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What's the best random number generator?

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Henry Choy

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Feb 3, 1995, 11:43:06 AM2/3/95

I found that the basic rand() in C is not entirely uniformly
distributed, but it's not too bad.

Over all these years, what is the best random number generator?

--
Henry Choy "Math class is hard" - Barbie

e-mail: ch...@cs.usask.ca I AGREE!

Harry Smith

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Feb 3, 1995, 4:29:59 PM2/3/95

Try this one:

/* Start of file Random.C */

#include <stdio.h> /* Definitions for stream input/output */
#include <math.h> /* Definitions for the math floating point package */

#define MAXLONG 0x7fffffffL

char *Name = "Random - Test a version of Minimal Standard random number generator";
char *Version= "C Version 1.0, last revised: 1994-09-03, 1600 hours";
char *Author = "Copyright (c) 1981-1994 by author: Harry J. Smith,";
char *Address= "19628 Via Monte Dr., Saratoga, CA 95070. All rights reserved.";

/* Notes: See article "RANDOM NUMBER GENERATORS: GOOD ONES ARE HARD TO FIND"
* by Strphen K. Parks and Keath W. Miller in Communications of the ACM,
* Oct 1988 pg 1192. It defines the Minimal Standard random number generator
* as f(z) = 16807 * z mod 2147483647. 16807 = 7^5, 2147483647 = 2^31 - 1.
* The seed must be between 1 and 2^31 - 2 inclusive. Period is 2^31 - 2.
*/

/* Developed in the C subset of Borland C++ Version 4.0 */

/* Global variables */
long SeedL = 1L;

/* Procedure prototypes */
void init( void); /* Initialize */
void ReadReal( char *Mess, double Min, double Max, double Nom, double *I); /*
Read in a Real from keyboard */
void RandomizI2( void); /* Integer Version 2 */
double RandomI2( void); /* Integer Version 2 */

/* -------------------------------------- */
void init( void) /* Initialize */
{
printf("\r\n%s\r\n%s\r\n%s\r\n%s\r\n", Name, Version, Author, Address);
} /* End of init */

/* -------------------------------------- */
void ReadReal( char *Mess, double Min, double Max, double Nom, double *R) /*
Read in a Real from keyboard */
{
char St[129];
char *ior;
int Stat;

do {
do {
printf("%s", Mess);
printf(" [%.1lf, %.1lf] (ENTER => %.1lf): ", Min, Max, Nom);
ior = gets( St);
} while (ior == NULL);
Stat = sscanf( St, "%lf", R);
} while (!(((Stat == 1) && (*R >= Min) && (*R <= Max)) || (St[0] == 0)));
if (St[0] == 0) *R = Nom;
printf("Input = %.1lf\n\n", *R);
} /* End of ReadReal */

/* -------------------------------------- */
void RandomizI2( void) /* Integer Version 2 */
{
double R;
do {
printf("\nInput a random number seed from 1 thru 2147483646, or 0 to exit\n");
ReadReal("Seed = ", 0, 2147483646L, (double)SeedL, &R);
if (fabs(R) <= MAXLONG)
SeedL = R;
else
SeedL = -1;
} while ((SeedL < 0) || (2147483646L < SeedL));
} /* End RandomizI2 */

/* -------------------------------------- */
double RandomI2( void) /* Integer Version 2 */
/* MaxLongInt must be 2^31 - 1 or larger, it is 2^31 - 1, method works */
/* Run time = 0.292 milliseconds / call, 16 mHz 80386, 10 mHz 80387 */
/* The returned double is in the range 0.0 < d < 1.0 (never = 0.0 or = 1.0) */
{
static long A = 16807L;
static long M = 2147483647L;
static long Q = 127773L; /* M div A */
static long R = 2836L; /* M mod A */
long Lo, Hi, Test;

Hi = SeedL / Q;
Lo = SeedL % Q;
Test = A * Lo - R * Hi;
if (Test > 0)
SeedL = Test;
else
SeedL = Test + M;
return (double)SeedL / M;
} /* End of RandomI2 */

/* -------------------------------------- */
int main(void) {
double Ra; /* Local variables */
int i, n;

init();
RandomizI2();
for( ; SeedL != 0; ) {
printf("Seed(0) = %ld\n", SeedL);
if (SeedL == 1) {
n = 10000;
printf(
"For an initial seed(0) = 1, the correct value of seed(10000) = 1043618065\n");
} else n = 1;
for (i = 0; i < n; i++) {
Ra = RandomI2();
}
printf("Seed(%d) = %ld\n", n, SeedL);
printf("Ra = %.18f\n", Ra);
RandomizI2();
}
printf("Random: Exit");
return 0; /* Exit main */
} /* End of main */

/* End of file Random.C */

--
| Harry J. Smith
| 19628 Via Monte Dr., Saratoga, CA 95070-4522, USA
| Home Phone: 408 741-0406, Work Phone: 408 235-5088 (Voice Mail)
| EMail: HJS...@ix.netcom.com on the Internet via Netcom NetCruiser
--

David C. Jones

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Feb 4, 1995, 2:54:58 AM2/4/95

In <3gu78n$f...@ixnews3.ix.netcom.com> HJS...@ix.netcom.com (Harry Smith) writes:

>In <3gtmeq$b...@tribune.usask.ca> ch...@cs.usask.ca (Henry Choy) writes:

>>
>>I found that the basic rand() in C is not entirely uniformly
>>distributed, but it's not too bad.
>>
>>Over all these years, what is the best random number generator?
>>

>Try this one:

>* Oct 1988 pg 1192. It defines the Minimal Standard random number generator
>* as f(z) = 16807 * z mod 2147483647. 16807 = 7^5, 2147483647 = 2^31 - 1.
>* The seed must be between 1 and 2^31 - 2 inclusive. Period is 2^31 - 2.

