A Modification of Talbot’s Method for the Simultaneous Approximation of Several Values of the Inverse Laplace Transform MARIAROSARIA
RIZZARDI
Istituto Universitario
Navale
In recent years many results have been obtained in the field of the numerical inversion of Laplace transforms. Among them, a very accurate and general method is due to Talbot: this method approximates the value of the inverse Laplace transform f(t), for t fixed, using the complex values of the Laplace transform IVs ) sampled on a suitable contour of the complex plane. On the basis of the interest raised by Talbot’s method implementation, the author has been induced to investigate more deeply the possibilities of this method and has been able to generahze Talbot’s method, to approximate simultaneously several values of f( t) using the same sampling values of the Laplace transform. In this way, the only unfavorable aspect of the classical Talbot method, that is, that of recomputing all of the samples of IVs) for each t, has been eliminated. .Aualysis]: General—error analysis; Subject Descriptors: G. 1.0 [Numerical G. 1.2 [Numerical Analysis]: Approxirnation-nonhnear approximation; G, 1.4 [Numerical Analysis]: Quadrature and Numerical Differentiation-equal interval integration; error analysis; G.1.9 [Numerical Analysis]: Integral Equations—Fredholm equations
Categories
numerical
General
and
algorithms;
Terms: Algorithms
Additional numerical
Key Words and Phrases: Complex method, TALBOT, trapezoidal rule
inversion
formula,
inverse
Laplace
transform,
1. INTRODUCTION The Laplace absolutely
transform integrable
of a function on any finite
F(s)
= /me-s’f(t) o
f(t), interval
dt,
defined (O, a],
on the interval is defined
(O, + CJ) and
as follows:
Res>aO,
where a. is the Laplace transform abscissa of convergence. The inverse f(t) from known values of ~(s). Laplace problem is that of reconstmcting
Author’s address: Istituto Universitario Navale, Facolta di Scienze Nautiche, Istituto di Matematicaj via A. De Gasperi, 5-80133 Napoli, Italy; email:
[email protected]. Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of ACM. Inc. To copy otherwise, to republish. to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. 01995 ACM 0098-3500/95/12000-0347 $03.50 ACM Transactions
on Mathematical
Software,
Vol. 21, No. 4, December
1995, Pages 347-371.
348
Marlarosaria
.
The
Laplace
problems
in
Rlzzardi
transform the
represents
fields
of science
a very and
effective
tool
engineering;
for solving
however,
the
several
numerical
inversion of Laplace transforms still remains a very difficult problem to be solved in general. Among the methods of inversion based on the complex contour integration of the Riemann Inversion Formula, 1983; 1990; Talbot 1979] has proved Laplace
transform
to compute
functions
~(t).
computational pends If
very
need
to
transform
desirable
that,
mate just
a single
range
inefficient.
compute
the
of f(t),
same
for
may
also be stated
a particular
suitable possible? In this cation form
without
to Talbot’s
considerations
that
several
uses the values
in the
The Riemann
theory,
same
results
question, been chosen
for
any
Moreover,
of the
t in a
how
is it
a modifi-
Laplace
transform.
of
Some
transpractical
too.
BACKGROUND
Inversion
of its Laplace
Formula
transform
gives
an integral
function
F(s),
that
1
f(t)
‘
—
27Ti
where
B is the Bromwich
to the
imaginary
ties
have
we introduce
are reported
be
any value
again
samples
inverse
to approxi-
previous
result?
of its inverse
and some numerical
2. THEORETICAL
to the
de-
it would
computed
to use them
loss of accuracy
method
method,
parameters
And what is the cost? article, starting from the underlying
to approximate
terms
answer
if Talbot’s
of t, is it possible
value
interval
as follows:
the
algorithm
f(tk ) of an
of 3’(s),
for
that
evaluations.
can also be used to approximate
when t belongs to a suitable interval. This work attempts to give a positive
which
values
to use Talbot’s
values
known
inversion function
several
and we want
for efficiency, value
In fact, it is well
transform
numerically
function,
we need
all of the computations
transform
of Laplace
Rizzardi range of
of t where
of values
to repeat
of a Laplace
on the number
Laplace
f(t)
it needs
may become complexity
heavily we
and to a wide
Unfortunately,
each t,and this
Talbot’s method [Murli and to be applicable both to a wide
axis.
