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

• To: mathgroup at smc.vnet.net
• Subject: [mg23170] fastest way to do pair-sum / make pair-list
• From: Wijnand Schepens <Wijnand.Schepens at rug.ac.be>
• Date: Thu, 20 Apr 2000 23:48:42 -0400 (EDT)
• Organization: RUG
• Sender: owner-wri-mathgroup at wolfram.com

```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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