I've got one very similar to it. In fact, I think Turbo Pascal uses
one very similar to it. However, there are better ones.

I don't have the code with me off hand, but for my assembly class we
designed a generator with three seeds. Each seen had a range from
1 - p, where p was a very high prime number close to 35565 and
each range was different. This effectively gave you a generator
that could go close to (2^16)^3 times without ever repeating the
same seed. If you've ever done any work on this stuff (I spent
about 8 weeks on an IS in it high school) you'll know that
the more individual seeds you have, the smoother your distrubtion
is. With this algorithm, you could put in as many seeds as you
wanted to and increase the smoothness of the distribution by
any precision you wanted to. If you want to see it I'll try to
dig it up out of my notes.

davy J.

Harry J. Smith

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Feb 4, 1995, 7:08:04 PM2/4/95

In <3gvbsi$o...@hermes.unt.edu> djo...@ponder.csci.unt.edu (David C. Jones) writes:

>
>In <3gu78n$f...@ixnews3.ix.netcom.com> HJS...@ix.netcom.com (Harry Smith) writes:
>
>>In <3gtmeq$b...@tribune.usask.ca> ch...@cs.usask.ca (Henry Choy) writes:
>
>>>
>>>I found that the basic rand() in C is not entirely uniformly
>>>distributed, but it's not too bad.
>>>
>>>Over all these years, what is the best random number generator?
>>>
>
>>Try this one:
>
>>* Oct 1988 pg 1192. It defines the Minimal Standard random number generator
>>* as f(z) = 16807 * z mod 2147483647. 16807 = 7^5, 2147483647 = 2^31 - 1.
>>* The seed must be between 1 and 2^31 - 2 inclusive. Period is 2^31 - 2.
>
>I've got one very similar to it. In fact, I think Turbo Pascal uses
>one very similar to it. However, there are better ones.

I agree, try this one:

#include <stdlib.h> /* defines exit() and malloc() */
#include <string.h> /* defines strlen etc. */
#include <stdio.h> /* defines NULL, EOF, stdin, stdout, stderr, */
/* fprintf(), fileno(), ... for standard i/o */

extern int errno;

char name[] =
"Rando - Random Number Generater from a data encryption system written in C";
char version[] = "Version 6.0, 1995-02-04, 0900 hours";
char author[] = "Copyright (c) 1987-1995 by author: Harry J. Smith";
char address[] =

char usage1[] = "usage: RANDO key";
char usage2[] = " key is alphanumeric key to generate the seed";

#define POOLSIZE 2048 /* Number of integers in random number pool */
#define MAXKEY 24 /* Number of characters in short key */
#define TRUE 1
#define FALSE 0

int ri[9], /* Current random integers from the 9 generators */
rs[9], /* Initial random seeds for the 9 generators */
rr, /* Random running combination of ri[0] thru ri[8] */
*pool, /* Pool of POOLSIZE random integers */
next = 0, /* Next random no. routine to use, cycles 0 thru 8 */
poolmask = /* Mask for index to random pool = 2**N - 1 */
POOLSIZE-1;
char *buf, /* Buffer for key input */
key[MAXKEY+1]; /* The short key string in standard form */

void init( char *argv[]); /* Initialize the process */
void standard( char *cp); /* Standardize a key string */
void convkey( void); /* Convert key to 9 seeds */
int rn( int n); /* random integer from ri[n], n<9 */
int rc9( void); /* Combination of rn(0) thru rn(8) */
int Rando( void); /* Return next 16 random bits */

/* --------------------------------------------------------------------- */
void main( int argc, char *argv[])
{
int i, j, ran;
char st[5];

if(argc != 2) { /* Validate arguments */
fprintf( stderr, "\n%s\n%s\n%s\n\n", name, version, author);
fprintf( stderr, "%s\n%s\n", usage1, usage2);
exit(1);
}
init( argv); /* Initialize the process */
for (i = 0; i < 340; i++) {
ran = Rando();
sprintf(st, "%4X", ran);
for (j = 0; j < 4; j++)
if (st[j] == ' ') st[j] = '0'; /* change leading blanks to 0 */
printf("%4s", st);
}
exit(0); /* Exit main */
} /* End of main (RANDO) */

/* --------------------------------------------------------------------- */
void init( char *argv[]) /* Initialize the process */
{
int i, j, len;

pool = (int *) malloc( 2 * POOLSIZE);
if (pool == (int *) NULL) {
fprintf( stderr, "RANDO: not enough memory\n");
exit(2);
}
buf = (char *) pool;
strcpy( buf, argv[1]);
standard( buf); /* Standardize long key */
fprintf( stderr, "%s = input key in standard form\n", buf);
if( (len = strlen( buf)) > MAXKEY) {
for( i = 1; i < MAXKEY; ++i) /* Hash long key */
buf[i] += buf[i-1];
for( i = len-2; i >= MAXKEY; --i)
buf[i] += buf[i+1];
}
for( i = 0; i <= MAXKEY; ++i) /* Clear short key */
key[i] = 0;
for( i = j = 0; i < len; ++i) { /* Compress long key */
key[j] += buf[i];
if( ++j == MAXKEY)
j = 0;
}
for( i = 0; i < MAXKEY; ++i) /* Fill out key replacing */
if( !key[i]) /* nulls with spaces */
key[i] = ' ';
standard( key); /* Standardize short key */
convkey(); /* Convert short key to random */
/* number seeds */
rr = rc9(); /* Init first word of running */
/* combination of ri[0]- ri[8] */
for( i = 0; i < POOLSIZE; ++i) { /* Initialize pool, random int */
pool[i] = rr += rn( next);
if( ++next == 9)
next = 0;
}
rr = rc9(); /* Init rr for first random */
/* index into pool */

fprintf( stderr, "%s = short key in standard form\n", key);
for( i = 0; i < 9; ++i)
fprintf( stderr, "rs[%d] = %-7d ", i, rs[i]);
for( i = 0; i < 9; ++i)
fprintf( stderr, "ri[%d] = %-7d ", i, ri[i]);
fprintf( stderr, "\n");
return;

