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Laguerre-Gauss Quadrature

Laguerre-Gauss quadrature, also called Gauss-Laguerre quadrature or Laguerre quadrature, is a Gaussian quadrature over the interval [0,infty) with weighting function W(x)=e^(-x) (Abramowitz and Stegun 1972, p. 890). It fits all polynomials of degree 2m-1 exactly (Chandrasekhar 1960, p. 61).

The abscissas for quadrature order n are given by the roots of the Laguerre polynomials L_n(x). The weights are

w_i=-(A_(n+1)gamma_n)/(A_nL_n^'(x_i)L_(n+1)(x_i))
(1)
=(A_n)/(A_(n-1))(gamma_(n-1))/(L_(n-1)(x_i)L_n^'(x_i)),
(2)

where A_n is the coefficient of x^n in L_n(x). For Laguerre polynomials,

A_n=((-1)^n)/(n!),
(3)

where n! is a factorial, so

(A_(n+1))/(A_n)=-1/(n+1)
(4)
(A_n)/(A_(n-1))=-1/n.
(5)

Additionally,

gamma_n=int_0^inftyW(x)[L_n(x)]^2dx=1,
(6)

so

w_i=1/((n+1)L_n^'(x_i)L_(n+1)(x_i))
(7)
=-1/(nL_(n-1)(x_i)L_n^'(x_i)).
(8)

Using the recurrence relation

xL_n^'(x)=nL_n(x)-nL_(n-1)(x)
(9)
=(x-n-1)L_n(x)+(n+1)L_(n+1)(x)
(10)

which, since x_i is a root of L_n(x), gives

nL_n(x)=(x-n-1)L_n(x)=0,
(11)

so (10) becomes

x_iL_n^'(x_i)=-nL_(n-1)(x_i)=(n+1)L_(n+1)(x_i)
(12)

gives

w_i=1/(x_i[L_n^'(x_i)]^2)
(13)
=(x_i)/((n+1)^2[L_(n+1)(x_i)]^2).
(14)

The error term is

E=((n!)^2)/((2n)!)f^((2n))(xi)
(15)

(Abramowitz and Stegun 1972, p. 890).

Beyer (1987) gives a table of abscissas and weights up to n=6.

nx_iw_i
20.5857860.853553
3.414210.146447
30.4157750.711093
2.294280.278518
6.289950.0103893
40.3225480.603154
1.745760.357419
4.536620.0388879
9.395070.000539295
50.263560.521756
1.41340.398667
3.596430.0759424
7.085810.00361176
12.64080.00002337

The abscissas and weights can be computed analytically for small n.

nx_iw_i
22-sqrt(2)1/4(2+sqrt(2))
2+sqrt(2)1/4(2-sqrt(2))

For the generalized Laguerre polynomial L_n^beta(x) with weighting function w(x)=x^betae^(-x),

A_n=((-1)^n)/(n!)
(16)

is the coefficient of x^n in L_n^beta(x) and

gamma_n=int_0^inftyx^betae^(-x)[L_n^beta(x)]^2dx
(17)
=(Gamma(n+beta+1))/(n!),
(18)

where Gamma(z) is the gamma function. The weights are then

w_i=(Gamma(n+beta)x_i)/(n!(n+beta)[L_(n-1)^beta(x_i)]^2)
(19)
=(Gamma(n+beta+1)x_i)/(n!(n+1)^2[L_(n+1)^beta(x_i)]^2),
(20)

and the error term is

E_n=(n!Gamma(n+beta+1))/((2n)!)f^((2n))(xi).
(21)

See also

Gaussian Quadrature

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 890 and 923, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 463, 1987.Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 61 and 64-65, 1960.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 325-327, 1956.

Referenced on Wolfram|Alpha

Laguerre-Gauss Quadrature

Cite this as:

Weisstein, Eric W. "Laguerre-Gauss Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Laguerre-GaussQuadrature.html

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