Re: unexpected behaviour of Sum

*To*: mathgroup at smc.vnet.net*Subject*: [mg126511] Re: unexpected behaviour of Sum*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Wed, 16 May 2012 04:22:14 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <joq5h8$r20$1@smc.vnet.net> <201205150852.EAA10924@smc.vnet.net>*Reply-to*: murray at math.umass.edu

All this indicates that the behavior that surprised the O.P. is indeed a bit subtle. When defining sod, is there any good reason to be using Plus with Apply (Plus @@ IntegerDigits) instead of the more direct composite with Total (Total @ IntegerDigits)? I ask because for the example at hand, the ratio of Timing values for the two is roughly 3/2. On 5/15/12 4:52 AM, A Retey wrote: > Hi, > >> I do not want to say that this is a bug, however I did >> not find, in the Sum documentation, an explanation of this >> behaviour (v8.0.4) >> >> I defined an innocent function like >> sod[n_]:=Plus@@IntegerDigits[n] >> that compute the sum of the digits of a number. >> (I tried with other functions, so the culprit is not IntegerDigits) >> >> Then I compute two sums: >> Sum[sod[k],{k,10^6}] which gives 27000001 (ok) >> then >> Sum[sod[k],{k,1+10^6}] which gives 500001500001 (nonsense). >> then >> Sum[sod[k],{k,2,1+10^6}] which gives 27000002 (ok) >> >> so, it seems to me that Sum has a problem when the iterator >> works in a range greater than 10^6. >> >> Indeed, I made an experiment: >> I set L={}; and defined >> sod[n_]:=(AppendTo[L,n];Plus@@IntegerDigits[n]) >> Then Sum[sod[k],{k,1+10^6}] still gives 500001500001 >> and at the end L is equal to {k}, so it seems that >> Sum tried to do something symbolic (??). > > yes, Sum will try to something symbolic, and the default threshold is > indeed 1000000, as SystemOptions["SymbolicSumThreshold"] will show you. >> Is this a bug or a feature whose documentation I was not >> able to find? Say, is there an option to fix things ? > > You could in fact increase SymbolicSumThreshold, but I think there are > much better ways to achieve what you want. First you should understand > what happens: when switching to a symbolic algorithm, Sum will evaluate > the summand symbolically, if you do that you will find the following: > > Clear[k];sod[k] will return k > > What happend? IntegerDigits[k] will not evaluate, applying Plus will > replace the Head with Plus and the end result is k. Sum[k,{1,n}] will > return the well known Gauss formula and insert 10^6+1 and that's the > wrong result you see. > > One way to solve your problem would be to define sod so it only is > evaluated for input that it actually does handle, e.g.: > > ClearAll[sod] > sod[n_Integer] := Plus @@ IntegerDigits[n] > > Another, probably even better approach is to use functions that will not > even try to do anything symbolic. Actually Sum to some extend can be > seen as a function made to return symbolic sums and it will do explicit > summations just as an additional feature. A simple approach like this: > > Total[Table[sod[k], {k, 10^6 + 1}]] > > will immediately yield the correct result, independent on how you define > sod. Presumably because the table will create a large list which will > consume a lot of memory the last is slower than Sum in this case, but if > you are after performance, you could change the definition of sod to this: > > ClearAll[sod] > sod = Plus @@ IntegerDigits[#]& > > Total[Table[sod[k], {k, 10^6 + 1}]] // Timing > > which is now very fast since it now sees what sod actually is and will > compile it. If you also want to be more memory efficient you can compile > yourself: > > Timing[Compile[{}, > Module[{sum = 0}, Do[sum += sod[k], {k, 10^6 + 1}]; sum], > CompilationOptions -> "InlineExternalDefinitions" -> True][] > ] > >> Btw, using ParallelSum instead of Sum works fine (and faster). > > which probably is because it does not try to do the sum symbolically > (but probably not nearly as fast as the compiled versions)... > > hth, > > albert > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**Re: unexpected behaviour of Sum***From:*A Retey <awnl@gmx-topmail.de>