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From: efrank@truth.hep.upenn.edu (Ed Frank)
Newsgroups: comp.lang.fortran
Subject: Re: Random Number Functions--Generate versus Input
Summary: generation of direction cosines
Keywords: generation of direction cosines
Message-ID: <32565@netnews.upenn.edu>
Date: 9 Nov 90 15:54:06 GMT
Expires: 15 Dec 90 05:00:00 GMT
References: <1990Nov8.170334.24579@athena.mit.edu> <1990Nov8.215752.7075@athena.mit.edu>
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Reply-To: efrank@truth.hep.upenn.edu (Ed Frank)
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In article <1990Nov8.215752.7075@athena.mit.edu> oliver@athena.mit.edu (James D. Oliver III) writes:
>I guess I need to make this clear: the random number genertor itself is
>vectorizable, however, the values of interest to me are *functions* of
>these randome variables.  To be more specific, I'm generating a series of
>randomly oriented 3-D unit vectors.  The fastest way to do this is to
>generate two random numbers for x and y and calculate z as long as x**2 +
>y**2 <= 1.  This process, however, involves a conditional loop back to the
>ranf() function and thus is not vectorizable.  However, the fact that this
>method is not vectorizable is just of incidental interest, since it remains
>the fastest way of doing it.
>____________________________
>	Jim Oliver   
>	oliver@athena.mit.edu /	joliver@hstbme.mit.edu
>	oliver%mitwccf.BITNET@MITVMA.MIT.EDU

  First, it is not clear to me that this is the fastest method.  My 
method of generating random direction cosines is as follows.  Let DIR()
be the array of direction cosines, and let RANN() be the (uniform) random
number generator.  PI is 3.14.  Then
        Phi = 2.0 * PI * RANN( dum)       ! in polar coord. Phi and cos(theta)
        Dir(3) = -1.0 + 2.0*RANN( dum)    ! follow a flat distribution.
        Sth    = SQRT( 1.0 = Dir(3)**2)   ! convert cos(theta) to sin(theta)
        Dir(1) = Sth * COS( phi)          ! now get X and y
        Dir(2) = Sth * SIN( phi)          ! direction cosines
will produce the random directions for you.  You might want to include
this in your benchmark.  I think that this will beat your method with
respect to speed.

   Second, I do not completely understand your method. If I am reading
your description correctly, you will *NOT* obtain a uniform
distribution with your technique.  Have you histogrammed the direction
cosines produced by your method? The distributions should be flat.
Keep in mind that if you distribute points on the surface of a sphere
uniformly, and then project that distribution onto a plane, you will
not obtain a flat distrubution bounded by x**2+y**2 = 1.  Most of the
sphere's projected area will be near x**2 + y**2 = 1.

In short: Phi is flat from 0 to 2*pi
          Cos(theta) is flat from -1 to 1

Ed Frank
efrank@upenn5.hep.upenn.edu