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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