\ -B
estl’(s)
contour The
ds,
from
contour
representation
i.-
t>o,
to the
in
(2.1)
y – i~ to y + i~, with
is located
for ~(t)
is,
right
y > u-., parallel
of all
the
singulari-
of F’(s).
In Talbot’s
method
the Bromwich
contour
is replaced
by an equivalent
one,
which retains a proper balance between the two following effects: the new contour is moved as much as possible to the left so as to reduce in magnitude of the factor e ‘~ in (2.1), but it must not be moved too close to singularities F(s),
as to do so will
Referring
to Murli
result
in peaks
and Rizzardi
in the integrand
[1983;
1990]
function.
and Talbot
[1979]
for further
details about Talbot’s method and its implementation, here \\-e only outline those aspects that seem to be relevant to the method to be introduced. ACM
TransactIons
on Mathematical
Software,
Vol
21 Xo
4 December
1595
A Modification Talbot’s
method
transform
requires
function
that
3’(s)
the convex
be known
IimlF(s)l ~+.
F(s)
applicable
needs
Talbot’s
contour
in
Re(s ) 0,
Iim El(t,
n) = O
(2.llC)
n) = O.
(2.lld)
n
Vt >0,
limllz(t, n
The last two properties tell us that, for any possible value Talbot’s method is uniformly convergent for t >0. Nevertheless, it consists of a suitable choice, as a function
of A, a,
and
P,
of t, of all the
Talbot parameters for assuring a high rate of convergence, and this implies that, if several values of f(t) have to be evaluated, Talbot’s method needs to On the contrary, our idea is to look at a repeat aH of the computations. suitable which
choice is the
of
domain
To the previous
Talbot’s
parameters
where
truncation
as
defined by Talbot as the roundoff error sion, the middle term of the trapezoidal
E,(t, ACM
TransactIons
on Mathematical
a function
of
the
f(t) has to be evaluated. errors, we must add the roundoff
n) =&’ Software,
(
whole
error,
in computing, to the machine sum approximating (2.4):
10-C
interval,
which
is
preci-
Q(O, t) n’
(2.12) )
VOI 21, No 4, December
1995
A Modification where
c is the
computations This
decimal
will
formal
precision
be carried
definition
of the
of Talbot’s Method
floating-point
351
.
system
where
the
out.
of the roundoff
error
is justified
by the fact
that,
in
the classical method, usually the middle term in the sum is the largest; so the roundoff introduced in the summation process will be negligible in respect to the final result. But, as we will modified method,
see later, so we will
the estimate (2.12) is no longer reliable need to examine its stability closely.
for the
3. ERROR ANALYSIS Let us choose Talbot’s
parameters
A, a,
v, and
n at t* and use them
also for
Vt G [tmin, tmax]. In the following
sections,
we will
(i.e., the method modification the
that repeats all modified method
fied method
be built
will
call Talbot’s of the (which
step-by-step
method
computations avoids that
by examining
the
classical
for each repetition).
the error
method
t) and our This modi-
components.
3.1 Convergence First,
we want
to derive
the convergence
the classical method. Indeed, this assured by the last two properties emphasizes the following result: THEOREM
fied
dependence
3.1.1.
method
PROOF.
If we
m
- f(t)
the
term
[ f(t”
of the
If the classical
is convergent
of the modified
method
is not necessary since the of (2.11), but the following method
Talbot
t and
on
method
n. Let
converges,
from
that
of
convergence is analysis well us
then
prove
the
the modi-
too.
write
= [m
-m]
) – f(t )1 does not
+ [mm
depend
- f(t”)]
the term
ACM
[~
Transactions
– ~]
- f(t)],
(3.1)
on n, but only on t and ~(t). The
term [~ – f(t* )1 is the error of Talbot’s for n - cc by the last on t,and it vanishes Let us examine we have
+ [ f(t”)
method at t*; two properties with
on Mathematical
it does not dePend in (2.11).
respect
to t and
n. From
Software,
Vol. 21, No. 4, December
(2.8)
1995.
