Wave Propagation in Linear Media | re-examined

Chapter 6
Towards a quadrature routine
6Towards a quadrature routine
\Can you do Addition?" the White Queen asked. \What's one and one
and one and one and one and one and one and one and one and one?"
\I don't know," said Alice. \I lost count."
\She can't do Addition," the Red Queen interrupted. \Can you do Subtraction?
Take nine from eight."
\Nine from eight I can't, you know," Alice replied very readily: \but| "
\She can't do Subtraction," said the White Queen.
Lewis Carroll, Through the looking glass
With the discouraging result of the search for an o -the-shelf program to solve our problems,
there is nothing left but to write a dedicated quadrature routine that suits our needs. This
is, however, not at all a straightforward task, and we have to avoid carefully a number of
pitfalls associated with the in nitely oscillating integrals we wish to compute. This chapter is
thus devoted to some practical considerations that emerge in the course of the development
of such a quadrature routine. While it seems obvious that a combination of partitioning and
extrapolation is best suited to this purpose, there are some detailed questions yet to solve.
The most interesting one is certainly the choice of the partition itself, and we shall therefore
discuss which points are most suitable for this subdivision procedure | the zeros, extrema,
or any other points of the integrand. A rather implementation-speci c question is how the
partial integrals are to be computed e ciently. We shall also investigate a strategy how to
control the accuracy of the computation.
Throughout the subsequent sections, we shall present many examples programmed in Mathematica
to illustrate the various di culties that need to be considered. To understand these
pieces of code, a certain familiarity with Mathematica or any other functional programming
language is favourable, but not imperative. Since most of the used functions are selfexplaining,
we shall give further explanations only where they are deemed necessary.
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