Page 317 - Numerical Analysis Using MATLAB and Excel

Summary

7.9 Summary

• The first divided difference is defined as:

fx () – fx()
i
j
fx x ) ( i , j = -----------------------------
x
x –
j
i
where and are any two, not necessarily consecutive values of , within an interval.
x
x
x
j
i
• Likewise, the second divided difference is defined as:
fx x,( ) – fx x ) ( ,
j
k
i
j
fx x x,( i j , k ) = ---------------------------------------------
x
x –
i
k
and the third, fourth, and so on divided differences are defined similarly.
• If the values of are equally spaced and the denominators are all the same, these values are
x
referred to as the differences of the function.
h
x
• If the constant difference between successive values of is , the typical value of x k is
,,,
,
,
–
x = x + kh for k = … 2 – 1012 …
,
k
0
• We can now express the first differences are usually expressed in terms of the difference oper-
ator as
Δ
Δf = f k + 1 – f k
k
• Likewise, the second differences are expressed as
2
Δ f = ΔΔf ) ( k Δ = f k + 1 – Δf k
k
and, in general, for positive integer values of n
n
(
Δ f = ΔΔ n – 1 f ) k = Δ n – 1 f k + 1 – Δ n – 1 f k
k
Δ
• The difference operator obeys the law of exponents which states that
Δ ( m Δ f ) n k Δ = m + n f k
n
• The nth differences Δ f k are found from the relation

(
n
n
–
)
--------------------f
Δ f = f k + n – nf k + n – 1 + nn – 1 ) k + n – 2 + … + – ( 1 ) n 1 nf k + 1 – ( + 1 f k
k
2!
,,
For k = 0 , n = 1 2 3 and , the above relation reduces to
4
Numerical Analysis Using MATLAB® and Excel®, Third Edition 7−39
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