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Gauss-Hermite Integration Methods
¥
n
Here we consider the integral of the form 
 òe-x^2 f(x) dx = 
S
 lkfk
-¥
k = 0
The nodes xk's are found to be the roots of the Hermite polynomial
Hn+1(x) = (-1)n+1ex^2
 dn+1 (e-x^2)
dxn+1
We have
H0(x) = 1,
H1(x) = 2x
H2(x) = 2(2x2 - 1),
H3(x) = 4(2x3 - 3x)
The corresponding weights are obtained from the relation
lk =  ò ¥
e-x^2 Hn+1(x)
 dx
_¥
(x - xk) H'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 Hermite polynomials Hn(x) are orthogonal with respect to the weight function e-x^2 on (-¥,¥).
¥
òe-x^2Hm(x)Hn(x) dx = 0, m ¹ n
-¥

 
n
nodes xk
weights lk
 
0
0.0000000000
1.7724538509
 
1
± 0.7071067812
0.8862269255
 
2
0.0000000000
1.1816359006
± 1.2247448714
0.2954089752
 
3
± 0.5246476233
0.8049140900
± 1.6506801239
0.0813128354
 
4
0.0000000000
0.9453087205
± 0.9585724646
0.3936193232
± 2.0201828705
0.0199532421
 
 5
± 0.4360774119
0.7264295952
± 1.3358490740
0.1570673203
± 2.3506049737
0.0045300099
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