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

```On 17 Mar 2007, at 17:34, Andrzej Kozlowski wrote:

>
> On 15 Mar 2007, at 10:56, hemphill at hemphills.net wrote:
>
>> Andrzej Kozlowski <akoz at mimuw.edu.pl> writes:
>>
>>> Certianly for small vaues explicit addition is performed, as can be
>>> seen from:
>>>
>>> Trace[Sum[i, {i, 1, n}] /. n -> 5]
>>>
>>> Out=
>>> {{HoldForm[Sum[i, {i, 1, n}]],
>>>     HoldForm[(1/2)*n*(n + 1)]},
>>>    HoldForm[(1/2)*n*(n + 1) /. n -> 5],
>>>    HoldForm[(5*(1 + 5))/2], {HoldForm[1 + 5],
>>>     HoldForm}, HoldForm[(5*6)/2], HoldForm}
>>>
>>> In:=
>>> Trace[Sum[i, {i, 1, 5}]]
>>>
>>> Out=
>>> {HoldForm[Sum[i, {i, 1, 5}]], {HoldForm[i], HoldForm},
>>>    {HoldForm[i], HoldForm}, {HoldForm[i], HoldForm},
>>>    {HoldForm[i], HoldForm}, {HoldForm[i], HoldForm},
>>>    HoldForm[1 + 2 + 3 + 4 + 5], HoldForm}
>>>
>>> The second Trace certianly suggests explicit summation.
>>>
>>> On the other hand, indeed we have:
>>>
>>> In:=
>>> Trace[Sum[i, {i, 1, 10^9}]]
>>>
>>> Out=
>>> {HoldForm[Sum[i, {i, 1, 10^9}]], {HoldForm[10^9],
>>>     HoldForm}, HoldForm}
>>>
>>>
>>> If this is right, it sugges that the algorithm switches at some
>>> point
>>> form explicit addition to using a formula. One could obviously find
>>> out the point at which the algorithm switches form one approach to
>>> the other, I suspect the number would b eunder 1000.
>>
>> I have version 5.1 for Linux.  The algorithm switches approaches at
>> 10^6+1.
>>
>> Scott
>> -- Scott Hemphill	hemphill at alumni.caltech.edu
>> "This isn't flying.  This is falling, with style."  -- Buzz Lightyear
>>
>
>
> One interesting way to demonstrate this is:
>
> f[n_Integer] := n
>
>
> Timing[Sum[f[n], {n, 1, 10^6}]]
>
>
> {4.7203479999999995*Second, 500000500000}
>
>
> Timing[Sum[f[n], {n, 1, 10^9}]]
>
>
> {0.226001000000001*Second, Sum[f[n], {n, 1, 1000000000}]}
>
> Note that it took Sum a non-trivial amount of time to return the
>
> Now, this is also interesting:
>
>
> g[n_Real] := n
>
>
> Timing[NSum[g[n], {n, 1, 10^9}]]
>
>
> {0.2843939999999998*Second, 5.000000005*^17}
>
> It would be nice to have at least a reasonable hypothesis about
> what happened here.
>
> Andrzej Kozlowski

Of course it should have been:

Timing[Sum[f[n], {n, 1, 10^6 + 1}]]

{0.19891900000000007*Second, Sum[f[n], {n, 1, 1000001}]}

Andrzej Kozlowski

```

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