Wave Propagation in Linear Media | re-examined

5.1 Univariate numerical quadrature
Poincare-type asymptotic expansions
(x) x
1X
i=0
i
: (5.5)
xi Speci cally, (x) may be written as (x) = (x)+ (x), where its polynomial part (x) isof
degree and (x) P 1 i=0 +i=x i as x !1. The xl are taken to be the zeros of u , (x) ,so
they approach the zeros of f(x) only asymptotically. The W -algorithm in combination with
a Clenshaw-Curtis rule was used by Hasegawa and Torii [127] to compute in nite integrals
with circular functions as oscillating factors.
R xl
a f(x) dx by (xl) = R xl+1
xl
In an even later work [128], Sidi returns to the ideas of Lyness and replaces the integrals
f(x) dx, thus forming a sequence of integrals between successive
zeros of u , (x) , whose members are at least for large xl alternating in sign. Nonetheless, he
uses a modi cation of his W -algorithm to accelerate the convergence of the resulting series
P1i=,1 (xi), where (x,1) = Rx0 f(x)dx denotes the integral from the lower interval limit
a
to the rst zero that is greater than a. In contrast to the original W -transformation, the
modi ed version needs only analysis of the phase (x) of oscillations, whereas information
on the amplitude is not required. The choice of the partition points xl is simple, too, as it
involves nothing more than the determination of the largest zero of a given polynomial.
As a variant to Sidi's method, Ehrenmark [129] proposes not to use subdivision, but again
the sequence of truncated integrals F (xn) = Rxn f(x)dx. However, he chooses not the zeros
a
of f(x) as xn, but seeks those values of x at which the truncated integral equals its limit
limx!1 F (x).
Another approach by Khanh [130] is based on the direct evaluation of the tail of the integral.
It is applicable to integrands that satisfy either the linear di erential equation of rst order
f(x) =p(x)f 0 (x)orthat of second order f(x) =p(x)f 0 (x)+q(x)f 00 (x) with f(x), p(x), and
q(x) having asymptotic expansions at in nity. Then the tail R 1
f(x) dx may be approximated
x
by a linear combination of these expansions and their derivatives. The error can be controlled
by choosing the point where the tail is to be evaluated and the number of involved terms
su ciently large. The remaining nite integral R x
f(x) dx has to be calculated using some
a
other method.
5.1.2 Computer routines
Compared to the large number of scienti c papers that have addressed the problem of in nitely
oscillating integrals, there are astonishingly few ready-to-use computer programs available.
Many of them are public domain software and are contained in the comprehensive library
Netlib that is accessible via the World-Wide-Web under the address http://www.netlib.org.
Another entry point for seeking computer routines is the GAMS (Guide to Available Mathematical
Software) that is maintained by the US National Institute of Standards and Technology
at http://gams.nist.gov and provides a classi cation scheme allowing fast retrieval
of appropriate routines. For other software resources on the Internet, see also the book by
Uberhuber [131].
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