Thread Subject: question about complex line (Bromwich) integral

Hi all,

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

The key is to find the line "x=c", where the integration is done along the
vertical line x=c in the complex plane such that c is greater than the real
part of all singularities of F(s).

Suppose I have identified all the singularities, and determined that in a
large range of

c>max(RealPartOf(All Singularities)),

the location of "c" should not matter and I should be able to freely vary
"c" as long as it doesn't violate our rules above, and the integration
should yield precisely the same values.

Now I plot the numerical integratal results along with varying "c" in the
safe region.

I found that the results differ, by about 10%. The change is very smooth.
From one end of "c_min" to the other end of "c_max", the integral values go
from low to high.

Here is my headache: with no closed-form solution exist for such integral,
only numerical solutions exist.

Which value of the numerical integral results shall I trust? (I used
Matlab's quadl, which is Gaussian Lobatto).

Secondly, the speed of integration varies a lot with the change of c. Does
anybody have some suggestions/pointers on what is a good theoretical
guideline of choosing "c" when both accuracy and speed are needed?

Thanks!

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

> Hi all,
>
> 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
>
> The key is to find the line "x=c", where the integration is done along the
> vertical line x=c in the complex plane such that c is greater than the real
>
> part of all singularities of F(s).
>
> Suppose I have identified all the singularities, and determined that in a
> large range of
>
> c>max(RealPartOf(All Singularities)),
>
> the location of "c" should not matter and I should be able to freely vary
> "c" as long as it doesn't violate our rules above, and the integration
> should yield precisely the same values.
>
> Now I plot the numerical integratal results along with varying "c" in the
> safe region.
>
> I found that the results differ, by about 10%. The change is very smooth.
> From one end of "c_min" to the other end of "c_max", the integral values go
>
> from low to high.
>
> Here is my headache: with no closed-form solution exist for such integral,
> only numerical solutions exist.
>
> Which value of the numerical integral results shall I trust? (I used
> Matlab's quadl, which is Gaussian Lobatto).

Under the circumstances, I wouldn't trust any of them. It certainly looks
like there's something wrong here. Can you tell us what your function is?
Or better yet, find a simpler function that still gives you the same trouble.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

"Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20070726175607$14f3@news.ks.uiuc.edu...
> "Luna Moon" <lunamoonmoon@gmail.com> writes:
>
>> Hi all,
>>
>> 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
>>
>> The key is to find the line "x=c", where the integration is done along
>> the
>> vertical line x=c in the complex plane such that c is greater than the
>> real
>>
>> part of all singularities of F(s).
>>
>> Suppose I have identified all the singularities, and determined that in a
>> large range of
>>
>> c>max(RealPartOf(All Singularities)),
>>
>> the location of "c" should not matter and I should be able to freely vary
>> "c" as long as it doesn't violate our rules above, and the integration
>> should yield precisely the same values.
>>
>> Now I plot the numerical integratal results along with varying "c" in the
>> safe region.
>>
>> I found that the results differ, by about 10%. The change is very smooth.
>> From one end of "c_min" to the other end of "c_max", the integral values
>> go
>>
>> from low to high.
>>
>> Here is my headache: with no closed-form solution exist for such
>> integral,
>> only numerical solutions exist.
>>
>> Which value of the numerical integral results shall I trust? (I used
>> Matlab's quadl, which is Gaussian Lobatto).
>
> Under the circumstances, I wouldn't trust any of them. It certainly looks
> like there's something wrong here. Can you tell us what your function is?
> Or better yet, find a simpler function that still gives you the same
> trouble.
> --
> Robert Israel israel@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada

I thought they were all close to the correct value, the only problem is that
I don't know which one is the most correct.

Okay! Let me try to figure out how to put a huge formulae here, or a
simplified one, which pin-points the problem. I don't want people to lose
interest when seeing a huge function...

