Wave Propagation in Linear Media | re-examined

5.2 Convergence acceleration
one, these routines cannot be applied directly. It may be possible to extract an oscillatory
factor with constant period in order to obtain, say, sin( ~ f(x)) cos !x such that integration
can be carried out over individual `wave packets'. It is obvious that this requires the use
of a routine with an adaptive subdivision strategy like those from the Quadpack package.
Depending on the parameters, however, the number of zeros between the zeros of the periodic
envelope may be very high, which in turn increases the number of subdivisions needed and
the execution time of the program. For comparison, the example of wave propagation in the
transmission line lled with plasma (section 3.7) was computed with the DQAWF subroutine.
Although it worked very well (and fast) in general, it failed to produce correct results for
special cases like the boundary.
As pointed out before, Sidi investigated irregularly oscillating functions [128], which would
suit our purposes exactly. Unfortunately, there is no implementation of his work available in
a standard subroutine library. Thus the conclusions of this section are:
For the computation of in nitely oscillating integrals, a combination of subdividing the
integration range and extrapolating the sequence of partial sums is the method of choice.
Although many authors have addressed this topic, there is no ready-to-use software
package available for the case of irregular oscillations.
5.2 Convergence acceleration
One of the basic ideas in computing the value of an in nite integral is to de ne a sequence that
converges to this value. What remains is the problem of nding this limit numerically with
su cient accuracy. This section will introduce both the concept of series acceleration and
the notation used throughout subsequent chapters. Furthermore, we shall present two widely
used algorithms that are relevant for the implementation of our own quadrature strategy in
Mathematica.
Suppose that a sequence fSng is known to converge and only a nite set of its members is
available, then any member of it (usually the last one) may be taken as a rough estimate for
the limit S. If, however, the convergence is very slow, the computational e ort to determine
enough sequence members to achieve a reasonably accurate result may be too high. In such
cases, alternatives are sought to accelerate the convergence of the sequence. A very simple
possibility is to form the average of two consecutive sequence members
Tn = Sn + Sn+1
2
; (5.6)
which is particularly suitable for alternating sequences (see, for example, Lyness [121] or Boyd
[136] for an extension of this scheme). Such a sequence transformation T : fSng 7!fTngmust
have two properties in order to be of practical use [137, 138]:
It must be regular, that is, fTng must converge to the same limit as fSng.
fTng must converge faster than fSng, which is expressed by limn!1 Tn,S
Sn,S =0.
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