Page 318 - Numerical Analysis Using MATLAB and Excel

Chapter 7 Finite Differences and Interpolation

Δf = f – f 1
0
2
2
Δ f = f – 2f + f 0
2
0
1
3
Δ f = f – 3f + 3f – f 0
3
2
0
1
4
Δ f = f – 4f + 6f – 4f + f 0
2
1
3
0
4
• As with derivatives, the nth differences of a polynomial of degree are constant.
n
• The factorial polynomials are defined as
x () n () xx 1 x2 … – ) ( x – n + 1 )
(
)
–
( =
and
1
x () – n () = ----------------------------------------------------------
( x – 1 x – 2 … ) ( x + n )
)
(
Using the difference operator with the above relations we obtain
Δ
–
Δ x() n () = nx() ( n 1 )
and
Δ x() – n () = – n x() ( n – 1 ) –
n – n
These are very similar to differentiation of x and x .
• We can express any algebraic polynomial f x() as a factorial polynomial p x() . Then, in anal-
n
n
ogy with Maclaurin power series, we can express that polynomial as
p x() = a + a x() 1 () + a x() 2 () + … + a x() n ()
0
2
n
1
n
where
j
Δ p 0()
n
,,,
,
a = ------------------- for j = 0 1 2 … n
j
j!
• Factorial polynomials provide an easier method of constructing a difference table. The proce-
dure is as follows:
1. We divide p x() by to obtain a quotient q x() and a remainder which turns out to be
r
x
0
n
0
the constant term a 0 . Then, the factorial polynomial reduces to
p x() = r + xq x()
n
0
0
7−40 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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