TR/61 April 1976
THE ACCURATE NUMERICAL INVERSION OF
The accurate numerical inversion of Laplace transforms
Mathematics Department, Brunei University, Uxbridge
Inversion of almost arbitrary Laplace transforms is
effected by trapezoidal integration along a special
contour. The number n of points to be used is one
of several parameters, in most cases yielding absolute
errors of order 10-7 for n = 10, 10-11 for n = 20,
10-23 for n = 40 (with double precision working), and
so on, for all values of the argument from 0+ up to
some large maximum.
The extreme accuracy of which the method is capable
means that it has many possible applications of
various kinds, and some of these are indicated.
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The inversion of Laplace transforms is a topic of fundamental
importance in many areas of applied mathematics, as would be
evident by a glance at, for example, Carslaw and Jaeger (1948).
In the more standard applications the inversion can he accomplished
by the use of a dictionary of transforms, or in the case of
rational function transforms by partial fraction decomposition.
Where these methods are of no avail recourse may be had to the
inversion integral formula, which is likely to lead to an
intractable integral, or to an infinite series, often with
terms involving the roots of some transcendental function. It
is clear that in all but the simplest cases considerable effort
is needed to obtain an accurate numerical value of the inverse
for a specified value of the argument.
It is therefore natural that attention has been paid by
mathematicians, engineers, physicists and others to alternative
ways of evaluating the inverse. Early methods (e.g. Widder (1935),
Tricomi (1935), Shohat (1940)) involved expansion of the inverse
in series of Laguerre functions. Salzer (1955) evaluated the
inversion integral by Gaussian quadrature using an appropriate
system of orthogonal polynomials. Since 1955 et very large number
of methods for numerical inversion have been published: see for
example the partial bibliography in Piessens and Branders (1971)
or the fuller one in Piessens (1974) . A useful critical survey
of earlier work was given by Weeks (1966).
Many of the methods use either orthogonal series expansions, or
weighted sums of values of the transform at a set of points,
usually complex points. In either case considerable preliminary
work must be carried out. In the second type this may be done in
advance once and for all for each selected set of points, and the
points and weights stored in the computer. However, if more points
are desired for the sake of gaining increased accuracy,much further
computational effort must be expended first.
In general the methods hitherto published have been intended for
use with transforms of particular types, e.g. rational functions