352
.
Mariarosar!a
Rizzardr
then, for n + CO, e-n’ - () on Integral Theorem, we have that
=& and by (3.1), it follows lim
Ml
and
e-n’
+ m on
&f2;
)(a+As(2)) - I)cJz
jMQ(z,t*)(e(t-t*
So by
Cauchy’s
=f(~)
-f(~*)
that [~-~(t)]
V t > 0
uniformly
= O,
n-x since, 3.2
by the hypothesis,
Talbot’s
method
❑
is convergent.
Stability
To emphasize the role played by roundoff in our modified method, recall from Murli and Rizzardi [1983; 1990] that the numerical Talbot’s method may be expressed as
we must result of
Ae[’t m= —-Tn(t), n
(3.2)
and (3J= j(n/n). Formula (3.2) has been obtained rule approximation to (2.4), after taking symmetry substitution kfz G M, so that
Talbot’s
contour
z=2i6,
in (2.3) is parameterized
s(O)
=
CT+
by applying the trapezoidal into account and after the
AsL,(@),
as
/3=(
-T,7r),
where s,,(o) All have
of Talbot’s an error
(3.2) and (3.3),
parameters that
have
= Ocoto+ been
rewritten
for
t’,
TransactIons
on Mathematical
at
t*,
accuracy
so f( t* ) is supposed requirement,
we have
Ifil ACM
chosen
does not exceed the input
ivo.
Software,
S ftt.und(t’, Vol
n),
21, No 4, December
1995
to
and from
A Modification
of Talbot’s Method
.
353
where
Now,
from
(3.2) and (3.3), it follows
that
~emt Iml
= —lT.(t)/ n
~(a
+ A)l +
and
n–1
:eAt*+A(t–t*) 2 “+:,( The have
sequence Vj:
eAt’6Jcot
6,+
A(t–t*)~,
cOttIJ
6J)F(a+ AS,,(OJ))I.
{61 cot @j}, j = 1,...,
l O; more
U.
inverses
t
— ERRT.lLO$
~
ERRT.i.
~
ERRT.,.X
—%-+?---+3-
Fig.1.
Error
-l
plots fort
precisely,
the
10
5
t
~[1,
lO].
Laplace
transform
test
functions
are
Fl(s)
= 1/s2
F2(S)
= 1/(s
F3(s)
= atan(l/s)
~4(S)
=
S2/(S3
– 1)
= t
f2(t)
= e’
f,(t)= sin(t)/t +
8)
The behavior of the errors defined of test functions. More precisely, ACM
f,(t)
Transactions
f,(t)
= (e- 2~ + 2etcos(t@))\3.
above is surprisingly looking at the tests on Mathematical
Software,
homogeneous for a lot in the macrointeruals Vol
21, No. 4, December
1995.
356
Manarosaria
.
Rlzzardl
F1
Test function
Test function F2
102
104
10°
102
,0-2
10°
/
8 k
a)
I
&
10“4
$10-2
,0.4
,0-6
,0-6
108
, ~.8
,0.10 o
20
40
60
o
20
40
t
60
t
Test function F3
Test function
F4
~~m 102
8 k
al
10° ,0.2 ,0.4
,(--’”~
I
o
20
40
O“60L
60
20
40
60
t
IL
I Fig.2,
Error
plOtsfort=
I
[10,50],
(Figures 1-3), we can observe that creases, as expected, but it becomes t + tmax, when compared to the other
ERR~~,X decreases and ERR~~,m . . . .. inquickly very large (often overflow) for errors. (Figures 4–5), the On the contrary, looking at plots in the micromtervals swap the roles between them, remaining of a similar magnitude; two errors but usually Talbot’s parameters chosen at the lower bound of the interval give more accuracy at a minor computational-effort: this can be explained by the instability of the method that computes f~~,X. According to the previous analysis, from “small” error ERR ~~,,( t ) always occurs for the Talbot error analysis’): this phenomenon ACM
Transactions
on Mathematical
Softww-e,
Vol
all
of these
tests
we see that
a
t < t* = tmax(incontradiction to is due to the fact that for t < t*
21, No 4, December
1995
A Modification
of Talbot’s Method
357
Test function F1 1020 ~ L
1o 10
15 10
05 10° 0.5
1 1
:
0.10 ~
500
o
1000
t Test function F3 1040
Fig. 3.
