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Application 1
Using the Transfer Function formalism, we want to estimate the accuracy of
the three integrating schemes discussed in Chapter 4. We want to compare
the Transfer Function of each of those algorithms to that of the exact result,
jωt
obtained upon integrating exactly the function e .
e jtω
jωt
The exact result for integrating the function e is, of course, , thus giv-
jω
ing for the exact Transfer Function for integration the expression:

H = 1 (6.101)
exact jω

Before proceeding with the computation of the transfer function for the dif-
ferent numerical schemes, let us pause for a moment and consider what we
are actually doing when we numerically integrate a function. We go through
the following steps:
1. We discretize the time interval over which we integrate; that is, we
define the sampling time ∆t, such that the discrete points abscissa
are given by k(∆t), where k is an integer.
2. We write a difference equation for the integral relating its values
at the discrete points with its values and that of the integrand at
discrete points with equal or smaller indices.
3. We obtain the value of the integral by iterating the defining differ-
ence equation.
The test function used for the estimation of the integration methods accu-
racy is written at the discrete points as:

yk() = e jk (ω∆ t) (6.102)

The difference equations associated with each of the numerical integration
schemes are:

t ∆
Ik( + 1 ) = Ik( ) + ( yk( + 1 ) + yk( )) (6.103)
T T
2
I k ( + 1 ) = I k ( ) + ∆ ty k( + 1 / )2 (6.104)
MP MP
t ∆
Ik( + 1 ) = Ik( − 1 ) + ( yk( + 1 ) + 4 yk( ) + yk( − 1 )) (6.105)
S S
3
leading to the following expressions for the respective Transfer Functions:

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