} /* End of init */

/* --------------------------------------------------------------------- */
void standard( char *cp)
{
/* Standardize a key string
* Output characters are from 33 to 95 inclusive
*/

for( ; *cp; ++cp)
{
*cp &= 127;

if( *cp > 95)
*cp -= 32;

if( *cp < 32)
*cp += 32;

if( *cp == ' ')
*cp = '/';
}
} /* End of standard */

/* --------------------------------------------------------------------- */
void convkey( void)
{
/* Convert the encryption key to 9 seeds
*
* input: key[i], i = 0, ..., MAXKEY-1
* output: rs[j], j = 0, ..., 8
*
* [0] [1] [2] [3] [4] [5] [6] ... [22] [23]
* 111111 111111 111111 111111 111111 111111 111111 111111 111111
* 6 6 4 2 6 6 2 4 6 6 6
* ****** seed0 *****...... seed1 ......***** seed2 ... seed8 ******
* 16 BITS 16 BITS 16 BITS 16 BITS
*
* 144 bit key converted to 144 bits of seed,
* seeds of zero are changed to 1.
*/

int i, j,
keyi[MAXKEY]; /* Key in integer form, 6 bits each int */

for( i = 0; i < MAXKEY; ++i)
keyi[i] = key[i] & 63;

for( i = 0, j = 0; j < 9; i += 8, j += 3)
{
rs[j] = ( keyi[i] << 10) |
( keyi[i+1] << 4) |
( keyi[i+2] >> 2);

rs[j+1] = ( keyi[i+2] << 14) |
( keyi[i+3] << 8) |
( keyi[i+4] << 2) |
( keyi[i+5] >> 4);

rs[j+2] = ( keyi[i+5] << 12) |
( keyi[i+6] << 6) |
keyi[i+7];
}
for( j = 0; j < 9; ++j) { /* Do not allow a zero seed */
if( !rs[j]) rs[j] = 1;
ri[j] = rs[j];
}
} /* End of convkey */

/* --------------------------------------------------------------------- */
int rn( int n)
{
/*
* Returns the next random integer from ri[n], 0 <= n <= 8
*/

static int repeat = TRUE;
int i;

switch( n)
{
case 0:

/* 1st Congruential Generator, 16 bits
* Generates Random integers from -32768 to 32767 inclusive
* Cycle length = 65536 = 2**16
*/

return( ri[0] = 25173 * ri[0] + 6925);

case 1:

/* 1st Shift-register Generator, 16 bits
* Random integer from -32768 to 32767 inclusive, 0 not generated
* Generator = -22620, Cycle length = 65535 = 3 * 5 * 17 * 257
*/

if( ri[1] & 1)
return( ri[1] = ( (unsigned) ri[1] >> 1) ^ -22620);

return( (unsigned) ri[1] >>= 1);

case 2:

/* 2nd Congruential Generator, 16 bits
* Random integer from -32768 to 32767 inclusive
* Cycle length = 65537 = prime, zero repeats once
*/

if( !ri[2])
{
if( repeat)
{
repeat = FALSE;
return( 0);
}
else repeat = TRUE;
}

return( ri[2] = 23629 * ri[2] + 13849);

case 3:

/* 2nd Shift-register Generator, 16 bits
* Random integer from -32768 to 32767, not all generated
* Generator = -07493, Cycle length = 65521 = prime, (65536 - 15)
*/

if( ri[3] & 1)
ri[3] = ( (unsigned) ri[3] >> 1) ^ -7493;
else
(unsigned) ri[3] >>= 1;

if( ri[3] == 1)
for( i = 0; i < 14; ++i)
rn( 3); /* Throw 14 away */

return( ri[3]);

case 4:

/* 3rd Congruential Generator, 16 bits
* Random integer from -32768 to 32767, not all generated
* Cycle length = 65519 = prime, (65536 - 17)
*/

ri[4] = 4821 * ri[4] + 13001;
if( !ri[4])
for( i = 0; i < 17; ++i)
rn( 4); /* Throw 17 away */

return( ri[4]);

case 5:

/* 3rd Shift-register Generator, 16 bits
* Random integer from -32768 to 32767, not all generated
* Generator = -25501, Cycle length = 65497 = prime, (65536 - 39)
*/

if( ri[5] & 1)
ri[5] = ( (unsigned) ri[5] >> 1) ^ -25501;
else
(unsigned) ri[5] >>= 1;

if( ri[5] == 1)
for( i = 0; i < 38; ++i)
rn( 5); /* Throw 38 away */

return( ri[5]);

case 6:

/* 4th Congruential Generator, 16 bits
* Random integer from -32768 to 32767, not all generated
* Cycle length = 65479 = prime, (65536 - 57)
*/

ri[6] = 10349 * ri[6] + 7001;
if( !ri[6])
for( i = 0; i < 57; ++i)
rn( 6); /* Throw 57 away */

return( ri[6]);

case 7:

/* 4th Shift-register Generator, 16 bits
* Random integer from -32768 to 32767, not all generated
* Generator = -18916, Cycle length = 65449 = prime, (65536 - 87)
*/

if( ri[7] & 1)
ri[7] = ( (unsigned) ri[7] >> 1) ^ -18916;
else
(unsigned) ri[7] >>= 1;

if( ri[7] == 1)
for( i = 0; i < 86; ++i)
rn( 7); /* Throw 86 away */

return( ri[7]);

case 8:

/* 5th Congruential Generator, 16 bits
* Random integer from -32768 to 32767, not all generated
* Cycle length = 65447 = prime, (65536 - 89)
*/

ri[8] = 30133 * ri[8] + 14001;
if( !ri[8])
for( i = 0; i < 89; ++i)
rn( 8); /* Throw 89 away */

return( ri[8]);
}
return( 0); /* This line is never reached */
} /* End of rn */

/* --------------------------------------------------------------------- */
int rc9( void)
{
/*
* Combination of rn(0) thru rn(8), one each
*/

int i, rc;

for( rc = 0, i = 0; i < 9; ++i)
rc += rn( i);

return( rc);
} /* End of rc9 */

/* --------------------------------------------------------------------- */
int Rando( void) /* Return next 16 random bits */
{
int ran;
int ind; /* Random index into pool of random integers */

ind = rr & poolmask; /* Random index to random pool */
ran = pool[ind]; /* XOR data with random int */
pool[ind] ^= ( rr += rn( next)); /* XOR random int with next rr */
if( ++next == 9) /* Cycle to next generator */
next = 0;
return ran;
} /* End of Rando */
/* End of RANDO.C */

harm...@world.std.com

unread,

Feb 5, 1995, 3:47:45 PM2/5/95

In article <3h14t4$a...@ixnews1.ix.netcom.com>,

Harry J. Smith <HJS...@ix.netcom.com> wrote:
>In <3gvbsi$o...@hermes.unt.edu> djo...@ponder.csci.unt.edu (David C. Jones) writes:
>
>>
>>In <3gu78n$f...@ixnews3.ix.netcom.com> HJS...@ix.netcom.com (Harry Smith) writes:
>>
>>>In <3gtmeq$b...@tribune.usask.ca> ch...@cs.usask.ca (Henry Choy) writes:
>>
>>>>
>>>>I found that the basic rand() in C is not entirely uniformly
>>>>distributed, but it's not too bad.
>>>>
>>>>Over all these years, what is the best random number generator?
>>>>
>>
>>>Try this one:
>>
>>>* Oct 1988 pg 1192. It defines the Minimal Standard random number generator
>>>* as f(z) = 16807 * z mod 2147483647. 16807 = 7^5, 2147483647 = 2^31 - 1.
>>>* The seed must be between 1 and 2^31 - 2 inclusive. Period is 2^31 - 2.

The random number generator in O-Matrix is uniformly distributed
with a period of the order 2^144 whereas most algorithms are of the
order 2^32. If you are interested I can get forward the
source of the original algorithm.

Herman Rubin

unread,

Feb 6, 1995, 10:09:33 AM2/6/95

This keeps coming up, and many have posted various procedures.

There is NO procedure which can be "certifiably" good. A test which
it is guaranteed to fail is the test that it came from the procedure,
which truly random numbers will definitely not appear to be, at least
beyond the seed length or a version of it. Some have mentioned the
period, but one believed to be good was recently found to give bad
answers in a "real" problem, and its period is 2^1279 - 1, which
exceeds the total number of bits which can be constructed in many
models of the total life, including the future life, of the universe.

There are some procedures believed cryptographically strong, but they
are really time consuming. I would personally suggest taking some
fairly good procedures, with seeds in the thoudands of bits, and
XORing the results with physical random numbers, which may need to
be recycled. If the pseudo-random numbers are good, the result is
likely to be good, as the physical random nubers are extremely
unlikely to match the patterns in the pseudo-random ones, and this
same "non-matching" property is also likely to remove the weaknesses
of the pseudo-random nubers, while the pseudo-random ones will assure
the uniformity.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!a.stat!hrubin(UUCP)

Byron Bodo

unread,

Feb 8, 1995, 1:06:13 AM2/8/95

Keep in mind the old saw that "Random # generators are like sex.
Even when they're bad, their often still pretty good."

The correct question should be "what's the best random # generator
for your particular project?" The quick & dirty generators are often
adequate for many common needs.

Here's some FORTRAN source for what the authors claim is a superior
generator. I can't vouch for it as I haven't tried it.

-bb

****************************************************************
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<HR><TITLE>random number generator: here it is</TITLE>
<H2>random number generator: here it is</H2><I>
Phillip Helbig <phe...@hs.uni-hamburg.de><BR>
2 Nov 1994 12:46:45 GMT<BR>Hamburger Sternwarte, Germany<BR>Posted to: <B><A HREF="comp.lang.fortran">comp.lang.fortran</A>
</B><BR>

</I><HR>
<PRE>
Hello everyone!

Well, here's what you've all been waiting for. Again, would
someone please crosspost this to the relevant groups (numerical
analysis and so on)?

I am including, separated by a row of 72 stars, my response
(received today) from Fred James from CERN, who wrote the code,
and my original email from James from about a year ago. What
you should cut out and compile starts right after the row of
plus signs in this original email. This consists of the
SUBROUTINE RANLUX, up until the END statement, and then the
PROGRAM RLXTST, which you can use to make sure everything is
working correctly on you computer, which includes the output
which should be produced in the comment lines at the end.

All I did was delete James's opening remarks, separate the
subroutine and the program, and compile them separately.
Except for the C's at the beginning of the lines, of course,
the test output on my computer (VAX 3100) was identical to the
test output in the RLXTST comments. All in all, less than 2
minutes work. Which speaks a lot for coding in absolutely
standard FORTRAN, as well as for the VAX compiler. (While most
compilers, if not all, which offer extensions of course support
`only' standard FORTRAN, some, such as IBM AIX on a RISC 6000,
have to be given options in order _NOT_ to support extensions
which are INCOMPATIBLE with the standard (like folding to lower
case). If you start a compiler with the defaults, then my view
is that it is OK to support extensions, but it should at least
compile standard FORTRAN without having to be explicitly told that
this is standard FORTRAN or otherwise produce results incompatible
with the standard.)

