In their survey of algorithms for inverting the Laplace transform, Davies and Martin [16] note that the Fourier series method without accelerated convergence gives good accuracy on a wide variety of functions. Since the Laplace Transform is closely related to the Fourier transform it is not surprising that inversion methods based on a Fourier series expansion would yield accurate results. In fact the two sided Laplace transform can be derived from the Fourier transform in the following way. We can define the Fourier transform as

F { f ( t ) } = ∫ − ∞ ∞     f ( t ) e − 2 π i ν t d t (4)

Then letting v = 2 π ν we have

F { f ( t ) } = ∫ − ∞ ∞     f ( t ) e − i v t d t (5)

This Fourier transform exists provided f ( t ) is an absolutely integrable function, i.e.

∫ − ∞ ∞ | f ( t ) | d t < ∞ (6)

As many functions do not satisfy condition (6), f ( t ) is multiplied by the exponential dampening factor e − u t thus

F { f ( t ) e − u t } = ∫ − ∞ ∞     f ( t ) e − i v t e − u t d t (7)

and letting s = u + i v we obtain the two sided Laplace Transform of f ( t ) as

F { f ( t ) e − u t } = L { f ( t ) } = ∫ − ∞ ∞     e − s t f ( t ) d t (8)

Le Page [17] notes that the integral given by (8) can be written in two parts as follows:

∫ − ∞ ∞     e − s t f ( t ) d t = ∫ − ∞ 0     e − s t f ( t ) d t + ∫ 0 ∞     e − s t f ( t ) d t (9)

The second term on the RHS in the above expression is referred to as the one sided Laplace transform or simply the Laplace transform. Thus, s is defined as a complex variable in the definition of the Laplace transform.

As before the inverse Laplace transform is given as:

f ( t ) = 1 2 π i ∫ u − i ∞ u + i ∞     e s t F ( s ) d s (10)

With s = u + i v in (10) this leads to the result

f ( t ) = 2 e u t π ∫ 0 ∞ [ Re { F ( u + i v ) } cos ( v t ) − Im { F ( u + i v ) } sin ( v t ) ] d v (11)

[8], which can be replaced by the cosine transform pair

Re { F ( u + i v ) } = ∫ 0 ∞     e − u t f ( t ) cos ( v t ) d t (12)

f ( t ) = 2 e u t π ∫ 0 ∞ Re { F ( u + i v ) } cos ( v t ) d v (13)

or by the sine transform pair

Re { F ( u + i v ) } = − ∫ 0 ∞     e − u t f ( t ) sin ( v t ) d t (14)

f ( t ) = − 2 e u t π ∫ 0 ∞ Im { F ( u + i v ) } sin ( v t ) d v (15)

Dubner and Abate [18] applied a trapezoid rule to (13) resulting in the Fourier series approximation,

f ( t ) ≈ 2 e u t T [ 1 2 F ( u ) + ∑ k = 1 ∞ Re { F ( u + k π i T ) } cos ( k π t T ) ] (16)

where f ( t ) is expanded in the interval 0 ≤ t < T . For faster computation Simon et al. [19] proposed the following version of (16).

f ( t ) ≈ e u t t [ 1 2 F ( u ) + ∑ k = 1 ∞ Re { F ( u + k π i t ) } ( − 1 ) k ] (17)

This series can be summed much faster than (16) as there are no cosines to compute [20]. This algorithm is relatively easy to implement with u being the only real varying parameter.

However, as Crump [8] points out, for the expression in (17) the transform F ( s ) must now be computed for a different set of s-values for each distinct t. Since this type of application occurs often in practice in which the numerical computations of F ( s ) is itself quite time consuming this may not be an economical inversion algorithm to use. These drawbacks to some extent can be overcome by using the fast Fourier transform techniques in [20] [21].

Crump [8] also extends this method to one with faster convergence by making use of the already computed imaginary parts. There are several other acceleration schemes for example, those outlined by Cohen [10], however these acceleration methods in general require the introduction of new parameters which for the purposes of this investigation we wish to avoid.

Davies and Martin [16] cite the Stehfest [5] algorithm as providing accurate results on a wide variety of test functions. Since that time, this algorithm has become widely used for inverting the Laplace Transform, being favoured due its reported accuracy and ease of implementation.

Here we give a brief overview of the evolution of the algorithm from a probability distribution function to the Gaver functional whose asymptotic expansion leads to an acceleration scheme which yields the algorithm in its most widely used form.

Gaver [6] investigated a method for obtaining numerical information on the time dependent behavior of stochastic processes which often arise in queuing theory. The investigation involved examining the properties of the three parameter class of density functions namely

p n , m ( a , t ) = ( n + m ) ! n ! ( m − 1 ) ! ( 1 − e − a t ) n a e − m a t (18)

with n , m ∈ ℕ .

After the binomial expansion of the term ( 1 − e − a t ) n , Gaver went on to find the expectancy E [ f ( T n , m ) ] , where T n , m is the random variable with density (18). From this Gaver was able to express the inverse Laplace transform in terms of the functional

f n , m ( t ) = ln 2 t ( n + m ) ! n ! ( m − 1 ) ! ∑ j = 0 n ( n k ) ( − 1 ) k F ( ( k + m ) ln 2 t ) (19)

with certain conditions on n and m, Gaver makes n = m and Expresses (19) as

f n ( t ) = ln 2 t ( 2 n ) ! n ! ( n − 1 ) ! ∑ k = 0 n ( n k ) ( − 1 ) k F ( ( k + n ) ln 2 t ) (20)

While the expression in (20) can be used to successfully invert the Laplace transform for a large class of functions its rate of convergence is slow [9] [10]. However Gaver [6] has shown that (20), with t = ln 2 a has the asymptotic expansion

f n ( t ) ≈ f ( ln 2 a ) + α 1 n + α 2 n 2 + α 3 n 3 + ⋯ (21)

where the α j ’s are constant coefficients in the asymptotic series. Hence (21) in the limit converges to

f n ( t ) n → ∞ = f ( ln 2 a )

For the conditions on m and n and justification for the substitution for a referred to above see Gaver [6]. This asymptotic expansion provides scope for applying various acceleration techniques enabling a more viable application of the basic algorithm.