Error
plots for t ● [50, 1000].
1030 ~ 1020 L
$10’0 10°
,o-lo~~ o
1000
500
t
= we have chosen a contour unnecessarily close to the singularities of F(s), and this leads to a roundoff error in the trapezoidal sum that exceeds the E, in (2.12) as defined by Talbot. So, from results of the previous error analysis, we can conclude that the new method shows a similar performance (in terms of accuracy) over the whole
interval
[ t~,.,
t* such that
—we
choose
—we
maintain
The former
t~,X 1 if
the value
requirement
it minimizes
elt –‘* 1 and
of n A as small is obviously
as possible.
satisfied
by (3.7)
t* = tmld = :(tm,n + tma,). ACM Transactions
on Mathematical
Software,
Vol
21, No 4. December
1995
358
Manarosana
.
Rlzzardl
Test function F1 ‘0-4
Test function
~
’02
, ~.5
F2
~—
10°
,0-6
,0.2
z
s
:10”7
k
10”4
a)
,0-8
,0-6
,0-9
,0-8
,0-10
,0-10
2
1
3
4
I
2
1
3
t
4
t
Test function
F3
Test function
F4
105
104 ~ 102 10°
10° 8 k a)
,0.4 ,0.6
1
, ~.e 2
1
3
t
4
r== t .EzzzZ. I
ERRT,lLax
Fig.4
The latter into
is difficult
account
examine
them
4. ERROR Before
the
in
Error
pIotsfort=[l,4]
to be satisfied
other
the
error
next
because
components
as
the value well,
of n, in order
tends
to
large.
to take We
will
section.
ESTIMATION
introducing
components
the modified
of the classical
method,
method.
we need to examine
In this
way,
we will
Transactions
on Mathematical
Software.
Vol. 21, No 4, December
1995
closely
the error
be able to estimate
the number of function evaluations necessary to guarantee, method, the same accuracy the classical method has. ACM
be
in the
modified
A Modification
of Talbot’s Method
Test function F1
Test function
.
359
F2
1 , ~-5
1
~ 10-’ k al ,0.7
1
, ~-8
1 4
6
8
,0-9 ~ 4
6
4
1
t
Test function F3
Test function F4
8
‘ 0-3~
,0.4 ~ 10”5 k al ,0-6 ,0.7 ,&
~ 4
6
8
t
t
,
Fig.5.
Let us use the Talbot
Error plots fort
parameters
~[4,8].
A, u, v, and
n, chosen
at t*,for each t in
since the global error grows as t moves from t* to t~,n or the Talbot error components at the endpoints of the t~ax, let us consider t belongs to. We do not consider the roundoff error Er(t,n) interval that [%n>t
defined roundoff First,
~ax ], and
in (2.12) since, as shown in Section 3.2, Talbot’s estimation of error is no longer reliable. consider the error component E1(t, n) defined by (2.9). From (2.11)
the integrand
in (2.9) is regular
arc Ml may be any path from substituted by the parabola rl VZ
G
rl ACM
eZ
in the halfstrip
=2(7)
TransactIons
If
defined
in (2.6);
then
the
– 2 n i to 2 n i, and in particular, Ml may be from –2mi to 2wi with the vertex at z = p:
=x(T)
+ iy(~),
on Mathematical
Software,
Irl o
– 72),
Since rl is free to get deformed in the halfstrip (0, +~). Then, by the Darboux Theorem, we have Ilczl(t, Remembering on El,
Iength( rl ) ~m max z~r,
n)l
=x(T)
171s 1
and
But,
unlike
rl,
X(T)
= –p(l
y(~)
= 2rr7_.
rz cannot
be arbitrarily
be located on the right t+ +~, then p~O. Repeating function
the
of all
analysis
the
carried
deformed singularities
out
for
El,
o
in the halfstrip of
K; it must
I’( a + AsU(z)).
