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MathGroup Archive 2006

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Re: Sum problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65322] Re: Sum problem
  • From: Maxim <m.r at inbox.ru>
  • Date: Sun, 26 Mar 2006 05:44:06 -0500 (EST)
  • References: <e00283$pmg$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Fri, 24 Mar 2006 06:05:55 +0000 (UTC), Majka Arkadiusz  
<Arkadiusz.Majka at telekomunikacja.pl> wrote:

> Hi ,
>
> I have a problem that drives me crazy. In order to sum a (long) list of
> elements (random numbers), say tab
>
> tab = Table[Random[], {10^7}];
>
> we use
>
> In[3]:=
> Plus@@tab
>
> Out[3]=
> 4.999014220610286`*^6
>
> ....but to check Sum we get
>
> In[6]:=
> Sum[tab[[i]],{i,1,Length[tab]}]
>
> Part specification K$30 is neither an
>     integer nor a list of integers.
>
> There is no problem for
>
> Sum[tab[[i]], {i, 1, 10^6}]
>
> ...and for
>
> Sum[tab[[i]], {i, 1, 5*10^6}]
>
> So Sum is ok. for number of elements of tab less than certain number  
> that is
> < Length[tab]
>
> I also tested recursion methods (doSum[k_]:=doSum[k]=doSum[k-1] ......  
> Etc)
> and got the same : fine but not for many elements (of course I set
> $RecursiveLimit and $IterationLimit as Infinity)
>
> I understand that to sum a list of elements we use Plus, Total  etc but
> nevertheless I want understand what I explained above.
>
> Best,
>
> Arek
>

This issue was discussed on MathGroup some two years ago; basically the  
problem is that Sum works as described in chapter A.4.2 of The Mathematica  
Book (that is, as an "iteration function") only when the number of steps  
is not greater than 10^6; otherwise it calls SymbolicSum`SymbolicSum:

In[1]:= SeedRandom[1]; {Random[], Random[]}

Out[1]= {0.66869268, 0.83119987}

In[2]:= SeedRandom[1]; Sum[ArcTan[k + Random[]], {k, 10^6 + 1}]

Out[2]= Sum[ArcTan[0.83119987 + k], {k, 1000001}]

This is evaluated as a symbolic sum; the summand is evaluated twice, first  
with k renamed to K$n and then in its original form.

The recursive method should certainly work in principle; the problem is  
that it will crash the kernel if the recursion is nested too deeply:

doSum[k_] := doSum[k - 1] + tab[[k]]
doSum[0] := 0

Block[{$RecursionLimit = Infinity}, doSum[Length@ tab]]
(*crashes the kernel*)

This can be done if we rewrite doSum in a tail-recursive fashion:

doSumAux[k_, acc_] := doSumAux[k - 1, acc + tab[[k]]]
doSumAux[0, acc_] := acc
doSum[k_] := doSumAux[k, 0]

Block[{$IterationLimit = Infinity}, doSum[Length@ tab]]

which will give exactly the same result as Plus @@ Reverse@ tab.

Maxim Rytin
m.r at inbox.ru


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