Wave Propagation in Linear Media | re-examined

5.1 Univariate numerical quadrature
5.1 Univariate numerical quadrature
All e cient quadrature methods approximate the integrand by elementary functions (polynomials)
whose integrals can be determined analytically. This is fairly simple if the integrand is
smooth. Oscillatory functions, however, are di cult to approximate and require either higher
order polynomials or a subdivision of the integration interval. Things get even worse if the
interval is in nite, like it is for our wave integrals. In this section we shall thus give a brief
survey of both published algorithms and available computer routines that tackle the problem
of univariate quadrature of in nitely oscillating functions.
5.1.1 Algorithms
Many authors have considered nite integrals of the form
I =
Z b
a
f(x) e jq(x) dx : (5.1)
Such an approach may be applicable to in nite integration intervals if the integrand decays
fast enough (exponentially) that a truncation is justi ed, so a look at some of the methods
is reasonable (see also [113]). Levin [114] pursues the idea that if f were of the form f(x) =
jq 0 (x)p(x)+p 0 (x), then the integral could be evaluated directly. Thus the integration problem
is transformed into the problem of solving a di erential equation.
Evans [115] subdivides the integral
I =
NX
i=1
Z a+ih
a+(i,1)h
f(x) sin
cos
!q(x)dx (5.2)
and approximates both f(x) and q(x) in each subinterval by linear or quadratic forms, respectively.
These can be integrated analytically, which is equivalent to the trapezoidal and
Simpson's rules for general quadrature. The quadratic approximation in fact dates back to
a remarkably early idea of Filon [116]. An entirely di erent approach is that of Ehrenmark
[117] who proposes a three-point formula with weights chosen such that the formula is exact
for the functions x, sin kx, and cos kx. This method is fairly easy to implement and, according
to Evans [115], lends itself to an extension to a higher number of points.
Xu and Mal [118] compute wave number integrals of the form R b
f(x) cos(rx) dx by a modi ed
a
Clenshaw-Curtis scheme where they approximate only f(x) by Chebyshev polynomials. To
achieve the desired accuracy, they turn to adaptive interval subdivision if the given maximum
order polynomial approximation is not su cient. For certain integrands, specialised complexplane
techniques as descibed by Davies [119] may be employed. By transforming the contour
of integration, the integral can be evaluated easily.
If the integration interval is in nite and the decay of the integrand is such that the integral
converges too slowly, a common technique is to partition the integral and to extrapolate from
a nite series of subintervals to its limit. All proposed methods require some knowledge of
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