applied
in
this
So, for
case
to
the
eKc+As, (z)) e for sufficiently the error
large
Ez that
values
l~~(t,
‘
the dependence
n)l < O(A, of p, this
–1
of n, we are able to establish
emphasizes
lE2(f, For smal
values
–nz
a, v) .e~[~-~p[u-
of this
error
lJ/~+~p/(e’-l)]
an upper on t and
bound
to
n:
-nP$
becomes
n)l < @(A, u, v) .e’iU+’-’~(~-l)/2]-~P,
(4.5)
where
p s such that r, is located, in the halfstrip K, on the right of all the singularities of 3’( a + ~sp( z )). Also for Ez, choose the Talbot parameters A, u, v, and n at t“, and use them also for t~,. and t~,,. Writing (4.5) and t equal to tm,n,t*, and t~,X, at tm,n. respectively, we see again that the error is bigger at tmax and smaller ACM
Transactions
on Mathematical
Software,
Vol. 21, No, 4, December
1995.
362
Marlarosarla
.
Rlzzardl
Test function ‘0-2
Test function
F1 ’00
~
F2
~ I
, “.2
10-8
,o-lo~ o
10
5
t
t Test function
Test function
F3
F4
10°,
10°
1
I
10
t
t
I
‘RR’’’’’” ~ Fig,6,
Then,
if we wish
n(z) ml n for
t~,.,
to find
such
a suitable
plots
fort
value
=[0,1,
lO],
of n, namely,
n(#X
for
t~~,
and
as
lE,(tmln, from
Error
(4.5), it suffices
ng:n)l
=
to choose ~(a
m] n
= ~ — (4.6)
which ACM
once again TransactIons
returns
on Mathematical
n(~~~ < n. Software,
Vol. 21, No 4, December
1995
A Modification
,
Test function FI
~-6
lD
z k
10”6
al
,0.9
~ 0.7
, ~-lo
,0-8
0
20
40
L
60
‘
0
20
Test function F3
,0-4
Test function ‘ 0-2 ~
T
,0.3 r ~ 10-4[ k al ,0.5 ,0-6
,(--@~ o
-t
1O-’L
20
40
o
60
60
F4
x
3 20
40
60
t
t
F1g.7.
MODIFIED
40
t
t
5. THE
363
,0-5
10”8
.-
.
Test function F2
,0.4
, ~.7
8 k
of Talbot’s Method
TALBOT
Error
plots fort
~[10,50].
METHOD
Taking into account the previous considerations and both formulas for n&).X in (4.4) and (4.6), the simplest computational strategy consists of the following algorithm: —By
a call to the subroutine
rical parameters (according as stated in (3.7).
TAPAR
[Murli
to Talbot’s
and Rizzardi
method)
1990],
are computed
the geometfor
t = t~,d,
—A correction to the accuracy parameter n is then computed by choosing the formulas (4.4) and (4.6) when t“ ==tm,d. maximum value of n’~~, between ACM
TransactIons
on Mathematical
Software,
Vol
21, No. 4, December
1995
364
.
Marlarosaria
Rizzardl Test function F1 ‘0”2
~
,o-’”~ 1000
500
o
t Test function
F3
102 Fig.8.
Error
plots fort
G [50,1000]. 10°
,0.2 G k a) ,0.4
,0-6
10-8
500
0
1000
t
==J —Finally,
to compute
n~~X in (4.6),
we need
to know
a value
of p such
that
the corresponding parabola rz is located on the right side of all of the singularities of F( a + Asv( z )). A numerical value of p can be obtained, for example, by assuming that the image of the vertex of 17z, SU(—p), agrees with the midpoint between cro and m + A, the rightmost real part on the Talbot contour, i.e., mo+u+A ,o>O:u+As,,(-p)=
which
ACM
leads
TransactIons
to the following
on Mathematical
2’
nonlinear
Software,
Vol
equation:
21, No 4, December
1995
A Modification
of Talbot’s Method
Test function F1
.
