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Gauss-Laguerre Integration Methods
¥
n
Here we consider the integral of the form 
 òe-xf(x) dx = 
S
 lkfk
0
k = 0
The nodes xk's are found to be the roots of the Laguerre polynomial
Ln+1(x) = (-1)n+1ex
 dn+1 (e-x xn+1)
dxn+1
We have
L0(x) = 1,
L1(x) = (x-1)
L2(x) = x2 - 4x + 2,
L3(x) = x3 - 9x2 +18x - 6
The corresponding weights are obtained from the relation
lk =  ò ¥
e-x Ln+1(x)
 dx
0
(x - xk) L'n+1(xk)
The method produces exact results for polynomials of degree upto 2n+1. The nodes and weights for the method for n are given. The Laguerre polynomials Ln(x) are orthogonal with respect to the weight function e-x on (0,¥).
¥
òe-xLm(x)Ln(x) dx = 0, m ¹ n
0

 
n
nodes xk
weights lk
 
1
 0.5857864376
0.8535533906
 
3.4142135624
0.1464466094
 
2
0.4157745568
0.7110930099
2.2942803603
0.2785177336
6.2899450829
0.0103892565
 
3
0.3225476896
0.6031541043
1.7457611012
0.3574186924
4.5366202969
0.0388879085
9.3950709123
0.0005392947
 
4
0.2635603197
0.5217556106
1.4134030591
0.3986668111
3.5964257710
0.0759424497
7.0858100059
0.0036117587
12.6408008443
0.0000233700
 
 5
0.2228466042
0.4589646740
1.1889321017
0.4170008308
2.9927363261
0.1133733821
 
 5.7751435691
0.0103991975
9.8374674184
0.0002610172
15.9828739806
0.0000008985
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations