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```Hi all,

It is the same problem but a different question(equally challenging). I want
to use the numerical integration to do the Bromwich type integral, as shown
in the URL below:

http://en.wikipedia.org/wiki/Bromwich_integral

It's an integral on infinite interval.

I have tried several standard methods of transforming an infinite integral
to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of them
gives correct result.

The naive trunction, for example, setting MAX=100000, and then do:

at least give quite good results.

In real engineering applications, it is rarely true that I only need to deal
with one integral with fixed parameters. I often have to vary the parameters
programmatically and automatically. For some parameter sets, a large MAX
gives "inf" or "nan" resutls, because for too large MAX, the integrand
becomes overflowed and return non-numbers; a too small MAX might not have
sufficient coverage. And also the choice of MAX affects the speed.

Any more approaches I can try out? Please give me some pointers in

Thanks a lot!

```
 0
Reply lunamoonmoon (258) 7/26/2007 12:03:38 PM

See related articles to this posting

```On Jul 26, 3:03 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
> Hi all,
>
> It is the same problem but a different question(equally challenging). I want
> to use the numerical integration to do the Bromwich type integral, as shown
> in the URL below:
>
> http://en.wikipedia.org/wiki/Bromwich_integral
>
> It's an integral on infinite interval.
>
> I have tried several standard methods of transforming an infinite integral
> to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of them
> gives correct result.
>
> The naive trunction, for example, setting MAX=100000, and then do:
>
>
> at least give quite good results.
>
>
> In real engineering applications, it is rarely true that I only need to deal
> with one integral with fixed parameters. I often have to vary the parameters
> programmatically and automatically. For some parameter sets, a large MAX
> gives "inf" or "nan" resutls, because for too large MAX, the integrand
> becomes overflowed and return non-numbers; a too small MAX might not have
> sufficient coverage. And also the choice of MAX affects the speed.
>
> Any more approaches I can try out? Please give me some pointers in
>
> Thanks a lot!

Doetsch Gustav, 1956.
Anleitung zum praktishen gebrauch der Laplace-transformation.
R. Oldenbourg Munchen 1956. 198 pages.

Hannu

```
 0
Reply haporopu (2) 7/26/2007 4:44:41 PM

```"Luna Moon" <lunamoonmoon@gmail.com> writes:

> Hi all,
>
>
>
> It is the same problem but a different question(equally challenging). I
> want
> to use the numerical integration to do the Bromwich type integral, as shown
>
> in the URL below:
>
>
>
> http://en.wikipedia.org/wiki/Bromwich_integral
>
>
>
> It's an integral on infinite interval.
>
>
>
> I have tried several standard methods of transforming an infinite integral
> to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of them
>
> gives correct result.
>
>
>
> The naive trunction, for example, setting MAX=100000, and then do:
>
>
>
>
>
>
> at least give quite good results.

Actually this approach can easily lead to wrong results.  A case in point:
integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods will
return answers very close to 0, because they don't happen to sample the
function in the small interval around x=10 on which it is not close to 0.
--
Robert Israel              israel@math.MyUniversitysInitials.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada
```
 0
Reply israel1 (48) 7/26/2007 6:15:21 PM

```"Hannu Poropudas" <haporopu@luukku.com> wrote in message
> On Jul 26, 3:03 pm, "Luna Moon" <lunamoonm...@gmail.com> wrote:
>> Hi all,
>>
>> It is the same problem but a different question(equally challenging). I
>> want
>> to use the numerical integration to do the Bromwich type integral, as
>> shown
>> in the URL below:
>>
>> http://en.wikipedia.org/wiki/Bromwich_integral
>>
>> It's an integral on infinite interval.
>>
>> I have tried several standard methods of transforming an infinite
>> integral
>> to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of
>> them
>> gives correct result.
>>
>> The naive trunction, for example, setting MAX=100000, and then do:
>>
>>
>> at least give quite good results.
>>
>>
>> In real engineering applications, it is rarely true that I only need to
>> deal
>> with one integral with fixed parameters. I often have to vary the
>> parameters
>> programmatically and automatically. For some parameter sets, a large MAX
>> gives "inf" or "nan" resutls, because for too large MAX, the integrand
>> becomes overflowed and return non-numbers; a too small MAX might not have
>> sufficient coverage. And also the choice of MAX affects the speed.
>>
>> Any more approaches I can try out? Please give me some pointers in
>>
>> Thanks a lot!
>
> Doetsch Gustav, 1956.
> Anleitung zum praktishen gebrauch der Laplace-transformation.
> R. Oldenbourg Munchen 1956. 198 pages.
>
> Hannu
>