365
Test function F2 ,0-2
,0-4
& k
al
10-6 ,0-6
, ~-lo .-
,()-lo~
0
5 t
o
10
Test function F3
,0.3
5 t
10
Test function F4
,0.2
[
,0.4 ,0-4
~ 10-5 S k
k al ,0-6
a
10“6 10-8
,0-7 ,0-8
,(yo~
0
5
10
5 t
o
t
Fig. 9.
This
equation
6. NUMERICAL Numerical assuring First, classical
the
of the radix
solved just
by a Newton
using
have
shown
inverse
that
Laplace
the
modified
transform
over
Talbot
that
and with ACM
Talbot’s
Transactions
method
parameters
on Mathematical
method
a relatively
the same accuracy of Talbot’s method almost we report on some of the results of comparing method
process
gives
a
a few iterations.
EXAMPLES
results
approximate
plots for t ● [1, 10].
Error
may be numerically
good approximation
10
Software,
large
everywhere. our method always Vol
is able
to
interval, with
chosen
the
at t~,d
21, No 4, December
1995.
.
366
1
Manarosaria
Rizzard!
Test function FI
~.4
Test function
F2
‘“”’~
~-5
,0.4
1
0-6
1
~ 10”5 k a ,0-6
L ~.7
:1 al
1 1
~-8
%
:;L4
~.9 ~.lo
1
o
20
40
o
60
20
t
40
60
t
Test function
Test function
F3
‘ 0“4 ~—–
‘ 0“2 ~~
,0-5
1
F4
0-6
m
10’
I
d
L
1
, ~.t?
1
o
40
20
Fig
60
0
Error
plots fort
in Figures
—ERR~,lbO,
method;
of the classical
—ERR~~O~,~ : the error
of our modified
—ERR~~,~
obtained
: the
error
method;
by using
60
=[10,50]
for each t G [t ~,~, t~,, ]. In particular, ing errors: : the error
40
t
e, 10.
20
6-8
we compare
the follow-
and
V t Talbot’s
parameters
chosen
at
tmid. P1ots show that ‘RR Tmod,f iS always less than ER%~,d: both use the same geometrical parameters h, c, and v, but they differ in the value of the accuracy parameter n (which in “Tmodif” is larger than for “Tmid”) to take into account the whole interval that t belongs to. ‘hese
ACM
TransactIons
on Mathematical
Software,
Vol
21, No 4. December
1995,
A Modlflcatlon
of Talbot’s Method
.
367
Test function F1
, ~.2
, ~-4
& k 10-6 al ,0-8
,0.10 0
500
1000
t Test function F3 Fig. 11.
Error
plots fort
e[50j
1000].
,o-J_____4 o
500
1000
t
Second,
in Figures
9– 11 we compare
the errors
ERR~.lbOt,
ERR~~O~l~,
and
> where the first two are the same as before, and where ERR~~., (defined in Section 3) is related to Talbot’s method with parameters chosen at t~ax and using Vt e [tm,n,tmax]. These results are very encouraging. As expected, in the middle part of the interval the aim is always achieved. Often, the accuracy is better than the classical Talbot’s method. induces a But, sometimes, when t is small, the closeness of singularities However, when t is small, then “case 1“2 holds, loss of accuracy near t = tm,n.
ERRTmax
and
in “case
1“ the
values
of n in the
classical
method
“’Case 1“ occurs, in Talbot’s method, when F’(s) has only real singularities complex singularities for moderate t (usually t < 10). ACM Transactions
on Mathematical
Software,
are
so small
(e.g.,
‘dt or when F(s) has
Vol. 21, No. 4, December
1995.
368
.
Mariarosana
Rlzzardi
Test function
Test function
FI
14! 13 I
1
18
I
16
.
F2
n n
.
.
s
~ 14 fl) K 12 ~ 10
b *
8
‘~
246810
246810
t
t Test function
Test function
F3
F4
35
20 L-3-e-J
30 ~ 25 > :20 ‘ z
.