The Martin Luescher mentioned is a professor at DESY/University
of Hamburg who does theoretical elementary particle physics and
who wrote an article (DESY 93--133, September 1993, ISSN 0418-9833)
entitled `A Portable High-Quality Random Number Generator for
Lattice Field Theory Simulations' which I accidentally ran into
in the Hamburg Observatory library. Luescher directed me to Fred
James, and since then we've used the generator quite a bit, tested
it extensively just to be safe and have noticed no problems. For
most purposes, one could probably use the highest level with no
problems as far as CPU time goes.

Have fun!!

Phillip Helbig Tel. .............. +49 40 7252 4110
Hamburger Sternwarte Email .... phe...@hs.uni-hamburg.de
Gojenbergsweg 112 Fax ............... +49 40 7252 4198
D-21029 Hamburg Telex ............... 217884 hamst d

************************************************************************

Hello,
Thank you for your message about RANLUX. I am happy to hear of
satisfied users. Yes, I know a little about some of the various rng's
that are being proposed these days, in particular by the Numerical
Recipes people. I have told Bill Press about RANLUX and he has the
references. The area of rng's was always one of the weakest points in
Num. Recip., and I guess that will continue to be true, although their
algorithms will improve of course. They do a very good job in so many
areas, but you can't really expect them to be best in everything.

Press seems to be working closely with Marsaglia, which I also did at
one time, until I discovered Martin Luescher and Pierre L'Ecuyer who have
a more rigorous and more convincing approach to the problem. You may
know that Marsaglia once said: "Random numbers are like sex: ... Even
when they are bad they are still pretty good." However, we must remember
that the actual RANLUX algorithm owes much to several people including
Marsaglia:

1. Marsaglia invented the "subtract-with-borrow" algorithm which has
many practical advantages (fast and portable because all computations are
done in portable floating-point, very long period, and reasonably easy to
initialize, restart, etc.), but is not sufficiently random.
2. L'Ecuyer and Tuzaka (and perhaps others) recognized and proved that
Marsaglia's algorithm is in fact equivalent to a linear congruential
generator (which is bad) with a very big multiplier (which is good).
This gave a first basis for understanding the randomness properties and
in particular the flaws.
3. Luescher, using Kolmogorov's chaos theory, showed rigorously how to
improve the algorithm by skipping, and how much you have to skip to make
it lose all memory of its past. He also can calculate the period
exactly, with or without skipping.
4. I wrote the Fortran version of the generator.

Feel free to propagate RANLUX, but make sure to give proper credit
always to Luescher and the two articles in Computer Physics Communications
where it all is published. Thanks again, Fred.

************************************************************************

I guess that theoretical cosmology is easier than experimental cosmology
(less time-consuming) but this random number generator should be good enough
for both. The references are Lu:scher, Computer Physics Communications
79 (1994) 100, and James, CPC 79 (1994) 111.