Any English translations?Thanks!

```
 0
Reply lunamoonmoon (258) 7/27/2007 4:22:03 AM

```"Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20070726180249\$0ff7@news.ks.uiuc.edu...
> "Luna Moon" <lunamoonmoon@gmail.com> writes:
>
>> Hi all,
>>
>>
>>
>> It is the same problem but a different question(equally challenging). I
>> want
>> to use the numerical integration to do the Bromwich type integral, as
>> shown
>>
>> in the URL below:
>>
>>
>>
>> http://en.wikipedia.org/wiki/Bromwich_integral
>>
>>
>>
>> It's an integral on infinite interval.
>>
>>
>>
>> I have tried several standard methods of transforming an infinite
>> integral
>> to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of
>> them
>>
>> gives correct result.
>>
>>
>>
>> The naive trunction, for example, setting MAX=100000, and then do:
>>
>>
>>
>>
>>
>>
>> at least give quite good results.
>
> Actually this approach can easily lead to wrong results.  A case in point:
> integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods
> will
> return answers very close to 0, because they don't happen to sample the
> function in the small interval around x=10 on which it is not close to 0.
> --
> Robert Israel              israel@math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada

I also felt that way. Any smart integrator around?

My function is mostly dense around 0, is that the region where most
integrators pay attention to?

```
 0
Reply lunamoonmoon (258) 7/27/2007 4:23:16 AM

```> > Doetsch Gustav, 1956.
> > Anleitung zum praktishen gebrauch der Laplace-transformation.
> > R. Oldenbourg Munchen 1956. 198 pages.
>
> > Hannu
>
> Any English translations?Thanks!

Unfortunately I found only French and no English translation:

Introduction a l'utilisation pratique de la transformation de Laplace.
by Gustav Doetsch
Language: French  Type:   Book
Publisher: Paris, Gauthier-Villars, 1959.
OCLC: 1302926

Hannu

```
 0
Reply haporopu (2) 7/27/2007 1:26:46 PM

```Robert Israel wrote:
>
> > I have tried several standard methods of transforming an infinite integral
> > to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of them
> > gives correct result.
> >
> > The naive trunction, for example, setting MAX=100000, and then do:
> >
> >
> > at least give quite good results.
>
> Actually this approach can easily lead to wrong results.  A case in point:
> integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods will
> return answers very close to 0, because they don't happen to sample the
> function in the small interval around x=10 on which it is not close to 0.

Would you mind identifying a few so we'll know what to avoid in the
future. If your broad claim has some validity then I must have been
quite fortunate as none of my quick picks returned anything close to
zero, but rather nailed the result to - sqrt{pi) - very nicely!?

- qagi, Gauss-Kronrod quadrature for inf intervals (netlib)
- rkm,  modified Merson (4,4) integrator (in-house stock)
- pid,  adaptive spline fit integral controller (sdx function)

You need to be careful with such claims w/o supporting it with a few
independent experiments[*]. Elsewhere *iirc*, someone suggested
Spellucci's tenure be yanked for a similar offense... *wadr*, it's
irresponsible for such posts to originate from edu addresses where the
uninitiated may consider them gospel, rather than a reality deficient

[*] entirely different algorithms, with pid controller further

---
sdx - modeling, simulation.
http://www.sdynamix.com

```
 0
Reply jrw (34) 7/31/2007 11:25:48 PM

```widmar <jrw@sdynamix.com> writes:

> Robert Israel wrote:
> >
> > > I have tried several standard methods of transforming an infinite
> > > integral
> > > to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of
> > > them
> > > gives correct result.
> > >
> > > The naive trunction, for example, setting MAX=100000, and then do:
> > >
> > > "quadl(integrand, -MAX, +MAX);"
> > >
> > > at least give quite good results.
> >
> > Actually this approach can easily lead to wrong results.  A case in
> > point:
> > integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods
> > will
> > return answers very close to 0, because they don't happen to sample the
> > function in the small interval around x=10 on which it is not close to 0.
>
> Would you mind identifying a few so we'll know what to avoid in the
> future. If your broad claim has some validity then I must have been
> quite fortunate as none of my quick picks returned anything close to
> zero, but rather nailed the result to - sqrt{pi) - very nicely!?