Much of the literature alludes to the fact that a Salzer [7] acceleration scheme is used on the Gaver functional in (20) which results in the Stehfest algorithm. In fact Stehfest’s approach was a little more subtle than a direct application of the Salzer acceleration.

Equations (4) to (8) showed that the Laplace transform can be seen as a Fourier transform of the function

e − u t f ( t ) ,     t > 0 (31)

i.e.

F { e − u t f ( t ) } = L { f ( t ) } = F ( s ) (32)

Hence the Fourier transform inversion formula can be applied to recover the function thus:

F − 1 { F ( s ) } = e − u t f ( t ) = 1 2 π ∫ − ∞ ∞     F ( s ) e i v t d v (33)

as s = u + i v we have that d s = i d v and so

f ( t ) = 1 2 π i ∫ u − i ∞ u + i ∞     F ( s ) e s t d s (34)

This result provides a direct means of obtaining the inverse Laplace transform. In practice the integral in (34) is evaluated using a contour

1 2 π i ∫ B     e s t F ( s ) d s (35)

with B here denoting the Bromwich contour [3]. The contour is chosen so that it encloses all the possible singularities of F ( s ) . The idea of the contour is introduced so that the Cauchy residue theorem can be used to evaluate the integral.

However, when f ( t ) is to be calculated using numerical quadrature it may be more appropriate to devise a new contour. To ensure that (35) is integrable we may wish to control the growth of the magnitude of the integrand e s t by moving the contour to the left and so giving the real part of s a large negative component [1] [22].

However, the deformed contour must not be allowed to pass through any singularities of F ( s ) . This is to ensure that the transform is analytic in the region to the right of B.

Table 4 and Table 5 show that when compared with the other four algorithms, the Fourier series method performs with the least accuracy on all the functions tested. It also fails to reconstruct functions 8, 15, 17 and 18, with poor results for functions 4, 5 and 12.

However, for the functions which it successfully reconstructs it does so with a RMS accuracy of between L= 3.6(−5) and 1.2(−2). We believe that this scheme will improve greatly when an acceleration scheme is applied. This is an issue we intend to investigate in our future work.

With the exception of function 7, J 0 ( t ) , Logan’s algorithm successfully inverts all the functions given in Table 2 and Table 3 with very good accuracy. We found that in SDP best results are obtained by equating a = 1 in (50). Table 3 and Table 4 show that for these functions the RMS error varies between 3.6(−8) to 8.4(−12).

Except for function 7 the J 0 ( t ) , the Abate-Valko scheme successfully inverts all the functions in Table 3 and Table 4. Moreover, it does so with greater

accuracy than the Logan scheme. The tables show that the RMS error varied between 6.5(−11) and 6.2(−12).

Table 3 and Table 4 show that the Stehfest algorithm shows poor accuracy when inverting functions 1, 7, 10 and 11. For these functions the RMS error varies between 2.01(−2) to 9.2(−3). Its poor performance is due to the fact that the Stehfest algorithm has difficulty inverting functions of a cyclic nature [5]. However, it inverts the remaining functions with good accuracy with a RMS error of between 0 to 2.9(−5). Table 3 and Table 4 shows that the Salzer-Gaver algorithm shows poor accuracy for functions 1, 7, 10 and 11. These are the very same functions that the Stehfest algorithm has problems inverting. Again this is due to the difficulties it encounters when inverting cyclic functions. It inverts the remaining functions with less accuracy than the Stehfest with an RMS error varying between 10(−15) to 10(−5).

With the exception of function 7, the Logan and Abate-Valko algorithms successfully inverted the remaining functions to a high degree of accuracy, see Table 6 and Table 7. Duffy [14] also remarks that when using the Talbot contour he had difficulties accurately inverting the Bessel function. This may related to the combination of the singularity on the imaginary axis and the branching nature of the square root function.

Abate-Valko [1] stated that they were able to overcome this by increasing the weights and hence the precision as a function of t. However, we were unable to replicate their results for this function.

Overall, the Abate-Valko scheme showed far greater accuracy than Logan’s across all the functions tested. However, Logan’s algorithm was still able to produce highly accurate results with RMS errors varying between 10(−60) to 10(−63). Moreover, Table 8 shows that Logan’s scheme was able to perform the inversion of these functions with shorter elapsed times.

The Stehfset and Salzer-Gaver algorithms were able to invert all the functions to a high degree of accuracy. The Salzer-Gaver scheme was in general about twice as accurate as the Stehfest algorithm which, was less accurate than Abate-Valko’s scheme. Nevertheless, the Stehfest scheme inverted the functions

well within any generally desired accuracy with the RMS error varying from 0.0 to 10(−41). Moreover, as Table 8 shows it in terms of the elapsed time it was the fastest of all the algorithms for the most part twice as fast as the Abate-Valko scheme which in turn was at least twice as fast as the Salzer-Gaver scheme.