🚀 go-pugleaf

In article <1992Jul27.124120.29511@cl.cam.ac.uk>, nmm@cl.cam.ac.uk (Nick Maclaren) writes:
> In article <1992Jul27.103659.11820@kth.se>, md87-mpe@hemul.nada.kth.se
> (Magnus Pettersson) writes:
>  
> |> 3. G.E.P.Box and M.E.Mueller
> |>    A Note on the Generation  of Random Normal Deviates
> |>    Ann.Math.Stat. (1958) 29, 610-611
> 
> Don't use this method in single precision - it has a very serious
> flaw.  While it is OK in double precision, there are faster and simpler
> methods available (e.g. the Polar Method as described in Knuth).  The
> reference for the flaw is by H.R. Neave round about 1970.
> 
> 
> Nick Maclaren
> University of Cambridge Computer Laboratory,
> New Museums Site, Pembroke Street,
> Cambridge CB2 3QG, England.


Wait! . . .  Stop! . . . Hold on a moment! . . .

There is nothing wrong with the Box-Muller method.  It is theoretically sound
and changing from single precision to double precision isn't going to affect
anything.  (For that matter, it isn't going to effect anything, either!)

The Box-Muller transformation is based on the simple fact of distribution
functions that if U and V are iid U(0,1), then 

                       sqrt(-2 log U) sin (2 pi V)
                       sqrt(-2 log U) cos (2 pi V)

are iid N(0,1).

I don't have the article in front of me but I recall that (as explained in a
followup letter) Neave's argument was flawed.  The problem was with a
uniform random number generator that used too small a modulus and didn't "kick
over" fast enough.  

I agree with Mclaren that the Neave article was published in the 70's.  I
can add that it appeared in Applied Statistics.  The followup letter appear
within the next year or two.  I'll leave it to someone else to look up the
exact references.  

(As a practical matter, I prefer to use the inverse of the standard normal
distribution to generate my random numbers.  I know it is computationally
intensive, but it relieves you from having to worry (as much)
about the joint distribution of values in your pseudo-random number stream.)