Divided Difference

GOAL

To understand what “divided difference” is.

POINTS

The divided difference is the difference between the values ​of functions divided by the difference of the arguments. The higher-order divided difference is the divided difference of the divided difference.

DEFINITION

0th divided difference

$$f[x_0] = f(x_0)$$

1st divided difference

$$f[x_0, x_1] = \frac{ f(x_0)-f(x_1)}{x_0-x_1}$$

2nd divided difference

$$f[x_0, x_1, x_2] = \frac{ f[x_0, x_1] – f[x_1, x_2] }{x_0-x_2}$$

Generalized nth divided difference

$$f[x_0, … x_n] = \frac{ f[x_0, … x_{n-1}] – f[x_1, … x_n] }{x_0 – x_n}$$

PROPERTIES

Linearity

$$(f+g)[x_0, … x_n] = f[x_0, … x_n] + g[x_0, … x_n]$$

Symmetry

σ is one of the permutations.

$$σ = \begin{pmatrix} 0 & 1 & … & n \\ σ(0) & σ(1) & … &σ(n)\end{pmatrix}$$

$$f[x_0, x_1, …, x_n] = f[σ(0), σ(1), … ,σ(n)]$$

Relationship with the derivative

$$f'(x) = \frac{df}{dx} = \lim_{h \to 0}f[x + h, x]$$