Page 279 - Numerical Analysis Using MATLAB and Excel

Chapter 7

Finite Differences and Interpolation

T his 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. These applications are discussed in this and the next three chapters.

7.1 Divided Differences

,
,
,
,
Consider the continuous function y = f x() and let x x x … x n – 1 , x n be some values of
0
2
1
x
x in the interval x ≤ 0 x ≤ x n . It is customary to show the independent variable , and its corre-
sponding values of y = f x() in tabular form as in Table 7.1.
TABLE 7.1 The variable x and y = f x() in tabular form
x fx()

x 0 fx ) ( 0

x 1 fx ) ( 1

x fx ) (
2 2
… …

(
x n – 1 fx n – 1 )
x n fx ) ( n

Let and be any two, not necessarily consecutive values of , within this interval. Then, the
x
x
x
i
j
first divided difference is defined as:
fx () – fx()
i
j
fx x,( i j ) = ----------------------------- (7.1)
x –
x
i
j
Likewise, the second divided difference is defined as:
fx x,( ) – fx x ) ( ,
k
i
j
j
fx x x,( i j , k ) = --------------------------------------------- (7.2)
x –
x
k
i
Numerical Analysis Using MATLAB® and Excel®, Third Edition 7−1
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