356 Introduction to numerical methods
Suppose we apply this algorithm to the first-order equation
˙
y = λy (6.213)
where λ is a constant. Assuming solutions of the form
y
n
= Aρ
n
(6.214)
the characteristic equation associated with (6.212) is
ρ
2
−
1 +
3
2
λh
ρ +
1
2
λh = 0 (6.215)
Assuming that λh is real, the AB-2 algorithm is numerically stable for −1 ≤ λh ≤ 0.
As a specific example, let λh =−0.1. The corresponding characteristic equation is
ρ
2
− 0.85ρ − 0.05 = 0 (6.216)
and the roots are ρ
1,2
= 0.9052, −0.0552. If the numerical procedure were exact, then we
should have ρ = e
λh
= 0.9048. Thus, we see that the root −0.0552 is extraneous.
As another example, let λh =−1 which is on the boundary of the stable region. The
characteristic roots in this case are ρ = 0.5, −1. The root −1 is on the stability boundary
and is extraneous. Thus, it is the extraneous root which causes numerical instability for
more negative values of λh.
In general, the number of extraneous roots is determined by the number of time increments
in the past at which data are used in the algorithm. For example, the AB-3 algorithm involves
˙
y
n−2
. This results in two extraneous roots for each variable, assuming first-order differential
equations.
For multiple-pass algorithms such as the various predictor–corrector methods, the algo-
rithm as a whole determines the number of extraneous roots. For example, the AB-3, AM-4
combination involves
˙
y
n−2
and will introduce two extraneous roots per variable.
6.4 Frequency response methods
Transfer functions
Consider a linear time-invariant system which has a sinusoidal input. The steady-state
output will be sinusoidal with the same frequency as the input but, in general, with a
different amplitude and phase. If one uses complex notation, the amplitude and phase of
the output relative to the input are expressed by the transfer function
G(i ω) = Me
iφ
(6.217)
where M(ω) is the relative amplitude and φ(ω) is the relative phase.
The system under consideration may be originally linear or may be a linearized system
which is represented by a set of perturbation equations. The data resulting from a numerical
integration of the linear equations can be considered to be samples taken from a continuous