Wave Propagation in Linear Media | re-examined

6.4 Asymptotic partition
consuming preparation of the internal sampling points and weights needed for the Gauss-
Kronrod formula.
There are two conclusions to be drawn from these experiments with respect to the implementation
of numerical quadrature for an automatic algorithm:
The rst integral between the lower integration bound and the rst partition point is
best evaluated with the setting Method->DoubleExponential, which yields an optimum
performance in case of a strongly oscillating integrand in the interval and is still
reasonable in all other cases.
The subsequent integrals that are then used for extrapolation ought to be computed
with Method->GaussKronrod, because here the integrand is smooth enough that the
simplicity of the Gauss-Kronrod rule will be an advantage.
6.4 Asymptotic partition
The zeros or extrema of the integrand are a perfect choice for the de nition of the subintervals
and the best one as far as the suitability for series acceleration is concerned. In terms of
computing time, this is true only as long as they are easily computable. If the oscillating
factor is su ciently complex so that these points can only be determined numerically, itmay
be more e cient tolook for a di erent partitioning strategy. A good idea is to consider the
integrand's behaviour at in nity and choose a partition that approaches the zeros as x !1.
This way we make sure that the resulting series is at least ultimately alternating and can be
treated by a convergence acceleration algorithm. More formally, ifwehaveanintegral of the
kind
I =
Z 1
a
(x) e j (x) dx ; (6.9)
where the argument function (x) has an asymptotic expansion as x !1
(x)=x p
with p being a non-negative integer, we can take its polynomial part
(x) =x p
1X
i=0
pX
i=0
ai
; (6.10)
xi ai
x i
(6.11)
to calculate the subdivision points, which maybemuch easier because we need to solve only
a polynomial equation. This strategy has been pursued by Sidi [128].
Again we present an example to make the application of the basic idea clear. The integral
Z 1
x
I =
x2 p
sin x2 , a2 dx = (6.12)
, a2 2
a
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