Re: fastest way to do pair-sum / make pair-list

• To: mathgroup at smc.vnet.net
• Subject: [mg23185] Re: [mg23170] fastest way to do pair-sum / make pair-list
• From: "Ralf D. Pluch" <aon.912308612 at aon.at>
• Date: Mon, 24 Apr 2000 01:11:56 -0400 (EDT)
• References: <200004210348.XAA19772@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```ladies & gentlemen,

duke_of_snowboard at gmx.at

thank's a lot in advance.

Philipp Pluch

Wijnand Schepens schrieb:

> 1.
>
> What is the most efficient way to calculate the sum
>
> Sum f [ (x[[i]]-x[[j]])^2 ]
> i<j
>
> ??
>
> Example:
>
> I have a vector of real numbers:
> lst = N[Range[100]];
> and a function operating on a number:
> f[r2_]:=(r2-1.0)^2
>
> Is
>
> Sum[  Sum[  f [ (lst[[i]]-lst[[j]])^2 ] , {j,i+1,Length[lst]}],
> {i,1,Length[lst-1]}]
>
> the fastest way??
>
> This kind of sum is very common in Molecular Modelling, where the total
> energy of a system is often a sum of pair-energies, which only depend on
> the distance between atoms.
> I was surprised that I didn't find anything on sums over pairs in
> Mathematica...
>
> 2.
>
> What is the most efficient way to generate a list of pairs starting from
> a list??
> Is there a standard Mathematica routine which does this?
>
> e.g.  {a,b,c,d} ----> {{a,b},{a,c},{a,d},{b,c},{b,d},{c,d}}
>
> or {x1,x2,...} -----> { {xi,xj} ...}
> with i<j
>
> Best solution I found was
> topairs[lst_] :=
>   Module[{l=Length[lst]},
>      Map[(Sequence @@ #1) &,
>              Table[ lst [[{i, j}]], {i, 1, l - 1}, {j, i + 1, l}]
>      ]
>   ]
>
> Another possibility would be
> topairs2[lst_] :=
>   Module[{l = Length[lst]},
>     Partition[
>        Flatten[
>            Table[ lst[[{i, j}]], {i, 1, l - 1}, {j, i + 1, l}]
>        ], 2
>     ]
>  ]
>
> but this doesn't have the same effect if operating on a list of lists
>   topairs[{{a1, a2}, {b1, b2}, {c1, c2}}]
> gives what we want,
>   topairs2[{{a1, a2}, {b1, b2}, {c1, c2}}]
> not

```

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