Page 284 - Numerical Analysis Using MATLAB and Excel

Chapter 7 Finite Differences and Interpolation

It is interesting to observe that the first difference in (7.10), is the difference quotient whose limit
defines the derivative of a continuous function that is defined as
fx )
)
Δx
(–
1
1
lim Δy lim fx +( -------------------------------------------- (7.11)
------- =
Δx → 0 Δx Δx → 0 Δx
n
As with derivatives, the nth differences of a polynomial of degree are constant.
7.2 Factorial Polynomials
The factorial polynomials are defined as

(
x () n () xx 1 x2 … – ) ( x – n + 1 ) (7.12)
)
–
( =
and
1
x () – n () = ---------------------------------------------------------- (7.13)
( x – 1 x – 2 … ) ( x + n )
)
(
These expressions resemble the power functions x n and x – n in elementary algebra.
Using the difference operator with (7.12) and (7.13) we obtain
Δ
Δ x() n () = nx() ( n – 1 ) (7.14)
and

Δ x() – n () = – n x() ( n – 1 ) – (7.15)

n – n
We observe that (7.14) and (7.15) are very similar to differentiation of x and x .
Occasionally, it is desirable to express a polynomial p x() as a factorial polynomial. Then, in anal-
n
ogy with Maclaurin power series, we can express that polynomial as

p x() = a + a x() 1 () + a x() 2 () + … + a x() n () (7.16)
n
1
2
0
n
and now our task is to compute the coefficientsa k .
For x = 0 , relation (7.16) reduces to
a = p 0() (7.17)
n
0
To compute the coefficient a 1 , we take the first difference of p x() in (7.16). Using (7.14) we
n
obtain

0
–
Δp x() = 1x a + 2a x() 1 () + 3a x() 2 () + … + na x() ( n 1 ) (7.18)
n
2
1
n
3
and letting x = 0 , we find that
7−6 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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