On Thu, 26 Jul 2007, Luna Moon wrote:

> Date: Thu, 26 Jul 2007 07:55:52 -0400
> From: Luna Moon <lunamoonmoon@gmail.com>
> Newsgroups: comp.soft-sys.math.maple, comp.soft-sys.matlab,
> sci.math.num-analysis, sci.physics, sci.math
> Subject: question about complex line (Bromwich) integral
>
> Hi all,
>
> 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
>
> The key is to find the line "x=c", where the integration is done along the
> vertical line x=c in the complex plane such that c is greater than the real
> part of all singularities of F(s).
>
> Suppose I have identified all the singularities, and determined that in a
> large range of
>
> c>max(RealPartOf(All Singularities)),
>
> the location of "c" should not matter and I should be able to freely vary
> "c" as long as it doesn't violate our rules above, and the integration
> should yield precisely the same values.
>
> Now I plot the numerical integratal results along with varying "c" in the
> safe region.
>
> I found that the results differ, by about 10%. The change is very smooth.
> From one end of "c_min" to the other end of "c_max", the integral values go
> from low to high.
>
> Here is my headache: with no closed-form solution exist for such integral,
> only numerical solutions exist.
>
> Which value of the numerical integral results shall I trust? (I used
> Matlab's quadl, which is Gaussian Lobatto).
>
> Secondly, the speed of integration varies a lot with the change of c. Does
> anybody have some suggestions/pointers on what is a good theoretical
> guideline of choosing "c" when both accuracy and speed are needed?
>
> Thanks!
>
>
>


You should take the value c where F(s) exp(st) has a minimum on the
real positive axis
(and of course c>max(RealPartOf(All Singularities))).
In general, at this minimum the analytic integrand has a saddle point.

Nico.

Nico Temme <Nico.Temme@cwi.nl> wrote in message <Pine.GSO.4.44.0707302133390.19017-100000@cwi.nl>...
>
>
>
>
> On Thu, 26 Jul 2007, Luna Moon wrote:
>
> > Date: Thu, 26 Jul 2007 07:55:52 -0400
> > From: Luna Moon <lunamoonmoon@gmail.com>
> > Newsgroups: comp.soft-sys.math.maple, comp.soft-sys.matlab,
> > sci.math.num-analysis, sci.physics, sci.math
> > Subject: question about complex line (Bromwich) integral
> >
> > Hi all,
> >
> > 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
> >
> > The key is to find the line "x=c", where the integration is done along the
> > vertical line x=c in the complex plane such that c is greater than the real
> > part of all singularities of F(s).
> >
> > Suppose I have identified all the singularities, and determined that in a
> > large range of
> >
> > c>max(RealPartOf(All Singularities)),
> >
> > the location of "c" should not matter and I should be able to freely vary
> > "c" as long as it doesn't violate our rules above, and the integration
> > should yield precisely the same values.
> >
> > Now I plot the numerical integratal results along with varying "c" in the
> > safe region.
> >
> > I found that the results differ, by about 10%. The change is very smooth.
> > From one end of "c_min" to the other end of "c_max", the integral values go
> > from low to high.
> >
> > Here is my headache: with no closed-form solution exist for such integral,
> > only numerical solutions exist.
> >
> > Which value of the numerical integral results shall I trust? (I used
> > Matlab's quadl, which is Gaussian Lobatto).
> >
> > Secondly, the speed of integration varies a lot with the change of c. Does
> > anybody have some suggestions/pointers on what is a good theoretical
> > guideline of choosing "c" when both accuracy and speed are needed?
> >
> > Thanks!
> >
> >
> >
>
>
> You should take the value c where F(s) exp(st) has a minimum on the
> real positive axis
> (and of course c>max(RealPartOf(All Singularities))).
> In general, at this minimum the analytic integrand has a saddle point.
>
> Nico.
>

You might want to look at:

http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=15725&objectType=file

which is a newly posted numerical *2D* inverse laplace transform program. It seems to have a 1D inverse laplace transform embedded in it.

