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Re: V8 slow like a snail
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127775] Re: V8 slow like a snail
*From*: Roland Franzius <roland.franzius at uos.de>
*Date*: Tue, 21 Aug 2012 05:00:12 -0400 (EDT)
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*References*: <k0fjaj$q71$1@smc.vnet.net> <k0fqto$r07$1@smc.vnet.net> <k0uos9$n7l$1@smc.vnet.net>
Am 21.08.2012 03:39, schrieb Dr. Wolfgang Hintze:
>>>>> Consider this integral for which we can safely expect Mathematica to
>>>>> be expert in solving:
>>
>>>>> f1[n_, m_] :=
>>>>> Integrate[n t^m Exp[-n t] (Exp[t] - 1)^(n - 1), {t, 0, \[Infin=
> ity]},
>>>>> Assumptions -> {{n, m} \[Element] Integers, m >= 0, n > 0}]
>>
>>>>> I carried out Timing[f1[n, m]] for m=0,1,2,3,10 in both versions. H=
> ere
>>>>> are the results in the format
>> But compare your results with thos after the standard substitution
>>
>> exp(-t) ->u, t->-Log[u], dt -> -du/u {t,0,oo}-> {u,0,1}
>>
>> h1[n_, m_] :=
>> Integrate[n (-Log[u])^m (1 - u)^(n - 1), {u, 0, 1},
>> Assumptions -> {{n} \[Element] Integers, n > 0}]
>
> I'm well aware of how to handle Mathematica i.e. to help it get to
> results (since 2003 in this group).
> However, as a conservative software user this was my first version
> change in Mathematica, and I didn't expect the differences be so grave
> when simply comparing the same commands.
>
> Your substitution is so obvious that exactly for this reason one could
> possibly expect Mathematica to do it by itself. But ok, in this form
> the integration in V8 is quick.
>
> And after all, one of the charming features of Mathematica is just the
> "human-like" behaviour to sometimes failing but with some help finally
> reaching the goal.
At least you found a real bug in vs 8.
Mathematica uses generalized hypergeometric series to express a wide
class of integrals containing Pochhammer symbols wich arise in partial
integration of expression(x)^n.
But some of these series dont work for integer parameters or need
special regularizations by Gamma factors.
In this case the indefinite integrals over the variable t are expressed
in terms of
HypergeometricPFQ[{-n,-n...},{1-n,1-n,...},, e^t]
Mathematica is not able to determine Limits for t->0 and t->oo. Moreover
these special hypergeometric series even seem to have no limits for
integer n.
So I suspect the time is vasted for determining the limits for real
noninteger n (Limit takes some 50-100 seconds to retun no result) and
after failing the standard substituition list is applied.
The general integration rule complex for "algebraic expression of powers
and exponentials" seems to miss a preprocessing for the exception of
integer n and m.
--
Roland Franzius
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