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{w }, respectively, and using the notation and results of Section 2.5, we have
2
for the identity operation the following weights:
w(0) = 1 (4.9a)

w(i) = 0 for i = 1, 2, 3, … (4.9b)

The Trapezoid numerical integrator, as given in Pb. 4.25, is a first-order sys-
tem with the following parameters:

∆ x
b 1 () = (4.10a)
0
2
∆ x
b 1 () = (4.10b)
1
2
a 1 () =− 1 (4.10c)
1
giving for its weight sequence, as per Example 2.4, the values:

∆ x
w 0() = (4.11a)
1
2
wi() = ∆ x for i = 12 3, ,… (4.11b)
,
1
The improved numerical differentiator’s weight sequence can now be
directly obtained by noting, as noted above, that if we successively cascade
integration with differentiation, we are back to the original function. Using
the results of Pb. 2.18, we can write:

k
wk() = ∑ w i w k i( ) 1 ( − ) (4.12)
2
i=0
Combining the above values for w(k) and w (k), we can deduce the following
1
equalities:

∆ x
w()0 == w ()0 (4.13a)
1
2 2
w()1 = 0 = ∆ x    1 w () 1 + w ( ) 0    (4.13b)
2
2
2
© 2001 by CRC Press LLC