Wave Propagation in Linear Media | re-examined

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6Towards a quadrature routine
Figure 6.10: Negative number of correct digits, log e S (k)
n , S , for varying n (ns) and k (ne), computed
with the 2 algorithm (WynnDegree->1) and the zeros of the integrand as partition points.
of sequence members used in the extrapolation as well as the precision of the internal
computations. We have already noticed that it is most promising to use the entire
sequence for this purpose rather than to shift the starting point for the extrapolation
to higher index values.
3. The parameter to be controlled is the accuracy of the result or | in other words
| its error. We thus need a function that gives us an a posteriori estimate of the
approximation error.
A simple way to obtain an error estimate in Mathematica is to use the functions Accuracy
and Precision on the result of a numerical computation. The former returns the number of
signi cant digits to the right of the decimal point and therefore is a measure for the absolute
error. The latter gives the total number of signi cant digits in a oating-point number. For
very small numbers, the zeros between the decimal point and the rst non-zero digit are not
counted, thus the precision of a number denotes its relative error (see also the explanations
in example 6.3.1).
Unfortunately, these two functions work only with variable-precision arithmetic, which cannot
make use of the fast hardware-based machine-precision operations. Instead, it is implemented
in software to keep track of the uncertainty associated with every number in the calculation,
which makes it much slower than machine-precision arithmetic. For this reason, when Mathematica
comes across a machine number in the course of the computation, all subsequent
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