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### Integrate Planck's law of radiation

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(sorry for the partial OT)

In your opinion what could be the best way to integrate the Planck's
law of radiation in an arbitrary interval?
The problem basically is that I know that if the nodes are misplaced the
result is completely wrong, expecially if the range of integration
is particullary large.
Are there some libraries to handle this problem?

--
Lurkos

 0
Reply lurkos.usenet (56) 2/5/2010 5:07:58 AM

Lurkos <lurkos.usenet@gmail.com> wrote:
> (sorry for the partial OT)

> In your opinion what could be the best way to integrate the Planck's
> law of radiation in an arbitrary interval?

This is not a bad place to find physicists.  There are groups
where you might find more of them, though.

> The problem basically is that I know that if the nodes are
> misplaced the result is completely wrong, expecially if the
> range of integration is particullary large.

I would probably do it with a few polynomials and then an
exponential for the tail.  I think most would not say "completely
wrong" though it might be off by a few percent.  Do you need it
much better than that?  It is a fairly smooth function, so it
should be possible to approximate to any degree of accuracy,
though it will get harder the better you want the result.

One possibility is a cubic spline calculated with fairly
closely spaced points.  You can then precompute the integral
between each pair of knots, and then compute the integrals between
the start and end point and the nearby knot from the spline
coefficients.

-- glen


 0

On 2/4/2010 11:07 PM, Lurkos wrote:
> (sorry for the partial OT)
>
> In your opinion what could be the best way to integrate the Planck's
> law of radiation in an arbitrary interval?
> The problem basically is that I know that if the nodes are misplaced the
> result is completely wrong, expecially if the range of integration
> is particullary large.
> Are there some libraries to handle this problem?
>

The integrand is singular at zero frequency, and the range of
integration is often infinite on one side. Use an appropriate routine
from Quadpack, available from Netlib. For the special case of
integration from zero to infinity, a series expansion of the integrand
and integration gives the series for Riemann's Zeta function \zeta(4)
multiplied by a factor.

HTH.

-- mecej4

 0
Reply mecej4_nospam (29) 2/13/2010 10:45:22 AM

Lurkos wrote:

> In your opinion what could be the best way to integrate the Planck's
> law of radiation in an arbitrary interval?
> The problem basically is that I know that if the nodes are misplaced the
> result is completely wrong, expecially if the range of integration
> is particullary large.

I would suggest to convert the Planck function (I guess you mean the
spectral intensity function of wavelength, and temperature as parameter)
into a logarithmic one using the substitution rule. I have done such
calculations before, but it's a long time ago.

If you have the Planckian spectral intensity as a function I(lambda,T)
then you may integrate an embedding function f(x) with x=log lambda,

f = I(exp(x),T) * lambda,

or, if you are using the decimal logarithm,

f = I(10.**x,T) * lambda * log(10.)

(double precision may be a good idea here.)
Then a standard integrator like Simpson's rule or a Gauss quadrature
(see Numerical Recipes...) should do.

Ingo

 0
Reply ingo.thies (51) 2/23/2010 2:50:37 PM

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