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w()2 = 0 = ∆ x    2 1 w () 2 + w ( ) 1 + w ( ) 0    (4.13c)
2
2
2
etc. …
from which we can directly deduce the following expressions for the weight-
ing sequence {w }:
2

w 0() = 2 (4.14a)
2 ∆ x

wi () = 4 (− 1) i for i = 1 2 3, ,… (4.14b)
,
2 ∆ x
From these weights we can compute, as per the results of Example 2.4, the
parameters of the difference equation for the improved numerical differenti-
ator, namely:

b 2 () = 2 (4.15a)
0 ∆ x

b 2 () =− 2 (4.15b)
1 ∆ x

a 2 () = 1 (4.15c)
1
giving for D(k) the following defining difference equation:

Dk() = 2 [ yk( ) − yk( − 1 )] − D k( − 1 ) (4.16)
t ∆

In Pb. 4.32 and in other cases, you can verify that indeed this is an
improved numerical differentiator. We shall, later in the chapter, use the
above expression for D(k) in the numerical solution of ordinary differential
equations.

In-Class Exercises

Pb. 4.31 Find the inverse system corresponding to the discrete system gov-
erned by the difference equation:

y k() = uk() − 1 uk( − 1 ) + 1 y k( − 1 )
2 3

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