Wave Propagation in Linear Media | re-examined

6Towards a quadrature routine
the best results that are in uenced chie y by the implementation details and the associated
numerical e ects. This region is characterised by extrapolation errors that are constant for
ne + ns = const and thus depend on the total number of sequence terms regardless of how
many are actually used for extrapolation.
The contour plot ( g. 6.6) also shows that it is di cult to decide where to begin the extrapolation
if not too many terms are to be used. For 40 terms, for example, the optimum lies
somewhere at ne = 15, ns = 25.
The results we obtain when choosing the -algorithm with WynnDegree->2 are shown in
g. 6.7. The inconvenient peaks have disappeared (which is the same for all even values of
WynnDegree, but not for the odd ones), yet it is hard to nd optimum values for ne and ns.
The setting WynnDegree->Infinity yields the best extrapolation results ( g. 6.8). It is still
true that there are pairs of sequences that give the same estimates for the limit S e(2m+1) n and
eS (2m+2)
n+1 , respectively, which can be seen from the bands running in parallel to the ns-axis.
The respective contour plot (6.9) shows that the best way to improve the extrapolation result
is to increase ne, perhaps with a few terms summed up explicitly. We have thus con rmed
the suspicion that the limiting process 1 from the last section is indeed superior.
For comparison we compute the extrapolation error if we do not use approximate partition,
but take the zeros of the integrand as partition points. Just the determination of the partial
integrals is di erent. The rest of the evaluation remains the same as with the asymptotic
partition before.
In[25]:= a = 20 Pi;
firstval = N[Integrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x,
fx,a,Sqrt[Pi^2+a^2]g],40];
seq = Table[NIntegrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x,
fx,Sqrt[(i Pi)^2+a^2],Sqrt[((i+1) Pi)^2+a^2]g],
fi,1,300g];
We nd that the results are far better and practically independent of the value we choose for
WynnDegree.
We thus conclude that when implementing an automatic quadrature routine with Mathematica,
we ought to consider the following points:
The setting WynnDegree->Infinity gives the best results for asymptotic partition and
reasonably good ones for standard (zero) partition, so it is a suitable choice for a default
value of this option.
To increase the accuracy of the extrapolation result, the number of terms included in
the extrapolation (NSumExtraTerms) should be increased, whereas NSumTerms may be
left unchanged.
Whenever feasible, the zeros of the integrand should be chosen as the partition points.
This yields more accurate results with fewer terms involved.
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