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

• To: mathgroup at smc.vnet.net
• Subject: [mg23257] Re: [mg23170] fastest way to do pair-sum / make pair-list
• From: Hartmut Wolf <hwolf at debis.com>
• Date: Sat, 29 Apr 2000 22:05:01 -0400 (EDT)
• Organization: debis Systemhaus
• References: <200004210348.XAA19772@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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

Dear Wijnand,

assume you have a list r of your atom distances, with length n, (n=4 in
the example) then perhaps you would like to try out the following:

Plus @@ Join @@ Array[
If[#1 < #2, f[(r[[#1]] - r[[#2]])^2], Unevaluated[Sequence[]]] &,
{4, 4}]

f[(r[[1]] - r[[2]])^2] + f[(r[[1]] - r[[3]])^2] +
f[(r[[2]] - r[[3]])^2] + f[(r[[1]] - r[[4]])^2] +
f[(r[[2]] - r[[4]])^2] + f[(r[[3]] - r[[4]])^2]

(BTW are you shure about your arguments to f? I would have assumed
f[(r[[i]] - r[[j]])] or else
f[Sqrt[(r[[i]] - r[[j]]).(r[[i]] - r[[j]])]]
)

Kind regards, Hartmut

```

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