Page 251 - Applied Numerical Methods Using MATLAB

240 NUMERICAL DIFFERENTIATION/ INTEGRATION
Given the N grid point t i ’s, we can get the corresponding weight w N,i ’s of the
N-point Gauss–Laguerre integration formula by solving the system of linear
equations like Eq. (5.9.7), but with the right-hand side (RHS) vector as

∞ ∞
RHS(1) = e −t −t = 1 (5.9.19a)
dt =−e
0
0
∞ ∞ ∞

−t n−1 −t n−1 −t n−2
RHS(n) = e t dt =−e t + (n − 1) e t dt
0 0 0
= (n − 1)RHS(n − 1) (5.9.19b)
5.9.4 Gauss–Chebyshev Integration
The Gauss–Chebyshev I integration formula is also expressed by Eq. (5.9.5) as

N

I GC1 [t 1 ,t 2 ,...,t N ] = w N,i f(t i ) (5.9.20)
i=1
√
2
and is supposed to give us the exact integral of 1/ 1 − t multiplied by a
polynomial f(t) of degree ≤ (2N − 1) over [−1, +1]
+1
1
I = √ f(t) dt (5.9.21)
−1 1 − t 2
The N grid point t i ’s are the zeros of the Nth-degree Chebyshev polynomial
(Section 3.3)
(2i − 1)π
t i = cos for i = 1, 2,... ,N (5.9.22)
2N

and the corresponding weight w N,i ’s are uniformly selected as

w N,i = π/N, ∀ i = 1,...,N (5.9.23)
The Gauss–Chebyshev II integration formula is also expressed by Eq. (5.9.5) as

N

I GC2 [t 1 ,t 2 ,...,t N ] = w N,i f(t i ) (5.9.24)
i=1
√
2
and is supposed to give us the exact integral of 1 − t multiplied by a polyno-
mial f(t) of degree ≤ (2N − 1) over [−1, +1]
+1
2
I = 1 − t f(t) dt (5.9.25)
−1