weiss, G. (1959). "Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form ∫ 0 ∞ exp ⁡ ( − x ) x n f ( x ) d x {\displaystyle \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

External links

Numerical integration
Newton–Cotes formulas
Gaussian quadrature
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