Gauss–Laguerre quadrature explained
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]
wi=
| xi |
| \left(n | +1\right)2\left[Ln+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
To integrate the function
we apply the following transformation
where
g\left(x\right):=exf\left(x\right)
. For the last integralone then uses Gauss-Laguerre quadrature. Note, that while this approach worksfrom an analytical perspective, it is not always numerically stable.
Generalized Gauss–Laguerre quadrature
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]
Further reading
- H. E. . Salzer . R. . Zucker . 1949 . Table of zeros and weight factors of the first fifteen Laguerre polynomials . . 55 . 10 . 1004–1012 . 10.1090/S0002-9904-1949-09327-8. free .
- P. . Concus . D. . Cassatt . G. . Jaehnig . E. . Melby . 1963 . Tables for the evaluation of
by Gauss-Laguerre quadrature . . 17 . 245–256 . 10.1090/S0025-5718-1963-0158534-9. free .
- T. S. . Shao . T. C. . Chen . R. M. . Frank . 1964 . Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials . . 18 . 88 . 598–616 . 10.1090/S0025-5718-1964-0166397-1 . 0166397 . 2002946. free .
- S. . Ehrich . 2002 . On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas . . 140 . 1–2 . 291–299 . 10.1016/S0377-0427(01)00407-1.
External links
Notes and References
- Equation 25.4.45 in Book: M. . Abramowitz . Milton Abramowitz . I. A. . Stegun . Irene Stegun . Handbook of Mathematical Functions . . 978-0-486-61272-0 . Abramowitz and Stegun. 10th reprint with corrections.
- Weisstein, Eric W., "Laguerre-Gauss Quadrature" From MathWorld--A Wolfram Web Resource, Accessed March 9, 2020
- P. . Rabinowitz . Philip Rabinowitz (mathematician) . G. . Weiss . Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form
. . 13 . 285–294 . 1959 . 10.1090/S0025-5718-1959-0107992-3. free.
