Wave Propagation in Linear Media | re-examined

6.5 Considerations for a Mathematica implementation
polynomial approximation causes the sequence of partial sums to take on the form of distinct
`wave packets' in the rhythm of the beat frequency @ ,
@x (x) , (x) . A reasonable criterion
could thus be to seek the abscissa value where such awavepacket comprises a given number
of k elements, that is
0 (x)
0 (x) , 0 (x) = k: (6.13)
There is, unfortunately, no general guideline how to choose k | the larger it is, the better
the result will be.
6.5 Considerations for a Mathematica implementation
Let us now put the considerations of the last section into a context that is closer to an
implementation in Mathematica. To evaluate an in nite sum by means of extrapolation,
we normally use the built-in function NSum with the option Method->SequenceLimit. This
function internally performs exactly the same operations as we did in the example above |
rst it computes a sequence of partial sums and then it passes an appropriate portion of this
list to the function SequenceLimit for extrapolation. The selection of this sublist is controlled
by the two parameters NSumTerms and NSumExtraTerms, respectively. The extrapolation
process itself may be in uenced by the option WynnDegree.
NSumTerms is the number n of terms that are summed up explicitly before the extrapolation
is performed.
NSumExtraTerms is the number k of terms that are used in the extrapolation.
WynnDegree sets something like the `degrees of freedom' for the extrapolation [144]. It is
also the only option to SequenceLimit. If its value is 1, Aitken's 2 algorithm is used,
for values greater than one the -algorithm is invoked. Apart from this distinction, the
actual function of this parameter is not documented further, but it seemingly speci es
how far the -array ( g. 5.1) is computed.
Remark (Sequence extrapolation) According to Keiper [144], the basic algorithm of
SequenceLimit transforms the sequence into a sequence of length n,2. In the triangular
scheme that is often used to illustrate the data dependencies of the -algorithm, this is
equivalent to the computation of the two columns to the right of the starting sequence.
As we have seen in section 5.2, the very rst execution of this algorithm yields the same
result as the 2 algorithm. The basic transformation is repeated WynnDegree times. If
the resulting sequence still has more than two elements, the whole process starts again
until the outcome is too short or | in terms of the -array |until the triangle is completed.
One could suspect that if the -array iscompleted anyhow, the result ought to
be independent of the parameter WynnDegree, but this is not true. The di erence is that
the column to the left of the starting sequence always consists of zeros, so for each time
the algorithm is restarted in the middle of the -array, the respective column to the left
of the previous result is replaced with zeros, which naturally a ects the nal outcome.
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