Page 250 - Applied Numerical Methods Using MATLAB

GAUSS QUADRATURE 239
Given the N grid point t i ’s, we can get the weight w N,i ’s of the N-point
Gauss–Hermite integration formula by solving the system of linear equations
like Eq. (5.9.7), but with the right-hand side (RHS) vector as

∞ 2 ∞ ∞
RHS(1) = e −t dt = e −x 2 dx e −y 2 dy
−∞ −∞ −∞

∞ ∞ ∞

2
2
= e −(x +y ) dx dy = e −r 2 2πr dr
−∞ −∞ −∞

∞ √

= −πe −r 2 = π (5.9.14a)
0

∞ 2 ∞ 2 1
RHS(n) = e −t t n−1 dt = (−2t)e −t t n−2 dt (= 0if n is even)
−2
−∞ −∞
1 −t 2 ∞ 1 ∞ −t 2 n−3 1
=− e t n−2 + (n − 2) e t dt = (n − 2)RHS(n − 2)
2 −∞ 2 2
−∞
(5.9.14b)
The procedure for finding the N grid point t i ’s and the corresponding weight
w N,i ’s of the N-point Gauss–Hermite integration formula is cast into the MAT-
LAB routine “Gausshp()”. Note that, even though the integrand function (g(t))
2
t
doesn’t have e −t 2 as a multiplying factor, we can multiply it by e −t 2 e = 1to
fabricate it as if it were like in Eq. (5.9.12):
∞ ∞ ∞

−t 2 t 2 −t 2
I = g(t) dt = e (e g(t)) dt = e f(t) dt (5.9.15)
−∞ −∞ −∞
5.9.3 Gauss–Laguerre Integration
The Gauss–Laguerre integration formula is also expressed by Eq. (5.9.5) as
N

I GLa [t 1 ,t 2 ,... ,t N ] = w N,i f(t i ) (5.9.16)
i=1
and is supposed to give us the exact integral of the exponential e −t multiplied
by a polynomial f(t) of degree ≤ (2N − 1) over [0, ∞)

∞

−t
I = e f(t) dt (5.9.17)
0
The N grid point t i ’s can be obtained as the zeros of the Nth-degree Laguerre
polynomial [K-1, Section 4.7]
N i i
(−1) N
i
L N (t) = t (5.9.18)
i! (N − i)! i!
i=0