+++++++++++++++++++++++++++++++++++
SUBROUTINE RANLUX(RVEC,LENV)
C Subtract-and-borrow random number generator proposed by
C Marsaglia and Zaman, implemented by F. James with the name
C RCARRY in 1991, and later improved by Martin Luescher
C in 1993 to produce "Luxury Pseudorandom Numbers".
C Fortran 77 coded by F. James, 1993
C
C references:
C M. Luscher, Computer Physics Communications 79 (1994) 100
C F. James, Computer Physics Communications 79 (1994) 111
C
C LUXURY LEVELS.
C ------ ------ The available luxury levels are:
C
C level 0 (p=24): equivalent to the original RCARRY of Marsaglia
C and Zaman, very long period, but fails many tests.
C level 1 (p=48): considerable improvement in quality over level 0,
C now passes the gap test, but still fails spectral test.
C level 2 (p=97): passes all known tests, but theoretically still
C defective.
C level 3 (p=223): DEFAULT VALUE. Any theoretically possible
C correlations have very small chance of being observed.
C level 4 (p=389): highest possible luxury, all 24 bits chaotic.
C
C!!! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C!!! Calling sequences for RANLUX: ++
C!!! CALL RANLUX (RVEC, LEN) returns a vector RVEC of LEN ++
C!!! 32-bit random floating point numbers between ++
C!!! zero (not included) and one (also not incl.). ++
C!!! CALL RLUXGO(LUX,INT,K1,K2) initializes the generator from ++
C!!! one 32-bit integer INT and sets Luxury Level LUX ++
C!!! which is integer between zero and MAXLEV, or if ++
C!!! LUX .GT. 24, it sets p=LUX directly. K1 and K2 ++
C!!! should be set to zero unless restarting at a break++
C!!! point given by output of RLUXAT (see RLUXAT). ++
C!!! CALL RLUXAT(LUX,INT,K1,K2) gets the values of four integers++
C!!! which can be used to restart the RANLUX generator ++
C!!! at the current point by calling RLUXGO. K1 and K2++
C!!! specify how many numbers were generated since the ++
C!!! initialization with LUX and INT. The restarting ++
C!!! skips over K1+K2*E9 numbers, so it can be long.++
C!!! A more efficient but less convenient way of restarting is by: ++
C!!! CALL RLUXIN(ISVEC) restarts the generator from vector ++
C!!! ISVEC of 25 32-bit integers (see RLUXUT) ++
C!!! CALL RLUXUT(ISVEC) outputs the current values of the 25 ++
C!!! 32-bit integer seeds, to be used for restarting ++
C!!! ISVEC must be dimensioned 25 in the calling program ++
C!!! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
DIMENSION RVEC(LENV)
DIMENSION SEEDS(24), ISEEDS(24), ISDEXT(25)
PARAMETER (MAXLEV=4, LXDFLT=3)
DIMENSION NDSKIP(0:MAXLEV)
DIMENSION NEXT(24)
PARAMETER (TWOP12=4096., IGIGA=1000000000,JSDFLT=314159265)
PARAMETER (ITWO24=2**24, ICONS=2147483563)
SAVE NOTYET, I24, J24, CARRY, SEEDS, TWOM24, TWOM12, LUXLEV
SAVE NSKIP, NDSKIP, IN24, NEXT, KOUNT, MKOUNT, INSEED
INTEGER LUXLEV
LOGICAL NOTYET
DATA NOTYET, LUXLEV, IN24, KOUNT, MKOUNT /.TRUE., LXDFLT, 0,0,0/
DATA I24,J24,CARRY/24,10,0./
C default
C Luxury Level 0 1 2 *3* 4
DATA NDSKIP/0, 24, 73, 199, 365 /
Corresponds to p=24 48 97 223 389
C time factor 1 2 3 6 10 on slow workstation
C 1 1.5 2 3 5 on fast mainframe
C
C NOTYET is .TRUE. if no initialization has been performed yet.
C Default Initialization by Multiplicative Congruential
IF (NOTYET) THEN
NOTYET = .FALSE.
JSEED = JSDFLT
INSEED = JSEED
WRITE(6,'(A,I12)') ' RANLUX DEFAULT INITIALIZATION: ',JSEED
LUXLEV = LXDFLT
NSKIP = NDSKIP(LUXLEV)
LP = NSKIP + 24
IN24 = 0
KOUNT = 0
MKOUNT = 0
WRITE(6,'(A,I2,A,I4)') ' RANLUX DEFAULT LUXURY LEVEL = ',
+ LUXLEV,' p =',LP
TWOM24 = 1.
DO 25 I= 1, 24
TWOM24 = TWOM24 * 0.5
K = JSEED/53668
JSEED = 40014*(JSEED-K*53668) -K*12211
IF (JSEED .LT. 0) JSEED = JSEED+ICONS
ISEEDS(I) = MOD(JSEED,ITWO24)
25 CONTINUE
TWOM12 = TWOM24 * 4096.
DO 50 I= 1,24
SEEDS(I) = REAL(ISEEDS(I))*TWOM24
NEXT(I) = I-1
50 CONTINUE
NEXT(1) = 24
I24 = 24
J24 = 10
CARRY = 0.
IF (SEEDS(24) .EQ. 0.) CARRY = TWOM24
ENDIF
C
C The Generator proper: "Subtract-with-borrow",
C as proposed by Marsaglia and Zaman,
C Florida State University, March, 1989
C
DO 100 IVEC= 1, LENV
UNI = SEEDS(J24) - SEEDS(I24) - CARRY
IF (UNI .LT. 0.) THEN
UNI = UNI + 1.0
CARRY = TWOM24
ELSE
CARRY = 0.
ENDIF
SEEDS(I24) = UNI
I24 = NEXT(I24)
J24 = NEXT(J24)
RVEC(IVEC) = UNI
C small numbers (with less than 12 "significant" bits) are "padded".
IF (UNI .LT. TWOM12) THEN
RVEC(IVEC) = RVEC(IVEC) + TWOM24*SEEDS(J24)
C and zero is forbidden in case someone takes a logarithm
IF (RVEC(IVEC) .EQ. 0.) RVEC(IVEC) = TWOM24*TWOM24
ENDIF
C Skipping to luxury. As proposed by Martin Luscher.
IN24 = IN24 + 1
