Chapter 4 Numerical Differentiation And Integration

I = 4I n − I n/2
− 12d 4
+ ...
3 n 4
Define Ĩ = (4I n − I n/2 )/3 then the error convergence order increasing from O(1/n 2 ) to
O(1/n 4 ).
• Romberg integration
The Richardson’s extrapolation process can be continued inductively. Define
I (k)
n
with n a multiple of 2 k , k ≥ 1.
= 4k I n
(k−1)
I (0)
1
4 k − 1
I (0)
2 I (1)
2
− I (k−1)
n/2
I (0)
4 I (1)
4 I (2)
4
I (0)
8 I (1)
8 I (2)
8 I (3)
8
... ... ... ...
4.7 Numerical Differentiation
From the Newton’s interpolation polynomial,
, n ≥ 2 k
f(x) − p n (x) = Φ n (x)f[x 0 ,x 1 ,...,x n ,x], Φ n (x) = (x − x 0 )...(x − x n )
f ′ (x) − p ′ n(x) = Φ ′ n(x)f[x 0 ,x 1 ,...,x n ,x] + Φ n (x)f[x 0 ,x 1 ,...,x n ,x,x]
= Φ ′ n(x) f(n+1) (ξ 1 )
(n + 1)!
Thus let x j = x 0 + jh, j = 0, 1,...,n. Then
+ Φ n (x) f(n+2) (ξ 2 )
(n + 2)!
Φ n (x) = O(h n+1 ), Φ ′ n(x) = O(h n )
f ′ (x) − p ′ n(x) =
{
O(h n ) Φ ′ n(x) ≠ 0
O(h n+1 ) Φ ′ n(x) = 0
10