Wave Propagation in Linear Media | re-examined

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6Towards a quadrature routine
Figure 6.3: Extrapolation error e S (k)
n , S for n = 0 and varying k, computed with the 2 algorithm
( ) and the -algorithm ( ).
There is one exception to the aforementioned operation principle of the sequence transformation:
If WynnDegree is set to Infinity, as many transformations as possible are carried
out in a row without restarting the process. This is now the case where the -array is really
computed from left to right, which matches the descriptions of the -algorithm known
from the literature.
It is worthwile to review the results of the previous example (6.12) with a view to the extrapolation
algorithm used. Being indi erent to this detail in the last chapter, we left the
settings of SequenceLimit at their defaults, which selected the 2 algorithm. If we choose
the -algorithm (for example with WynnDegree->Infinity) and carry out the computation
anew, we can compare both algorithms directly. Fig. 6.3 shows this comparison for the case
where the number of terms subject to extrapolation are varied, and g. 6.4 the alternative
with a xed number of extrapolation terms and a di erent starting index. In either case, the
-algorithm yields better results for a given number of sequence members.
Apart from the algorithm used, which is likely to be chosen once and for all in a series of
computations, there are in principle two knobs for the ne tuning of the extrapolation results.
So far, we only twiddled either one or the other, exploring the boundaries of this parameter
space. Yet it might be possible to nd a good compromise for the values of n and k, and
therefore we want to study their e ects in some detail now. We again take the integral (6.12),
this time with a = 20, and vary both NSumTerms and NSumExtraTerms (here denoted as ns
and ne). We compute this array for di erent values of WynnDegree and plot the logarithm to
the base 10 of the approximation error, which is essentially the number of correct digits.
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