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From: Ian Bush on 8 Apr 2010 13:19
On 8 Apr, 14:13, Gordon Sande <Gordon.Sa...(a)EastLink.ca> wrote:
> On 2010-04-08 00:08:46 -0300, Michael <ma...(a)athenavisual.com> said:
>
> > In thermodynamics we often have to evaluate expressions like the
> > following:
>
> > Sum_i Sum_k (x_i * x_k * A_ik * (1-K_ik))   where A_ik=sqrt(A_i * A_k)
>
> > The indices i and k run from 1 to the number of components in a
> > mixture. For a mixture of 200 components (very typical) we often have
> > to do this calculation around 1000 times. If one writes two regular DO
> > loops to perform the calculation the code is very slow. I am
> > soliciting ideas as what is the optimal way to implement the sum shown
> > here.
>
> In n-body calculations there is a similar formal structure of an interaction
> between pairs. But the interaction is known to be small for many pairs so
> they can be dropped. If you problem has similar structure then you could
> use similar approximations. But you have not indicated that this might be
> the case beyond indicating that your problem has a physical basis.
>

And even when you can't neglect the "small" terms, e.g. for the
coulomb
interaction, there are ways to reduce it to something like O(NlogN).
See e.g.

http://www.unc.edu/~perera/PAPERS/JChemPhys_1995_103_8577.pdf

which may be adaptable to your situation.

>
> A_ik=sqrt(A_i*A_k)=sqrt(A_i)*sqrt(A_k) so one might want to use x_i*sqrt(A_i)
> insted of your currect expression. That leaves K_ik as the great unkown that
> you forgot to describe. Is it a strength of an interaction? Would it allow
> terms to be ignored? ...?

Exactly. The form of K_ik is crucial here,

Ian

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