The Maple 11 default method does get it right (it cheats by looking for
critical points).

Maple 10 default method (this uses the NAG procedure d01ajc)...

> evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000));

0.

.6491348282e-39

> evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_Dexp));

.4591383855e-38

0.

> evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_Sinc));

.1563477515e-37

> evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_NCrule));

.1296493370e-39

Matlab Version 6.0.0.88 Release 12

ans =

9.9202e-40

ans =

1.7006e-39
--
Robert Israel              israel@math.MyUniversitysInitials.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada
```
 0
Reply israel1 (48) 8/1/2007 12:26:00 AM

```On Jul 31, 8:26 pm, Robert Israel
<isr...@math.MyUniversitysInitials.ca> wrote:
> widmar <j...@sdynamix.com> writes:
> > Robert Israel wrote:
>
> > > > I have tried several standard methods of transforming an infinite
> > > > integral
> > > > to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of
> > > > them
> > > > gives correct result.
>
> > > > The naive trunction, for example, setting MAX=100000, and then do:
>
> > > > "quadl(integrand, -MAX, +MAX);"
>
> > > > at least give quite good results.
>
> > > Actually this approach can easily lead to wrong results.  A case in
> > > point:
> > > integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods
> > > will
> > > return answers very close to 0, because they don't happen to sample the
> > > function in the small interval around x=10 on which it is not close to 0.
>
> > Would you mind identifying a few so we'll know what to avoid in the
> > future. If your broad claim has some validity then I must have been
> > quite fortunate as none of my quick picks returned anything close to
> > zero, but rather nailed the result to - sqrt{pi) - very nicely!?
>
> The Maple 11 default method does get it right (it cheats by looking for
> critical points).
>
> Maple 10 default method (this uses the NAG procedure d01ajc)...
>
> > evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000));
>
>                   0.
>
>
> > evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_CCquad));
>
>           .6491348282e-39
>
>
> > evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_Dexp));
>
>           .4591383855e-38
>
>
> > evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_Gquad));
>
>           0.
>
>
> > evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_Sinc));
>
>          .1563477515e-37
>
>
> > evalf(Int(exp(-(x-10)^2),x=-100000 .. 100000, method=_NCrule));
>
>          .1296493370e-39
>
> Matlab Version 6.0.0.88 Release 12
>
>
>
> ans =
>
>    9.9202e-40
>
>
>
> ans =
>
>    1.7006e-39
> --
> Robert Israel              isr...@math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada

HI folks,

This is really worrisome... what to do with these numerical issues?

I even don't know how to decide that 100000???

My program has to deal with all combinations of different parameter
values and there is no way to tell before-hand...

```
 0
Reply lunamoonmoon (258) 8/1/2007 1:38:44 AM

```On Jul 31, 4:25 pm, widmar <j...@sdynamix.com> wrote:
> Robert Israel wrote:
>
> > > I have tried several standard methods of transforming an infinite integral
> > > to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of them
> > > gives correct result.
>
> > > The naive trunction, for example, setting MAX=100000, and then do:
>
> > > "quadl(integrand, -MAX, +MAX);"
>
> > > at least give quite good results.
>
> > Actually this approach can easily lead to wrong results.  A case in point:
> > integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods will
> > return answers very close to 0, because they don't happen to sample the
> > function in the small interval around x=10 on which it is not close to 0.
>
> Would you mind identifying a few so we'll know what to avoid in the
> future. If your broad claim has some validity then I must have been
> quite fortunate as none of my quick picks returned anything close to
> zero, but rather nailed the result to - sqrt{pi) - very nicely!?
>
>    - qagi, Gauss-Kronrod quadrature for inf intervals (netlib)

Somehow I didn't notice this the first time: this is a procedure for
_infinite intervals_.  The integral I was discussing was on a finite
interval:
-100000 to 100000.  I'm not disputing the fact that typical numerical
integrators for infinite intervals might work in this case from -
infinity to infinity.

