TABLE OF CONTENTS This chapter begins with finite differences and interpolation which is one of its most important applications. Finite Differences form the basis of numerical analysis as applied to other numerical methods such as curve fitting, data smoothing, numerical differentiation, and numerical integration. These applications are discussed in this and the next three chapters.
Consider the continuous function y = f(x) and let x 0, x 1, x 2, , x n?1, x n be some values of x in the interval x 0 ? x ? x n. It is customary to show the independent variable x, and its corresponding values of y = f(x) in tabular form as in Table 7.1.
| x | f(x) |
|---|---|
| x 0 | f(x 0) |
| x 1 | f(x 1) |
| x 2 | f(x 2) |
|
|
|
| x n?1 | f(x n?1) |
| x n | f(x n) |
Let x i and x j be any two, not necessarily consecutive values of x, within this interval. Then, the firstdivided difference is defined as:
Likewise, the second divided difference is defined as:
The third, fourth, and so on divided differences, are defined similarly.
The divided differences are indicated in a difference table where each difference is placed between the values of the column immediately to the left of it as shown...
