Wave Propagation in Linear Media | re-examined

6.5 Considerations for a Mathematica implementation
0.001
-6
1. 10
-9
1. 10
-12
1. 10
-15
1. 10
0 50 100 150 200
Figure 6.4: Extrapolation error e S (k)
n , S for k = 12 and varying n, computed with the 2 algorithm
( ) and the -algorithm ( ).
In[24]:= a = 20 Pi;
firstval = N[Integrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x,
fx,20 Pi,21 Pig],40];
seq = Table[NIntegrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x,
fx,i Pi,(i+1) Pig],
fi,21,350g];
partial = FoldList[Plus,firstval,seq];
errtab = Table[fne,ns,Log[10,Abs[SetPrecision[
SequenceLimit[Take[partial,fns+1,ns+ne+1g],
WynnDegree->1],40]-N[Pi/2,40]]]g,
fne,6,50,1g,fns,0,60,1g];
The results for WynnDegree->1 are shown in g. 6.5 . It is interesting to notice that if only a
small number of terms are used for extrapolation, peaks appear where the extrapolation error
becomes unexpectedly high. These peaks seem to be related to the areas of the sequence
where the alternation is not very pronounced | the `nodes' of its wave-like envelope. A
contour plot of the results ( g. 6.6) reveals that these peaks follow the lines 2 ne + ns = const.
This is not surprising if we remember that each pass of the sequence transformation reduces
the length of the sequence by two. In the end there may be either one or two members left
depending on whether we started with an odd (2m +1) or even (2m +2) number of sequence
members. Both results | after exactly the same number m of transformations | are not
supposed to di er very much. If su cient terms are used for the extrapolation or if it is
started where the wave packets are broad enough (for low ne but high ns), then we obtain
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