>    - rkm,  modified Merson (4,4) integrator (in-house stock)
>    - pid,  adaptive spline fit integral controller (sdx function)

Are these also on an infinite interval?

--
Robert Israel              israel@math.MyUniversitysInitials.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada

```
 0
Reply israel (123) 8/2/2007 5:06:15 AM

```On Aug 2, 1:06 am, isr...@math.ubc.ca wrote:
> On Jul 31, 4:25 pm, widmar <j...@sdynamix.com> wrote:
>
>
>
> > Robert Israel wrote:
>
> > > > I have tried several standard methods of transforming an infinite integral
> > > > to finite interval, such as tan(x), atanh(x), exp(x), etc. But none of them
> > > > gives correct result.
>
> > > > The naive trunction, for example, setting MAX=100000, and then do:
>
> > > > "quadl(integrand, -MAX, +MAX);"
>
> > > > at least give quite good results.
>
> > > Actually this approach can easily lead to wrong results.  A case in point:
> > > integrate exp(-(x-10)^2) from -100000 to 100000.  Most numerical methods will
> > > return answers very close to 0, because they don't happen to sample the
> > > function in the small interval around x=10 on which it is not close to 0.
>
> > Would you mind identifying a few so we'll know what to avoid in the
> > future. If your broad claim has some validity then I must have been
> > quite fortunate as none of my quick picks returned anything close to
> > zero, but rather nailed the result to - sqrt{pi) - very nicely!?
>
> >    - qagi, Gauss-Kronrod quadrature for inf intervals (netlib)
>
> Somehow I didn't notice this the first time: this is a procedure for
> _infinite intervals_.  The integral I was discussing was on a finite
> interval:
> -100000 to 100000.  I'm not disputing the fact that typical numerical
> integrators for infinite intervals might work in this case from -
> infinity to infinity.
>
> >    - rkm,  modified Merson (4,4) integrator (in-house stock)
> >    - pid,  adaptive spline fit integral controller (sdx function)
>
> Are these also on an infinite interval?
>
>  --
> Robert Israel              isr...@math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada

I have tested both infinite and finite integrators for my application.

It turns out infinite integrators are a lot more unstable than the
finite integrators -- many times the infinite integrators return NAN,
while the finite ones return correct values... of course the finite
ones are not stable sometimes too . That's the major headache...

```
 0
Reply lunamoonmoon (258) 8/2/2007 12:30:13 PM

```israel@math.ubc.ca wrote:
>
> >    - qagi, Gauss-Kronrod quadrature for inf intervals (netlib)
>
> Somehow I didn't notice this the first time: this is a procedure for
> _infinite intervals_.  The integral I was discussing was on a finite
> interval:
> -100000 to 100000.  I'm not disputing the fact that typical numerical
> integrators for infinite intervals might work in this case from -
> infinity to infinity.
>
> >    - rkm,  modified Merson (4,4) integrator (in-house stock)
> >    - pid,  adaptive spline fit integral controller (sdx function)
>
> Are these also on an infinite interval?

The "finite" bounds in your example are "infinite" for all practical
intent and purposes, thus a compression - when successful - of the
interval is inconsequential.

No, in both cases and for the same reason as the above. Nevertheless,
Merson integrator was tested on +/-10^5 to check if there's a long term
roundoff accumulation - there was none; pid controller was simulated for
an hour although a 30 sec run would have been more than sufficient.

In short, applied numerics and the extreme numbers do not mix well -
examples abound, your calamitous Maple/Matlab tests notwithstanding.

---
sdx - modeling, simulation.
http://www.sdynamix.com

```
 0
Reply jrw (34) 8/8/2007 11:11:48 PM

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