Numerical Analysis Using MATLAB and Spreadsheets
This concise text provides complete, clear, and detailed explanations of the principal numerical analysis methods and well known functions used in science and engineering.
TABLE OF CONTENTS Numerical Analysis Using MATLAB and Spreadsheets
Chapter 7: Finite Differences and Interpolation
This chapter begins with finite differences and interpolation which is one of its most impor- tant 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. We will discuss these applications in this and the next three chapters.
7.1 Divided Differences
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 first divided 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...
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