Wave Propagation in Linear Media | re-examined

6.6 Controlling the accuracy of the extrapolation result
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Figure 6.9: Contour plot of g. 6.8, abscissa: ne, ordinate: ns.
6.6 Controlling the accuracy of the extrapolation result
As we have seen in the last sections, we can improve the result of the extrapolation under
most circumstances by increasing the number of sequence elements being subject to the
extrapolation. The severe problem that still remains is that this is only a qualitative nding
and that we have no guideline whatsoever as to how this parameter is to be chosen. From a
user's point of view, the exact number of sequence members necessary to achieve a certain
accuray is entirely irrelevant, provided that the goal is reached. On the contrary, it is much
more convenient to let the user specify only the desired accuracy and to leave the choice of
the other parameters to the algorithm, which then has to be somehow adaptive.
If we want to adapt only the parameters of a basic numerical algorithm, then we in turn need
a meta algorithm that starts from a given parameter set, calls the basic algorithm, judges the
results and changes the parameters. This procedure must be repeated iteratively until either
the outcome meets the requirements or a maximum number of iterations is reached | just
to prevent the algorithm from being lost in an in nite loop or wasting precious computing
time. From the previous sections, we can easily conclude what the three major elements of
such a meta algorithm must look like:
1. The basic algorithm consists of the computation of the sequence of partial sums for a
given integrand together with the extrapolation. It returns an estimate of the integral.
2. The parameters that can be in uenced directly by the meta algorithm are the number
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