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:

In this case

where xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by[1]

The following Python code with the SymPy library will allow for calculation of the values of and 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 = [rt.evalf(20) for rt in roots]
    w_i = [(rt / ((n + 1) * laguerre(n + 1, rt)) ** 2).evalf(20) for rt in roots]
    return x_i, w_i

print(lag_weights_roots(5))

For more general functions

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To integrate the function   we apply the following transformation

 

where  . 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

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More generally, one can also consider integrands that have a known   power-law singularity at x=0, for some real number  , leading to integrals of the form:

 

In this case, the weights are given[2] in terms of the generalized Laguerre polynomials:

 

where   are the roots of  .

This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.[3]

References

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  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. ^ 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  ". Mathematical Tables and Other Aids to Computation. 13: 285–294. doi:10.1090/S0025-5718-1959-0107992-3.

Further reading

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