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)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*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