Wave Propagation in Linear Media | re-examined

6.4 Asymptotic partition
of the integrand. The terms `zero' and `extremum' lose their meanings as they now refer to
the approximation rather than to the real integrand. Thus we would actually bene t from
the e ect we discussed in the last section only for x !1.
In the computation of the limit we used the complete sequence of partial sums with almost 250
members. Normally, wewould not begin with that large a number because their determination
could be time consuming. As the rst few sequence members look fairly irregular, the burning
question is how many wemust take in order to obtain a reliable result for the limit and how
we can improve the accuracy if needed. This obviously requires that more sequence members
be taken into account, however, there are two possibilities to do so:
1. We can increase the number of sequence members used in the extrapolation, that is, we
compute limk!1 e S (k)
n with n = 0 if we start with the very rst member.
2. We leave the length k of the subset subject to extrapolation xed and start the extrapolation
at a member with a higher index, which islimn!1 e S (k)
n .
Both limiting processes have not received much attention in the literature. Sidi [126] found
that for his W -transformation, method 1 gives better results and is more e cient.
We now examine these two possibilities brie y for our example. For the rst one, we take
the rst k members of the sequence to determine the limit and plot the absolute value of the
approximation error e S (k)
n , S depending on k.
In[22]:= approxerr1 = Table[Abs[SequenceLimit[Take[partial,f1,kg]] - N[Pi/2]],
fk,10,240g];
LogListPlot[approxerr1,PlotStyle->PointSize[0.006],PlotRange->All];
0.001
-6
1. 10
-9
1. 10
-12
1. 10
0 50 100 150 200
147