Wave Propagation in Linear Media | re-examined

5 Numerical quadrature and extrapolation
Searching these indices, one nds essentially three di erent quadrature routines, all written
in FORTRAN:
OSCINT is part of the sublibrary Toms (toms/639) in Netlib and was written by
Lyness and Hines [122]. It is aimed at integrands that may be separated into an oscillating
factor with an at least asymptotically constant period (i. e. circular or Bessel
functions) and an ultimately positive factor. The user has to specify the function to
be integrated, its (ultimate) period, and a starting point from which onwards the extrapolation
is performed. The algorithm then carries out a sequence of nite interval
quadratures, each over an integral having the length of a half period, and accelerates
the resulting sequence of partial sums by means of the Euler transformation [113]. By
default the quadrature is performed using the trapezoidal rule with a number of sampling
points given by the user. The quadrature error is not controlled, and the user
may as well replace the trapezoidal by a more sophisticated algorithm. The routine
uses double precision arithmetic.
DQAINF by Espelid and Overholt [123] can be found in the sublibrary Numeralgo
(numeralgo/na6). They extend the approach of Lyness and Hines in that the integrand
may be a linear combination of periodic functions p(x), all with the same (ultimate)
period, thus f(x) = Pk i=1 pi(x)gi(x). The gi(x) are supposed to have similar asymptotic
expansions as x ! 1, gi(x) = x , P1 j=0 c (i)
j =xj , with > 0. The user must specify
basically the period of the functions, the rst subdivision point, and of course an error
tolerance. Then the algorithm computes integral approximations over half-period
intervals using a 21-node Gauss rule and estimates the true integral value by means
of extrapolation. In addition, the quadrature error in the intervals is estimated and
if necessary, the interval is subdivided into three equal parts. There are both single
(SQAINF) and double precision versions available.
DQAWF is a routine of the Quadpack package [132], which is included in many
sublibraries of Netlib (Slatec, Cmlib). This double precision routine has also a single
precision counterpart (QAWF) and is designed for calculating Fourier transforms. Thus
the integrand needs to be of the form f(x) sin !x or f(x) cos !x. With f(x) and ! given
by the user, the algorithm integrates between the zeros of the oscillating factor. The
quadrature is based on a Clenshaw-Curtis rule in combination with a globally adaptive
subdivision strategy, so there is virtually no limitation imposed on the function to
be integrated. The convergence of the resulting series is then accelerated using the
-algorithm [113, 133].
Variants of this routine with slight modi cations may also be found in proprietary
libraries. In the NAG library (The Numerical Algorithms Group) it is called D01ASF
[134], in the IMSL (International Mathematical and Statistical Libraries) [135] it appears
as DQDAWF.
All these subroutines are best suited to integrands with a constant period as x !1. If this is
not the case, like for sin (f(x)) with f(x) having a polynomial part with degree greater than
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