An examination of some properties of 
computer generated "random" numbers.
 - Jim Mahoney, Dec 5 2002  

----------------------------------------------------------

The generators I'm using are

 (1) rand(), provided by the linux system, and

 (2) jims1Rand(), a "linear conguential" one
     based on the discussion in Numerical Recipes,
     namely  Rand(i+1) = ( a * Rand(i) + c ) mod m
     where the three constants (m,a,c) are constants.

----------------------------------------------------------

The files are

  readme-jim.txt	you're looking at it

  contents.html		html file with links to the pictures and what-not

  jims1Rand.c		code for alternatives to srand(), rand()
  jims1Rand.h		header for jims1Rand.c

  *.out			various output files from the programs
  *.gnu		        gnuplot files to create plots from .out files

  test1.c		main() routine to exercise jims1Rand and
			the sytem rand(), finding mean and std-dev
  test1.out		output from test1
  run			compile and run test1

  test2.c		main() to generate random (x,y) points
    Usage: ./test2 randRoutine nPoints nDim
      where randRoutine is an integer, 0=>rand(), 1=>jims1Rand()
            nPoints is the number of lines to output.
            nDim = number of randoms per line


  ./run2-0		creates test2-0.out and test2-0.png
			showing 2-D scatter of system rand()

  ./run2-1		creates test2-1.out and test2-1.png
			showing 2-D scatter of jim1Rand()

  run2-0-3		same with 3-D, sys rand()
  run2-1-3		same with 3-D, jim1Rand

  test3.c		main() to generate slice of (x,y) points
			where zMin<z<zMax from (x,y,z) triples

  run3-0		slice plot for sys rand()
  run3-1		slice plot for jim1Rand   (shows non-random pattern)



----------------------------------------------------------

What it does as of Dec 5

 The executable test1 produces test1.out which shows 
 some statistics of two random number generators.  
 At this level of detail (looking at the lists, comparing
 mean and standard deviation) they both look pretty random.
 
 The test2 programs and run2-* scripts show 2-dimensonal scatter
 plots of several different random number generators.

 The test3 programs run3-* scripts show 2-dim slices of 
 3-dimensional scatter plots of several random number generators.
 In other words, 

 
----------------------------------------------------------
 
Added on Dec 9

 * Look at predictability of rand(n+1) by rand(n)
   by plotting consecutive pairs.
   (DONE, with test2.c and files run2-0 and run2-1)

 * Look at predictability of r(n+2) from (n,n+1) by 
   plotting consecutive tripes as (x,y,z).
   (DONE, with test2.c and files run2-0-3 and run2-1-3)
   
 * Look at a *slice* of that (x,y,z) plot, say only
   the points (x,y) where 0.4<z<0.41
   ( *NOTE* I think this one is going to be the most interesting.)
   (DONE, with test3.c and files run3-0 and run3-1)

   The test3-0.png still looks random, 
   but the test3-1.png sure doesn't.  So my prediction worked.

-------------------------------------------------

Not done

 * Do all of the above for high-order bits vs 
   low-order bits

 * Add "shuffling" to jim1 routines to 
   see if things improve (see Numerical Recipes)
   
 * Test Knuth's "subtractive" algorithm

-------------------------------------------------

All the plotting can be done easily with gnuplot, 
using these C routines to output 2 or 3 columns of 
numbers 

  x1 y1
  x2 y2
  ...

and then just something like

 $gnuplot
 > set terminal png color
 > set output "file.png"
 > plot "test.out" with points

See the gnuplot manual in ../DiscreteCosineTransform/


-----------------------------------------------------------