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

*To*: mathgroup at smc.vnet.net*Subject*: [mg23225] Re: fastest way to do pair-sum / make pair-list*From*: "Allan Hayes" <hay at haystack.demon.co.uk>*Date*: Tue, 25 Apr 2000 01:40:29 -0400 (EDT)*References*: <8dok75$jds@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Wijnand , First using Sum sds1[data_] := Sum[(-data[[i]] + data[[j]])^2, {i, 1, Length[data]}, {j, 1, i - 1}] But it is quicker to use matrix operations, particularly Dot to sum squares. sds2[data_] := #.# &[{1, -1}.{Flatten[ NestList[Drop[#, -1] &, Drop[data, -1], Length[data] - 2], 1], Flatten[NestList[Drop[#, 1] &, Drop[data, 1], Length[data] - 2], 1]}] Tests data = Table[Random[], {300}]; sds1[data] // Timing {9.99 Second, 7678.5} sds2[data] // Timing {1.32 Second, 7678.5} Extra check sds2[{a, b, c, d}] (a - b)^2 + (a - c)^2 + (b - c)^2 + (a - d)^2 + (b - d)^2 + (c - d)^2 -- Allan --------------------- Allan Hayes Mathematica Training and Consulting Leicester UK www.haystack.demon.co.uk hay at haystack.demon.co.uk Voice: +44 (0)116 271 4198 Fax: +44 (0)870 164 0565 "Wijnand Schepens" <Wijnand.Schepens at rug.ac.be> wrote in message news:8dok75$jds at smc.vnet.net... > 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 > >