Re: Re: Re: fastest way to add up a billion numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg73998] Re: [mg73973] Re: [mg73907] Re: fastest way to add up a billion numbers*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Mon, 5 Mar 2007 04:50:42 -0500 (EST)*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst*References*: <es92b2$3oj$1@smc.vnet.net> <200703030555.AAA02643@smc.vnet.net> <200703040453.XAA25305@smc.vnet.net> <A35F0ECC-2DB3-4260-8C66-8BC9008E14DA@mimuw.edu.pl>*Reply-to*: murray at math.umass.edu

Very interesting -- and then I'm surprised to conclude that Mathematica does NOT, apparently, attempt to use the Gauss formula even in the first case. I should think efficiency would dictate using it; but then perhaps the extra processing needed to check whether a particular invocation of Sum could use that special case result overall would decrease efficiency of Sum. Andrzej Kozlowski wrote: > If you compare these three timing (and particularly the last one) you > will see that your suspision is wrong. > > In[1]:= > Timing[Sum[i, {i, 1, 10^9}]] > > Out[1]= > {0.11999199999999999*Second, 500000000500000000} > > In[2]:= > Timing[NSum[i, {i, 1, 10^9}]] > > Out[2]= > {0.0862520000000001*Second, 5.000000005*^17} > > In[3]:= > Timing[Sum[i, {i, 1, n}] /. n -> 10^9] > > Out[3]= > {0.02608200000000005*Second, 500000000500000000} > > Only the last method (the fastest) does what you suspected (which is > why I do not think it a "legitimate" solution to the posted problem). > The fastest method that actually adds up numbers uses NSum (which of > course gives only an approaximate answer). > > Andrzej Kozlowski > > On 4 Mar 2007, at 05:53, Murray Eisenberg wrote: > >> I wonder what the timing for a sum of the form Sum[n,{n,nstart,nend}] >> really indicates. I suspect that Mathematica is not actually summing >> but rather just uses Gauss' well-known closed-form formula for such a >> sum of consecutive positive integers. >> >> dimitris wrote: >>> There is recent post of mine asking something relevant here. >>> >>> You can find also a relevant thread here: >>> >>> http://forums.wolfram.com/mathgroup/archive/2005/May/msg00600.html >>> >>> One billion numbers? >>> Here is what I get to my PC >>> >>> In[3]:= >>> Range[10^9] >>>> From In[3]:= >>> No more memory available. >>> Mathematica kernel has shut down. >>> Try quitting other applications and then retry. >>> >>> In[1]:= >>> Table[i, {i, 1, 10^9}]; >>> >>>> From In[1]:= >>> No more memory available. >>> Mathematica kernel has shut down. >>> Try quitting other applications and then retry. >>> >>> Anyway... >>> >>> lst = Range[10^6]; >>> >>> Timing[Plus @@ lst] >>> {0.34299999999999997*Second, 500000500000} >>> >>> Timing[Total[lst]] >>> {0.2659999999999999*Second, 500000500000} >>> >>> Timing[Fold[Plus, 0, lst]] >>> {0.703*Second, 500000500000} >>> >>> Timing[lst /. {x_, y___} -> x + y] >>> {0.45300000000000007*Second, 500000500000} >>> >>> Timing[Tr[lst]] >>> {0.3899999999999997*Second, 500000500000} >>> >>> Timing[Sum[lst[[i]], {i, 1, 10^6}]] >>> {1.4379999999999997*Second, 500000500000} >>> >>> >>> But for your case I should simply suggest >>> >>> Timing[Sum[i,{i,1,10^9}]] >>> {0.047 Second,500000000500000000} >>> >>> >>> Regards >>> Dimitris >>> >>> >>> >>> >>> =CF/=C7 Raj =DD=E3=F1=E1=F8=E5: >>>> hi! >>>> >>>> Could somebody tell me what would be the fastest way to add up a >>>> billion numbers(from 1 to 10^9 i.e the first billion numbers) in >>>> Mathematica? >>>> >>>> Ofcourse the answer is n(n+1)/2, but is there any other way in >>>> Mathematica other than the following one: >>>> >>>> Total@@Range[10^9] >>>> >>>> Thanks, >>>> >>>> Raj >>> >>> >> >> --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 >> > -- 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: fastest way to add up a billion numbers***From:*"dimitris" <dimmechan@yahoo.com>

**Re: Re: fastest way to add up a billion numbers***From:*Murray Eisenberg <murray@math.umass.edu>