IF (IN24 .EQ. 24) THEN
IN24 = 0
KOUNT = KOUNT + NSKIP
DO 90 ISK= 1, NSKIP
UNI = SEEDS(J24) - SEEDS(I24) - CARRY
IF (UNI .LT. 0.) THEN
UNI = UNI + 1.0
CARRY = TWOM24
ELSE
CARRY = 0.
ENDIF
SEEDS(I24) = UNI
I24 = NEXT(I24)
J24 = NEXT(J24)
90 CONTINUE
ENDIF
100 CONTINUE
KOUNT = KOUNT + LENV
IF (KOUNT .GE. IGIGA) THEN
MKOUNT = MKOUNT + 1
KOUNT = KOUNT - IGIGA
ENDIF
RETURN
C
C Entry to input and float integer seeds from previous run
ENTRY RLUXIN(ISDEXT)
TWOM24 = 1.
DO 195 I= 1, 24
NEXT(I) = I-1
195 TWOM24 = TWOM24 * 0.5
NEXT(1) = 24
TWOM12 = TWOM24 * 4096.
WRITE(6,'(A)') ' FULL INITIALIZATION OF RANLUX WITH 25 INTEGERS:'
WRITE(6,'(5X,5I12)') ISDEXT
DO 200 I= 1, 24
SEEDS(I) = REAL(ISDEXT(I))*TWOM24
200 CONTINUE
CARRY = 0.
IF (ISDEXT(25) .LT. 0) CARRY = TWOM24
ISD = IABS(ISDEXT(25))
I24 = MOD(ISD,100)
ISD = ISD/100
J24 = MOD(ISD,100)
ISD = ISD/100
IN24 = MOD(ISD,100)
ISD = ISD/100
LUXLEV = ISD
IF (LUXLEV .LE. MAXLEV) THEN
NSKIP = NDSKIP(LUXLEV)
WRITE (6,'(A,I2)') ' RANLUX LUXURY LEVEL SET BY RLUXIN TO: ',
+ LUXLEV
ELSE IF (LUXLEV .GE. 24) THEN
NSKIP = LUXLEV - 24
WRITE (6,'(A,I5)') ' RANLUX P-VALUE SET BY RLUXIN TO:',LUXLEV
ELSE
NSKIP = NDSKIP(MAXLEV)
WRITE (6,'(A,I5)') ' RANLUX ILLEGAL LUXURY RLUXIN: ',LUXLEV
LUXLEV = MAXLEV
ENDIF
INSEED = -1
RETURN
C
C Entry to ouput seeds as integers
ENTRY RLUXUT(ISDEXT)
DO 300 I= 1, 24
ISDEXT(I) = INT(SEEDS(I)*TWOP12*TWOP12)
300 CONTINUE
ISDEXT(25) = I24 + 100*J24 + 10000*IN24 + 1000000*LUXLEV
IF (CARRY .GT. 0.) ISDEXT(25) = -ISDEXT(25)
RETURN
C
C Entry to output the "convenient" restart point
ENTRY RLUXAT(LOUT,INOUT,K1,K2)
LOUT = LUXLEV
INOUT = INSEED
K1 = KOUNT
K2 = MKOUNT
RETURN
C
C Entry to initialize from one or three integers
ENTRY RLUXGO(LUX,INS,K1,K2)
IF (LUX .LT. 0) THEN
LUXLEV = LXDFLT
ELSE IF (LUX .LE. MAXLEV) THEN
LUXLEV = LUX
ELSE IF (LUX .LT. 24 .OR. LUX .GT. 2000) THEN
LUXLEV = MAXLEV
WRITE (6,'(A,I7)') ' RANLUX ILLEGAL LUXURY RLUXGO: ',LUX
ELSE
LUXLEV = LUX
DO 310 ILX= 0, MAXLEV
IF (LUX .EQ. NDSKIP(ILX)+24) LUXLEV = ILX
310 CONTINUE
ENDIF
IF (LUXLEV .LE. MAXLEV) THEN
NSKIP = NDSKIP(LUXLEV)
WRITE(6,'(A,I2,A,I4)') ' RANLUX LUXURY LEVEL SET BY RLUXGO :',
+ LUXLEV,' P=', NSKIP+24
ELSE
NSKIP = LUXLEV - 24
WRITE (6,'(A,I5)') ' RANLUX P-VALUE SET BY RLUXGO TO:',LUXLEV
ENDIF
IN24 = 0
IF (INS .LT. 0) WRITE (6,'(A)')
+ ' Illegal initialization by RLUXGO, negative input seed'
IF (INS .GT. 0) THEN
JSEED = INS
WRITE(6,'(A,3I12)') ' RANLUX INITIALIZED BY RLUXGO FROM SEEDS',
+ JSEED, K1,K2
ELSE
JSEED = JSDFLT
WRITE(6,'(A)')' RANLUX INITIALIZED BY RLUXGO FROM DEFAULT SEED'
ENDIF
INSEED = JSEED
NOTYET = .FALSE.
TWOM24 = 1.
DO 325 I= 1, 24
TWOM24 = TWOM24 * 0.5
K = JSEED/53668
JSEED = 40014*(JSEED-K*53668) -K*12211
IF (JSEED .LT. 0) JSEED = JSEED+ICONS
ISEEDS(I) = MOD(JSEED,ITWO24)
325 CONTINUE
TWOM12 = TWOM24 * 4096.
DO 350 I= 1,24
SEEDS(I) = REAL(ISEEDS(I))*TWOM24
NEXT(I) = I-1
350 CONTINUE
NEXT(1) = 24
I24 = 24
J24 = 10
CARRY = 0.
IF (SEEDS(24) .EQ. 0.) CARRY = TWOM24
C If restarting at a break point, skip K1 + IGIGA*K2
C Note that this is the number of numbers delivered to
C the user PLUS the number skipped (if luxury .GT. 0).
KOUNT = K1
MKOUNT = K2
IF (K1+K2 .NE. 0) THEN
DO 500 IOUTER= 1, K2+1
INNER = IGIGA
IF (IOUTER .EQ. K2+1) INNER = K1
DO 450 ISK= 1, INNER
UNI = SEEDS(J24) - SEEDS(I24) - CARRY
IF (UNI .LT. 0.) THEN
UNI = UNI + 1.0
CARRY = TWOM24
ELSE
CARRY = 0.
ENDIF
SEEDS(I24) = UNI
I24 = NEXT(I24)
J24 = NEXT(J24)
450 CONTINUE
500 CONTINUE
C Get the right value of IN24 by direct calculation
IN24 = MOD(KOUNT, NSKIP+24)
IF (MKOUNT .GT. 0) THEN
IZIP = MOD(IGIGA, NSKIP+24)
IZIP2 = MKOUNT*IZIP + IN24
IN24 = MOD(IZIP2, NSKIP+24)
ENDIF
C Now IN24 had better be between zero and 23 inclusive
IF (IN24 .GT. 23) THEN
WRITE (6,'(A/A,3I11,A,I5)')
+ ' Error in RESTARTING with RLUXGO:',' The values', INS,
+ K1, K2, ' cannot occur at luxury level', LUXLEV
IN24 = 0
ENDIF
ENDIF
RETURN
END
C +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
PROGRAM LUXTST
C Exercise for the RANLUX Pseudorandom number generator.
C
DIMENSION RVEC(1000)
DIMENSION ISDEXT(25)
C
C check that we get the right numbers (machine-indep.)
WRITE (6,'(/A)') ' CALL RANLUX(RVEC,100)'
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX default numbers 1- 5:',
+ (RVEC(L),L=1,5)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX default numbers 101-105:',
+ (RVEC(L),L=1,5)
C
WRITE (6,'(/A)') ' CALL RLUXGO(0,0,0,0)'
CALL RLUXGO(0,0,0,0)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX luxury level 0, 1- 5:',