/’.
n
n
o
it may
be more
L
15
246810
t
t 1
Fig. 12.
n s 11 for
a required
Number
accuracy
of function
of five
evaluations
decimal
fort
digits)
=[1,
lO]
that
convenient to repeat the whole computation for each value of t. Or a simple trick may be used: enlarge the interval [ t~,~, t~~, ] toward O when establishing the method’s parameters. Figures 12– 14 report the number of function evaluations (n) used by the three different strategies: respectively, the classical method, Talbot’s method at t~,,, and the modified method; more precisely, we report the number of function evaluations required to approximate just a single value of an inverse Laplace transform: if we need to compute, for example, 100 samples of ~1(t) 12 we get that the classical for t G (O, 10], then from the first plot in Figure method will require 1000 function evaluations; Talbot’s method at tmax will ACM
TransactIons
on Mathematical
Software,
Vol. 21, No 4, December
1995
A Modification
of Talbot’s Method
369
.
Test function F2
Test function F1 35
I
13
30 3“”
15 1, 10
I
20
30
40
U”””
”””3”
I
I
I
1.
50
10
20
30
40
t
t
Test function F3
l-est function F4
50
4’ ~ 40 t
I
200 150
n
n
250
—1 I
t
I
t
20
15 I
L
10
20
30
40
50
10
20
30
40
50
t
t ~Talbot
?aTn,odif
“
u
“
nTmax Fig. 13.
Number
of function
evaluations
fort
=[10,50].
require 10 function evaluations; and the modified method will require only 12 function evaluations (giving more accuracy). This means that the global number of function evaluations, in the classical method, is the sum of all the function evaluations carried out for each value of t, while in the other two strategies, this is constant. Pictures report the number of function evaluations of Talbot’s method only to give
an idea
of how
much
the value
of n has been
raised
by the modified
method, since it is meaningless to compare the classical method and our modification from a computational point of view. Finally, from these plots we can see that the modified method produces, for gives at tmax; small values of t, a value of n larger than the Talbot method ACM
Transactions
on Mathematical
Software,
Vol. 21, No 4, December
1995.
370
Marlarosaria
.
Rizzardi
F1
Test function
14 13 9
12 ~ 11 & & & A .k *
10 ~~~~
*-
9 8 200
400
600 t
800
1000
Test function F3 Fig ?!=
14
Number
of function
evaluations
for
[50, 1000].
~-200
400
600 t
800
1000
n~ajbo~ n’h.
d,f
“
i nTnl.x
this
occurs
when
the
methods.
Otherwise,
a minor
computational
7. CONCLUDING A modification
transform
method
is based
on an
required. In this also ACM
way,
TransactIons
method
that
function
evaluations
is
is able to produce
not only
function
is the high
strategy on Mathematical
has
formula
F(s)
for
accuracy
is reached Software,
approximates
~(t)
approximation
transform
an efficient
method
small more
for
both
accuracy
at
cost.
Talbot’s
Laplace
Laplace
of function
1
+--+-+
REMARKS of
inverse the
number
the modified
“
Vol
that
been that
all
several introduced. uses
of the of Talbot’s
applies
the
values
well
21, No. 4, December
values The
same
of ~(t)
method
of
samples
that
retained,
to multidimensional 1995
an
modified of
are but
A Modification Laplace
inversion
tional
efficiency
problems. of the
algorithm,
or it
transform,
otherwise
Moreover,
summation
is possible
not usable
it is possible algorithm,
to use
of Talbot’s Method
for
prescribed
to enhance example,
equally
by the classical
the computavia
spaced
371
.
a parallel
values
of the
method.
REFERENCES A. AND RIZZA~DI, M. 1983. Sull’implementazione numei-ica dells Trasformata di Laplace. Rep. SOFMAT
MURIJ,
del metodo di Talbot per la inversion 10.83, Progetto
Finalizzato
Informatica
del CNR, Rome, Italy. MURLI,
A. AND RIZZARDI,
problem.
M.
1990.
Algorithm
ACM Trans. J4ath. SoftLV. 16,
TALBOT, A. 1979.
The accurate
numerical
682:
Talbot’s
method
for
the
Laplace
inversion
2. inversion
of Laplace
transforms.
J.
Inst. Math. Appl.
23. Received May 1992; revised July
1993 and August
ACM Transactions
1994; accepted September
on Mathematical
1994
Software, Vol. 21, No. 4, December
1995.
