Wave Propagation in Linear Media | re-examined

Chapter 5
Numerical quadrature and
extrapolation
5 Numerical quadrature and extrapolation
Someone has recently de ned an applied mathematician as an individual
enclosed in a small o ce and engaged in the study of mathematical problems
which interest him personally; he waits for someone to stick his head
in the door and introduce himself by saying, \I've got a problem." Usually
the person coming for help may beaphycicist, engineer, meteorologist,
statistician, or chemist who has suddenly reached a point in his investigation
where he encounters a mathematical problem calling for an unusual
or nonstandard technique for its solution. [...] The range of mathematical
topics from which queries may arise is all-inclusive. However, a topic which
arises frequently enough to merit some discussion is one which particularizes
the statement \I've got a problem," to \I've got an integral."
Milton Abramowitz [112]
In the course of the case studies in the rst part, we encountered exclusively solutions in the
form of in nite wave integrals. Unfortunately, these functions cannot be integrated analytically,
and therefore we must resort to numerical quadrature. The term numerical quadrature
is generally used in numerical mathematics to distinguish the process of nding a numerical
value for a de nite integral from the numerical solution of di erential equations, which is then
called numerical integration.
The objective of this chapter is to give an overview of relevant aspects of this part of numerics.
In particular, these are univariate quadrature and the problem of convergence or
series acceleration, which is almost inevitable if one tries to compute integrals over an in nite
range. Many of the presented algorithms have been incorporated in specialised computer routines
that provide ready-to-use solutions. Particular emphasis is placed upon these computer
routines and their applicability to the class of integrals we are interested in.
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