[Haskell-cafe] Squashing space leaks

[Daniel Fischer]

Am Montag, 9. Mai 2005 05:38 schrieb Greg Buchholz:
> Daniel Fischer wrote:
> > One question: the energy of the system should be constant - that little
> > physics I know. What I don't know is whether the change in energy in this
> > simulation is within usual, reasonable bounds or not. If not, is it due
> > to a too long time interval or should one use a more sophisticated
> > algorithm?
>
>     The change in energy is from round-off error caused by floating point
> arithmetic.  For this particular test, the shootout requires everyone to
> use the same algorithm.  So the error is expected.
>
> Greg Buchholz

Well, I think most of the change is due to the algorithm, which gives a rather 
crude approximation to the actual solution of the ODE. This is backed by 
trying different values of dt (actually tested with Java, because that uses 
hardly any memory at all and is much much faster).
For the tests, dt * k = 20, dt is the argument:
dafis at linux:~/Documents/haskell/dmitri/Greg/Bodies> java PlanSys 0.1
-0.1690751638285245
-0.16877178579767274
dafis at linux:~/Documents/haskell/dmitri/Greg/Bodies> java PlanSys 0.05
-0.1690751638285245
-0.16899653420881677
dafis at linux:~/Documents/haskell/dmitri/Greg/Bodies> java PlanSys 0.025
-0.1690751638285245
-0.1690547211965566
...
dafis at linux:~/Documents/haskell/dmitri/Greg/Bodies> java PlanSys 0.00001
-0.1690751638285245
-0.16907516338916015
dafis at linux:~/Documents/haskell/dmitri/Greg/Bodies> java PlanSys 0.000005
-0.1690751638285245
-0.16907516360969982
dafis at linux:~/Documents/haskell/dmitri/Greg/Bodies> time java PlanSys 
0.0000025
-0.1690751638285245
-0.16907516371932937

real    1m42.464s
user    1m39.900s
sys     0m2.540s

To me it looks like finer granularity on the time scale is not yet consumed by 
rounding errors (and since we never divide by small numbers, I wouldn't 
expect any dramatic effects for a while). 
If the algorithm - including dt - is prescribed, fine, but I wonder what sort 
of deviation physicists would consider acceptable.
For dt = 0.01, k = 2000 we have a relative error of about 2*10^(-5), is that 
within accepted bounds or not? (Any physicists hang about here?)

Cheers,
Daniel