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Chapter 7 Finite Differences and Interpolation

and, in general, for positive integer values of n

n
(
Δ f = ΔΔ n – 1 f ) k = Δ n – 1 f k + 1 – Δ n – 1 f k (7.6)
k
Δ
The difference operator obeys the law of exponents, that is,
Δ ( m Δ f ) n k Δ = m + n f k (7.7)

We construct the difference table in terms of the difference operator as shown in Table 7.4.
Δ

TABLE 7.4 Divided differences table in terms of the difference operator Δ

Function Differences

x f First Second Third Fourth …
x 0 f 0

Δf 0

x 1 f 1 Δ f 0
2

3
Δf 1 Δ f 0
2
4
x 2 f 2 Δ f 1 Δ f 0
Δf 2 Δ f 1
3
x 3 f 3 Δ f 2
2

Δf 3

x 4 f 4

…

x n f n

Example 7.2

Construct a difference table showing the values of x given as 1 234567,,,,, , and , the values of
8
fx() corresponding to y = f x() = x 3 , and the first through the fourth differences.

7−4 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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