Re: Wolfram, meet Stefan and Boltzmann
- To: mathgroup at smc.vnet.net
- Subject: [mg117316] Re: Wolfram, meet Stefan and Boltzmann
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Tue, 15 Mar 2011 06:03:02 -0500 (EST)
AES wrote:
> Responses to all:
>
> Many of you will recognize that the integral in question
>
> Integrate[x^3/(Exp[x] - 1), {x, 0, Infinity}]
>
> arises in the derivation of the Stefan-Boltzmann constant.
>
> I was reviewing this derivation in Wikipedia. Wikipedia's analytical
> evaluation of the integral itself, given in an Appendix to the
> Stefan-Boltzmann article, takes a couple of screens of text and a dozen
> intermediate formulas, with excursions into contour integration and
> infinite series expansions.
>
> I became curious as to whether Mathematica could "just do" this integral
> so I opened Mathematica, typed in the line above, and evaluated it. I
> was surprised at how long it took the first time, although reevaluation
> of the same cell was essentially instantaneous.
>
> So I shut down Mathematica; re-Opened it; and repeated the evaluation
> using Timing[-]. Did this several times, and found that from a "cold
> start", that is, when I Quit and then re-Open Mathematica, it always
> takes 18 or 19 seconds to do the first evaluation.
>
> My Mac is a 2007 or 2007 vintage MacBook with hardware overview given
> below, running Snow Leopard in a generally vanilla setup (only about 60
> GB occupied on the 160 GB HD).
>
> So, I'm still a bit curious: What's Mathematica doing all that time? --
> loading a lot of other stuff? actually going through some lengthy
> algorithm to evaluate the integral? ???
>
> Hardware Overview:
>
> Model Name: MacBook
> Model Identifier: MacBook2,1
> Processor Name: Intel Core 2 Duo
> Processor Speed: 2.16 GHz
> Number Of Processors: 1
> Total Number Of Cores: 2
> L2 Cache: 4 MB
> Memory: 1 GB
> Bus Speed: 667 MHz
> Boot ROM Version: MB21.00A5.B07
> SMC Version (system): 1.17f0
>
Offhand I do not know why it is so slow for that first evaluation. I
will make several comments on the questions raised here.
* There is a startup time for loading some mx files used in Integrate.
It should be negligeable in comparison to the time to integrate this
example. On my faster machine it takes around 1.5 seconds, or which i
would guess no more than .1 involves package loading. To check whether
loading might be a problem on the macBook (I doubt that it is), could do
as follows.
(i) Open a new kernel.
(ii) Evaluate
Integrate[x^4/(Exp[x] - 1), {x, 0, Infinity}]
(note different power of x). This will cause the same autoloading, but
not likely have anything cached that would se3rve to improve the next step.
(iii) Evaluate Timing[your integral].
* A second evaluation can be much faster because some of the key
computations needed were cached in the first computation. As to what
these are, more below.
* Integrate, for definite integrals, will check for convergence at the
endpoints and also look for interior path singularities. If any are
found, it will again do convergence checks.
* In the case of this integral the Newton-Leibniz method is used. That
is to say, we find an antiderivative and work out limits as needed.
So the steps are as follows.
(1) Check for convergence at 0 and infinity.
(2) Look for path singularities of the integrand. This often involves
hard core technology like solving transcendental equation. For this
example, the only candidate located is zero. We discard it because it is
an endpoint, hence already will get special handling (via Limit).
(3) Find the antiderivative. It is
In[22]:= InputForm[Integrate[x^3/(Exp[x] - 1), x]]
Out[22]//InputForm=
-x^4/4 + x^3*Log[1 - E^x] + 3*x^2*PolyLog[2, E^x] -
6*x*PolyLog[3, E^x] + 6*PolyLog[4, E^x]
(4) Check for path singularities of the antiderivative. Needed because
an off-path singularity in the integrand could lead to a branch cut
crossing on the path for the antiderivative. In this example I would
surmise a bit of time is spent checking the antiderivative.
(5) If singularities are found, check for convergence. If confergence is
fine, split the path. Exception: Not necessary if seeming singularities
have the same limits for antiderivative, when approaching from both
directions along integration path.
(6) Take limits at endpoints.
As for caching, I know that the antiderivative and the various limits
(and any calls to Series on which Limit might rely) will be cached.
Moreover intermediate simplifications might happen, and some parts of
those computations will be cahced e.g. polynomial gcds, if any such are
used.
One respondent commented to the effect that this was significantly
faster in version 4.2. This was because several of the above steps were
done in more heuristic, and less reliable, ways. That said, it is quite
possible that there are redundent steps, and steps that could be made
faster, in the framework of the present methodology. I an cautiously
hopeful that this will get some attention in future development.
Daniel Lichtblau
Wolfram Research