Ian Buckner <[EMAIL PROTECTED]> wrote in message
[EMAIL PROTECTED]">news:[EMAIL PROTECTED]...
> The Box-Muller algorithm rejects roughly 22.5% of the
> generated points. I'm not aware of any bound on the number
> of consecutive rejections, other than a statistical one, hence
> my statement. I would welcome correction if this is not the case.
>
> Regards
> Ian
>
> "Radford Neal" <[EMAIL PROTECTED]> wrote in message
> [EMAIL PROTECTED]">news:[EMAIL PROTECTED]...
> > >Box-Muller does not work for real time requirements.
> >
> > This isn't true, of course. A "real time" application is one where
> > one must guarantee that an operation takes no more than some
> specified
> > maximum time. The Box-Muller method for generating normal random
> > variates does not involve any operations that could take arbitrary
> > amounts of time, and so is suitable for real-time applications.
> >
> > This assumes that the time needed for Box-Muller is small enough,
> > which will surely often be true. If the time allowed is very small,
> > then of course one might need to use some other method.
> >
> > Rejection sampling methods would not be suitable for real-time
> > applications, since there is no bound on how many points may be
> > rejected before one is accepted, and hence no bound on the time
> > required to generate a random normal variate.
> >
> > Radford Neal
=========================
What is usually called the Box-Muller method is
the transformation of normal variates to polar coordinates
that we all owe to Laplace for showing us how to evaluate
\int_0^\infty e^{-x^2}. To get a pair of independent
standard normal variates x,y, use two [0,1) uniform variates
U1,U2 and put
x = sqrt(-2*ln(U1))*cos(2pi*U2)
y = sqrt(-2*ln(U1))*sin(2pi*U2).
This is a fixed-time procedure.
The random-time procedure, often mistakenly
called Box-Muller, was developed in 1956
by Marsaglia:
Generate uniform (-1,1) variates V1,V2 until
S = V1^2 + V2^2 <1
then return
x = V1*sqrt(-2ln(S)/S)
y = V2*sqrt(-2ln(S)/S).
George Marsaglia
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