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RNGs in Mathematica for Experimentation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg39617] RNGs in Mathematica for Experimentation
*From*: "flip" <flip_alpha at safebunch.com>
*Date*: Wed, 26 Feb 2003 02:42:41 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Hi All,
below, please find a discussion regarding several RNGs by Dr. Marsaglia. I
would really appreciate if one of you Mathematica experts could convert these very
short "C-code" RNGs and post the results. I would like to experiment with
these.
As a recommendation, it would be nice to have a user selectable RNG option
that allows you to choose simple LCG type, ICG, shift, Fibonacci, KISS, MWC,
CMWC, etc., et. al. for use in experiments (perhaps an option for Mathematica 5.0).
Anyway, I'd appreciate if someone could help out here.
Regards, Flip
To email me, remove "_alpha".
*** Newsgroup Discussion and C-Code for 3 RNG's by Marsaglia ***
"flip" <flip_alpha at safebunch.com> wrote in message
news:1046095684.571971 at news-1.nethere.net...
> Hi All,
> there are many RNG's out there with huge periods.
> For example:
> 1. Mersenne Twister (MT)
> 2. Add With Carry (AWC)
> 3. Subtract With Borrow (SWB)
> 4. KISS
> There is also Marsaglia's DIEHARD battery of tests to test the quality of
> these RNGs (in fact, 2 - 4 are his RNG's). Obviously, these RNGs are not
> good for crypto applications, but are good for anything Monte Carlo.
> Questions:
> 1. My question is this, which is the best overall RNG out there, where
best
> means that it passes all of the DIEHARD tests and has a huge period?
> 2. Is there a RNG which can be implemented in a FPGA to simulate "White
> Noise"?
> Thank you for any references (as I am looking through Google as we speak),
> but cannot believe that many haven't been down this road.
> Flip
>
It doesn't seem reasonable to consider that there is a best.
Just as the "Miss America Contest" judges have to view
talent, swim suit, evening gown and interview scores
before voting, for RNG's one should consider test results,
speed, simplicity, period length, size and possibly other
factors, depending on the application. (My own judging
tends to give extra weight to the equivalent of the
swim-suit competition---beauty of the underlying theory.)
Most of us are likely to give the most weight to
test results: the RNG should provide results
consistent with the underlying probability theory
for a wide variety of simulation problems, such
as those in the Diehard battery of tests.
The C procedure MWC256( ) listed below rates highly in all
categories. So does CMWC4096( ) below, with, so far,
record period.
The Mersenne Twister rates highly in test results and
period-length, but poorly in simplicity and swim-suit,
(beauty of underlying theory). C versions I have
seen take several dozens of statements.
Add-With-Carry (AWC) and Subtract-With-Borrow (SWB)
have been supplemented with Multiply-With-Carry (MWC)
and Complimentary-Multiply-With-Carry (CMWC).
I use SWB now only to provide IEEE standard 64-bit reals
in [0,1) directly, without the usual floating of integers. It was
the basis for the RNG in another system.
The KISS generator rates highly in test results and
simplicity, as Keep It Simple Stupid was the theme for
its name. Its swim suit rating is high (in each of the
three pieces). The KISS RNG is used to provide randomness
by several gaming companies. The third component has had
several versions, depending on whether it is implemented in
assembler, C or Fortran. Unfortunately, Fortran does
not provide an easy way to get the top and bottom halves
of the 64-bit product of two 32-bit integers.
Here is the latest version of KISS, in C:
unsigned long KISS(){
static unsigned long x=123456789,y=362436000,z=521288629,c=7654321;
unsigned long long t, a=698769069LL;
x=69069*x+12345;
y^=(y<<13); y^=(y>>17); y^=(y<<5);
t=a*z+c; c=(t>>32);
return x+y+(z=t);
}
Choosing random seed values for x,y,z,and c provides
an excellent source of 32-bit random integers, with
period >2^125---adequate, but much shorter than the
remarkable 2^19937 of the Mersenne Twister or the
2^8222 of MWC256( ). For a really really long period,
consider CMWC4096( ) with period 2^131086,
listed after MWC256( ):
static unsigned long Q[256],c=362436;
unsigned long MWC256(void){
unsigned long long t,a=1540315826LL;
unsigned long x;
static unsigned char i=255;
t=a*Q[++i]+c; c=(t>>32);
x=t+c; if(x<c){x++;c++;}
return(Q[i]=x); }
Choosing random values for the static array Q[256] and c
puts you at a random starting place in the base b=2^32-1
expansion of 1/p, where p is the prime 1540315826*b^256-1.
The expansion of 1/p has cycles of more than 10^2475 base-b
'digits', and they are returned in reverse order, from a
random starting point in the cycle. One attraction of
MWC256 is that it allows us to index the Q-array with
a byte.
Now for the really really big:
static unsigned long Q[4096],c=362436;
unsigned long CMWC4096(void){
unsigned long long t, a=18782LL;
static unsigned long i=4095;
unsigned long x,r=0xfffffffe;
i=(i+1)&4095;
t=a*Q[i]+c;
c=(t>>32); x=t+c; if(x<c){x++;c++;}
return(Q[i]=r-x); }
This RNG has period > 2^131086, some 10^33459 times as
long as that of the Mersenne Twister and rates highly in
all categories. It provides the more than 10^39460 base-b digits
in the expansion of (p-1)/p, where p is the prime
p=18782*b^4096+1. , b=2^32-1. Those base-b 'digits' are
returned in reverse order from a random starting point
determined by the random choice of the initial values in Q[4096] and c.
The above are my own ratings. As a judge in the contest,
you may want to rate them yourself, and compare .
George Marsaglia
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