Page 17 - Laplace transform numerical inversion
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Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 -
p 15
One can see on our tested function in Figure
that inversion performance is impressive. Indeed,
the reconstruction error is close to machine precision. Furthermore, as can be seen on the sign
test curve, the inverted function nicely follows the analytical result and always remains positive.
This is precisely the behaviour that has been sought.
2
3
4
5
6
0
0
1
0.5
1
0
1
2
3
4
5
6
-1e-014
-5e-015
0
5e-015
1e-014
1.5e-014
Error: f(t) - Fp_inv(t)
Test: Fp_inv(t) < 0
Figure 14: Inversion of
f
(
t
) =
e
−
µt
by Den Iseger algorithm.
Undoubtedly, if Den Iseger algorithm did not require the Laplace argument
p
to be complex, it
would be a serious candidate to solve the issue that is discussed here.
5 Discussion
5.1 A word about algorithms running time
All along this study, we have not focused on algorithms running time. The main reason is that
they could not directly be compared. Indeed, no implementation optimisations have been sought,
but depending on the algorithm specifications, a variety of shortcuts could be found. Further-
more, several heterogeneous libraries have been used that contributed in either slowing down or
speeding up processing times.
The only element that can be compared is the number of times the function
ˆ
F
(
p
)
, defined in the
Laplace domain, has been scanned. This measurement is important since in industrial applications
this number of calls is at the heart of the processing. Before giving the figures, one first needs to
set a few parameters. The typical values given hereafter have be chosen in order to obtain the
best performance :
M
= 256
: number of points on which inverted function
f
(
t
)
is rebuilt
N
K
= 18
: number of terms
N
or
K
in the summation
Q
= 2
: Laguerre algorithm number of degrees of freedom
N
DFT
= 8
M
: number of points used for the Discrete Fourier Transform
N
L
= 8
: Den Iseger algorithm quadrature rule number points
Often, the number of calls to the function
ˆ
F
(
p
)
, defined in the Laplace domain, is proportional to
the number of points on which the function is, directly or indirectly, rebuilt in the usual space, i.e.
M
or
N
DFT
. This is not the case for the polynomials series expansion algorithms.