Scott

On Jul 30, 3:39 pm, Nico Temme <Nico.Te...@cwi.nl> wrote:
> On Thu, 26 Jul 2007, Luna Moon wrote:
> > Date: Thu, 26 Jul 2007 07:55:52 -0400
> > From: Luna Moon <lunamoonm...@gmail.com>
> > Newsgroups: comp.soft-sys.math.maple, comp.soft-sys.matlab,
> > sci.math.num-analysis, sci.physics, sci.math
> > Subject: question about complex line (Bromwich) integral
>
> > Hi all,
>
> > 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
>
> > The key is to find the line "x=c", where the integration is done along the
> > vertical line x=c in the complex plane such that c is greater than the real
> > part of all singularities of F(s).
>
> > Suppose I have identified all the singularities, and determined that in a
> > large range of
>
> > c>max(RealPartOf(All Singularities)),
>
> > the location of "c" should not matter and I should be able to freely vary
> > "c" as long as it doesn't violate our rules above, and the integration
> > should yield precisely the same values.
>
> > Now I plot the numerical integratal results along with varying "c" in the
> > safe region.
>
> > I found that the results differ, by about 10%. The change is very smooth.
> > From one end of "c_min" to the other end of "c_max", the integral values go
> > from low to high.
>
> > Here is my headache: with no closed-form solution exist for such integral,
> > only numerical solutions exist.
>
> > Which value of the numerical integral results shall I trust? (I used
> > Matlab's quadl, which is Gaussian Lobatto).
>
> > Secondly, the speed of integration varies a lot with the change of c. Does
> > anybody have some suggestions/pointers on what is a good theoretical
> > guideline of choosing "c" when both accuracy and speed are needed?
>
> > Thanks!
>
> You should take the value c where F(s) exp(st) has a minimum on the
> real positive axis
> (and of course c>max(RealPartOf(All Singularities))).
> In general, at this minimum the analytic integrand has a saddle point.
>
> Nico.

Sometimes the root of that saddle point is too costly to find. And
sometimes the root-finder even cannot find a root. Suppose I don't
find that root as the optimal "c", but why should that matter? Theory
tells us as long as "c" is away from any singularities, it should
always give back the same integration results... right?

On Jul 30, 3:39 pm, Nico Temme <Nico.Te...@cwi.nl> wrote:
> On Thu, 26 Jul 2007, Luna Moon wrote:
> > Date: Thu, 26 Jul 2007 07:55:52 -0400
> > From: Luna Moon <lunamoonm...@gmail.com>
> > Newsgroups: comp.soft-sys.math.maple, comp.soft-sys.matlab,
> > sci.math.num-analysis, sci.physics, sci.math
> > Subject: question about complex line (Bromwich) integral
>
> > Hi all,
>
> > 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
>
> > The key is to find the line "x=c", where the integration is done along the
> > vertical line x=c in the complex plane such that c is greater than the real
> > part of all singularities of F(s).
>
> > Suppose I have identified all the singularities, and determined that in a
> > large range of
>
> > c>max(RealPartOf(All Singularities)),
>
> > the location of "c" should not matter and I should be able to freely vary
> > "c" as long as it doesn't violate our rules above, and the integration
> > should yield precisely the same values.
>
> > Now I plot the numerical integratal results along with varying "c" in the
> > safe region.
>
> > I found that the results differ, by about 10%. The change is very smooth.
> > From one end of "c_min" to the other end of "c_max", the integral values go
> > from low to high.
>
> > Here is my headache: with no closed-form solution exist for such integral,
> > only numerical solutions exist.
>
> > Which value of the numerical integral results shall I trust? (I used
> > Matlab's quadl, which is Gaussian Lobatto).
>
> > Secondly, the speed of integration varies a lot with the change of c. Does
> > anybody have some suggestions/pointers on what is a good theoretical
> > guideline of choosing "c" when both accuracy and speed are needed?
>
> > Thanks!
>
> You should take the value c where F(s) exp(st) has a minimum on the
> real positive axis
> (and of course c>max(RealPartOf(All Singularities))).
> In general, at this minimum the analytic integrand has a saddle point.
>
> Nico.

