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TR/61 April 1976

THE ACCURATE NUMERICAL INVERSION OF

LAPLACE TRANSFORMS

by

A. Talbot

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W9261050

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The accurate numerical inversion of Laplace transforms

A. TALBOT

Mathematics Department, Brunei University, Uxbridge

Abstract

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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1.

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

Introduction