Wave Propagation in Linear Media | re-examined

6Towards a quadrature routine
The conclusion to be drawn from this example is that the rst partition point must be beyond
any extremum or even point of in exion of the argument function. The most rigorous way
to ensure this is to require that the argument function and its derivatives be monotonic for
x>x0. From this point of view, many authors seem to be rather sloppy in the choice of the
partition.
Sidi [128], for example, takes x0 to be the rst zero of the circular function sin ( (x)) that is
greater than the lower integration limit a. Thus x0 satis es the polynomial equation (x) =q
for some integer q. The consecutive partition points xi ; i>0 are then determined to be the
largest positive solution of (x) =(q+l) . As can be seen from our example for large values
of xm, this results in a large second partial integral I1. A solution would be to determine
also x0 as largest root of (x) =q and not only as the zero closest to a. But then again we
would obtain an unnecessarily large partial interval (this time I0).
Even worse is the fact that although the proposed method guarantees that the xi are of
ascending order, it does by no means ensure that problems like the aforementioned are avoided.
If there are, for example, extrema at large ordinate values, they might not be comprised in
the sequence of the partial integrals because the series seems to converge earlier and the
evaluation is truncated consequently.
Espelid and Overholt [123] also require the rst partition point to be supplied by the user.
They can of course provide no information other than to choose it small enough to keep the
computational costs for the rst interval low and large enough that the asymptotic properties
of the functions are satis ed.
There is still another side to the story: the non-oscillating factor of the integrand ought tobe
smooth compared to the oscillating part, otherwise the sequence of partial sums may again
show discontinuities. Not only do discontinuous factors like the step function provoke such
problems, also rather innocent functions like e ,x2 or tanh(x) can hamper the straightforward
use of extrapolation. Therefore these parts of the integrand, too, should be monotonic for
x>x0, as should their derivatives. This aspect is also rarely addressed in the literature.
Sidi [128] does not consider the properties of the modulating factor at all, he just states
that it must be non-oscillating at x ! 1 and have an asymptotic expansion such that
the entire function is integrable at in nity. Lyness [121] brie y discusses the problems of
unexpected peaks in the integrand that are too far out to be included in the sequence used
for extrapolation. Consequently, for his quadrature routine [122], he lets the user decide where
to start the extrapolation.
Remark (Connection to physics) We have already encountered the fact that wave
integrals are dominated by a rather small region of the integration range. In fact, the
method of stationary phase is based upon this nding. The mathematical formulation
that an extremum of the argument function yields a large partial integral is equivalent
to the concept of local wave number and frequency, both of which are determined by the
extrema of the wave's phase.
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