What's the procedure of finding such "c"? Do you mean taking
derivative of F(c)*exp(c*t) for c as a real number?

On Jul 30, 3:39 pm, Nico Temme <Nico.Te...@cwi.nl> wrote:
> On Thu, 26 Jul 2007, Luna Moon wrote:
> > Date: Thu, 26 Jul 2007 07:55:52 -0400
> > From: Luna Moon <lunamoonm...@gmail.com>
> > Newsgroups: comp.soft-sys.math.maple, comp.soft-sys.matlab,
> > sci.math.num-analysis, sci.physics, sci.math
> > Subject: question about complex line (Bromwich) integral
>
> > Hi all,
>
> > 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
>
> > The key is to find the line "x=c", where the integration is done along the
> > vertical line x=c in the complex plane such that c is greater than the real
> > part of all singularities of F(s).
>
> > Suppose I have identified all the singularities, and determined that in a
> > large range of
>
> > c>max(RealPartOf(All Singularities)),
>
> > the location of "c" should not matter and I should be able to freely vary
> > "c" as long as it doesn't violate our rules above, and the integration
> > should yield precisely the same values.
>
> > Now I plot the numerical integratal results along with varying "c" in the
> > safe region.
>
> > I found that the results differ, by about 10%. The change is very smooth.
> > From one end of "c_min" to the other end of "c_max", the integral values go
> > from low to high.
>
> > Here is my headache: with no closed-form solution exist for such integral,
> > only numerical solutions exist.
>
> > Which value of the numerical integral results shall I trust? (I used
> > Matlab's quadl, which is Gaussian Lobatto).
>
> > Secondly, the speed of integration varies a lot with the change of c. Does
> > anybody have some suggestions/pointers on what is a good theoretical
> > guideline of choosing "c" when both accuracy and speed are needed?
>
> > Thanks!
>
> You should take the value c where F(s) exp(st) has a minimum on the
> real positive axis
> (and of course c>max(RealPartOf(All Singularities))).
> In general, at this minimum the analytic integrand has a saddle point.
>
> Nico.

Min or Max?

I plotted the F(c)*exp(c*t),

sometimes there is a min, sometimes there is no min(min is at the left
bound or right bound of the inteval), but there is a max...

On Aug 2, 12:58 pm, Luna Moon <lunamoonm...@gmail.com> wrote:
> On Jul 30, 3:39 pm, Nico Temme <Nico.Te...@cwi.nl> wrote:
>
>
>
> > On Thu, 26 Jul 2007, Luna Moon wrote:
> > > Date: Thu, 26 Jul 2007 07:55:52 -0400
> > > From: Luna Moon <lunamoonm...@gmail.com>
> > > Newsgroups: comp.soft-sys.math.maple, comp.soft-sys.matlab,
> > > sci.math.num-analysis, sci.physics, sci.math
> > > Subject: question about complex line (Bromwich) integral
>
> > > Hi all,
>
> > > 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
>
> > > The key is to find the line "x=c", where the integration is done along the
> > > vertical line x=c in the complex plane such that c is greater than the real
> > > part of all singularities of F(s).
>
> > > Suppose I have identified all the singularities, and determined that in a
> > > large range of
>
> > > c>max(RealPartOf(All Singularities)),
>
> > > the location of "c" should not matter and I should be able to freely vary
> > > "c" as long as it doesn't violate our rules above, and the integration
> > > should yield precisely the same values.
>
> > > Now I plot the numerical integratal results along with varying "c" in the
> > > safe region.
>
> > > I found that the results differ, by about 10%. The change is very smooth.
> > > From one end of "c_min" to the other end of "c_max", the integral values go
> > > from low to high.
>
> > > Here is my headache: with no closed-form solution exist for such integral,
> > > only numerical solutions exist.
>
> > > Which value of the numerical integral results shall I trust? (I used
> > > Matlab's quadl, which is Gaussian Lobatto).
>
> > > Secondly, the speed of integration varies a lot with the change of c. Does
> > > anybody have some suggestions/pointers on what is a good theoretical
> > > guideline of choosing "c" when both accuracy and speed are needed?
>
> > > Thanks!
>
> > You should take the value c where F(s) exp(st) has a minimum on the
> > real positive axis
> > (and of course c>max(RealPartOf(All Singularities))).
> > In general, at this minimum the analytic integrand has a saddle point.
>
> > Nico.
>
> Min or Max?
>
> I plotted the F(c)*exp(c*t),
>
> sometimes there is a min, sometimes there is no min(min is at the left
> bound or right bound of the inteval), but there is a max...

