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Re: Generate random numbers with fixed sum and different constraints
Posted:
Nov 20, 2012 2:17 AM
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"Roger Stafford" wrote in message <k8ens1$kfr$1@newscl01ah.mathworks.com>...
> "David Epstein" <David.Epstein.spam@remove.warwick.ac.uk> wrote in message <k8d1g1$4gr$1@newscl01ah.mathworks.com>...
> > @Roger: what were your reasons for rejecting this approach in your randfixedsum package on FEX?
> - - - - - - - - - -
> I think you will find that as n increases the acceptance rate in such a procedure shrinks toward zero altogether too rapidly, thereby restricting one in practice to a rather small range for n.
>
> To get a feeling for this, consider an n-dimensional hypersphere of radius 1/2 enclosed in an n-dimensional cube with unit-length sides. The n-dimensional volume of the cube is 1 whereas that of the hypersphere for even n is (pi/4)^(n/2)/(n/2)! (See http://en.wikipedia.org/wiki/N-sphere.) For n = 50 this would be 1.53E-28, a small acceptance rate indeed.
IMHO, the difficulty is in the same order than solving linear programing with the constraints A*y <= b, which have to find accurately the vertices without ambiguity. At least for the convex part.
I have little experience in Delaunay in high dimensional space. But I could imagine it 's challenging too.
Bruno
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