If when I did the integral using the "optimal c" and I got a value
that is different than the theoretical integral value, whereas using a
c that is slightly different than the "optimal c", I got a value that
agrees with the theoretical integral value.

What might be the problem?
Can anybody show how is the "optimal c" obtained correctly? Thanks

"Luna Moon" <lunamoonmoon@gmail.com> wrote in message
<f8a28a$7nv$1@news.Stanford.EDU>...
> Hi all,
>
> 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
>
> The key is to find the line "x=c", where the integration
is done along the
> vertical line x=c in the complex plane such that c is
greater than the real
> part of all singularities of F(s).
>
> Suppose I have identified all the singularities, and
determined that in a
> large range of
>
> c>max(RealPartOf(All Singularities)),
>
> the location of "c" should not matter and I should be able
to freely vary
> "c" as long as it doesn't violate our rules above, and the
integration
> should yield precisely the same values.
>
> Now I plot the numerical integratal results along with
varying "c" in the
> safe region.
>
> I found that the results differ, by about 10%. The change
is very smooth.
> From one end of "c_min" to the other end of "c_max", the
integral values go
> from low to high.
>
> Here is my headache: with no closed-form solution exist
for such integral,
> only numerical solutions exist.
>
> Which value of the numerical integral results shall I
trust? (I used
> Matlab's quadl, which is Gaussian Lobatto).
>
> Secondly, the speed of integration varies a lot with the
change of c. Does
> anybody have some suggestions/pointers on what is a good
theoretical
> guideline of choosing "c" when both accuracy and speed are
needed?
>
> Thanks!
>
>

Why don't you try to solve the problem by another technique,
such as finite differencing, to compare your solution with?

On Apr 16, 3:22=A0pm, "David Lampert" <lampert_da...@hotmail.com> wrote:
> "Luna Moon" <lunamoonm...@gmail.com> wrote in message
>
> <f8a28a$7n...@news.Stanford.EDU>...> Hi all,
>
> > 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
>
> > The key is to find the line "x=3Dc", where the integration
> > is done along the vertical line x=3Dc in the complex plane
> > such that c is greater than the real part of all singularities
> > of F(s).
>
> > Suppose I have identified all the singularities, and
>
> determined that in a large range of
>
> > c>max(RealPartOf(All Singularities)),
>
> > the location of "c" should not matter and I should be able
> > to freely vary "c" as long as it doesn't violate our rules
> > above, and the integration should yield precisely the
> > same values.

You are approximating an infinite integral with a finite
integral

f(t) ~ exp(ct) *INT(w=3D-L,L){ dw F(c+jw) exp(jwt) }

Have you factored out the exp(ct) term?
How far are the endpoints s =3D c+/- jL from the nearest singularity?
Near the endpoints exp(jwL) oscillates wildly

You should be able to diagnose your problem
by varying c,L,a and w0 with

F(s) =3D 1/[(s-a)^2+w0^2]

I would think that L has to chosen so that

d =3D sqrt( (c-a)^2 + (L-w0)^2 )

is sufficiently large.

Hope this helps.

Greg