+ (RVEC(L),L=1,5)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX luxury level 0, 101-105:',
+ (RVEC(L),L=1,5)
C
WRITE (6,'(/A)') ' CALL RLUXGO(389,1,0,0)'
CALL RLUXGO(389,1,0,0)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX luxury p=389, 1- 5:',
+ (RVEC(L),L=1,5)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX luxury p=389, 101-105:',
+ (RVEC(L),L=1,5)
C
WRITE (6,'(/A)') ' CALL RLUXGO(75,0,0,0)'
CALL RLUXGO(75,0,0,0)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX luxury p= 75, 1- 5:',
+ (RVEC(L),L=1,5)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX luxury p= 75, 101-105:',
+ (RVEC(L),L=1,5)
C
WRITE (6,'(/A)') ' test restarting from the full vector'
CALL RLUXUT(ISDEXT)
WRITE (6,'(/A/(1X,5I14))') ' current RANLUX status saved:',ISDEXT
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX numbers 1- 5:',
+ (RVEC(L),L=1,5)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX numbers 101-105:',
+ (RVEC(L),L=1,5)
C
WRITE (6,'(/A)') ' previous RANLUX status will be restored'
CALL RLUXIN(ISDEXT)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX numbers 1- 5:',
+ (RVEC(L),L=1,5)
CALL RANLUX(RVEC,100)
WRITE (6,'(A/9X,5F12.8)') ' RANLUX numbers 101-105:',
+ (RVEC(L),L=1,5)
C
WRITE (6,'(/A)') ' test the restarting by skipping'
CALL RLUXGO(4,7674985,0,0)
CALL RLUXAT(I1,I2,I3,I4)
WRITE (6,'(A,4I10)') ' RLUXAT values =',I1,I2,I3,I4
DO 150 LI= 1, 10
150 CALL RANLUX(RVEC,1000)
CALL RLUXAT(I1,I2,I3,I4)
WRITE (6,'(A,4I10)') ' RLUXAT values =',I1,I2,I3,I4
CALL RANLUX(RVEC,200)
WRITE (6,'(A,2F10.6)') ' Next and 200th numbers are:',
+ RVEC(1), RVEC(200)
CALL RLUXGO(I1,I2,I3,I4)
CALL RANLUX(RVEC,200)
WRITE (6,'(A,2F10.6)') ' Next and 200th numbers are:',
+ RVEC(1), RVEC(200)
C
WRITE (6,'(/A)') ' The following should provoke an error message'
CALL RLUXGO(4,11111,31,0)
STOP
C
C OUTPUT FROM THE ABOVE TEST PROGRAM SHOULD BE:
C --------------------------------------------
C CALL RANLUX(RVEC,100)
C RANLUX DEFAULT INITIALIZATION: 314159265
C RANLUX DEFAULT LUXURY LEVEL = 3 p = 223
C RANLUX default numbers 1- 5:
C 0.53981817 0.76155043 0.06029940 0.79600263 0.30631220
C RANLUX default numbers 101-105:
C 0.43156743 0.03774416 0.24897110 0.00147784 0.90274453
C
C CALL RLUXGO(0,0,0,0)
C RANLUX LUXURY LEVEL SET BY RLUXGO : 0 P= 24
C RANLUX INITIALIZED BY RLUXGO FROM DEFAULT SEED
C RANLUX luxury level 0, 1- 5:
C 0.53981817 0.76155043 0.06029940 0.79600263 0.30631220
C RANLUX luxury level 0, 101-105:
C 0.41538775 0.05330932 0.58195311 0.91397446 0.67034441
C
C CALL RLUXGO(389,1,0,0)
C RANLUX LUXURY LEVEL SET BY RLUXGO : 4 P= 389
C RANLUX INITIALIZED BY RLUXGO FROM SEEDS 1 0 0
C RANLUX luxury p=389, 1- 5:
C 0.94589490 0.47347850 0.95152789 0.42971975 0.09127384
C RANLUX luxury p=389, 101-105:
C 0.02618265 0.03775346 0.97274780 0.13302165 0.43126065
C
C CALL RLUXGO(75,0,0,0)
C RANLUX P-VALUE SET BY RLUXGO TO: 75
C RANLUX INITIALIZED BY RLUXGO FROM DEFAULT SEED
C RANLUX luxury p= 75, 1- 5:
C 0.53981817 0.76155043 0.06029940 0.79600263 0.30631220
C RANLUX luxury p= 75, 101-105:
C 0.25600731 0.23443210 0.59164381 0.59035838 0.07011414
C
C test restarting from the full vector
C
C current RANLUX status saved:
C 16156027 16534309 15243811 2751687 6002207
C 7979506 1301976 4567313 4305996 5872599
C 12003090 2146823 12606367 4111505 5979640
C 12739666 10489318 14036909 11729352 8061448
C 7832659 6069758 3197719 1832730 75080216
C RANLUX numbers 1- 5:
C 0.22617835 0.60655993 0.86417443 0.43920082 0.23382509
C RANLUX numbers 101-105:
C 0.08107197 0.21466845 0.84856731 0.94078046 0.85626233
C
C previous RANLUX status will be restored
C FULL INITIALIZATION OF RANLUX WITH 25 INTEGERS:
C 16156027 16534309 15243811 2751687 6002207
C 7979506 1301976 4567313 4305996 5872599
C 12003090 2146823 12606367 4111505 5979640
C 12739666 10489318 14036909 11729352 8061448
C 7832659 6069758 3197719 1832730 75080216
C RANLUX P-VALUE SET BY RLUXIN TO: 75
C RANLUX numbers 1- 5:
C 0.22617835 0.60655993 0.86417443 0.43920082 0.23382509
C RANLUX numbers 101-105:
C 0.08107197 0.21466845 0.84856731 0.94078046 0.85626233
C
C test the restarting by skipping
C RANLUX LUXURY LEVEL SET BY RLUXGO : 4 P= 389
C RANLUX INITIALIZED BY RLUXGO FROM SEEDS 7674985 0 0
C RLUXAT values = 4 7674985 0 0
C RLUXAT values = 4 7674985 161840 0
C Next and 200th numbers are: 0.019648 0.590586
C RANLUX LUXURY LEVEL SET BY RLUXGO : 4 P= 389
C RANLUX INITIALIZED BY RLUXGO FROM SEEDS 7674985 161840 0
C Next and 200th numbers are: 0.019648 0.590586
C
C The following should provoke an error message
C RANLUX LUXURY LEVEL SET BY RLUXGO : 4 P= 389
C RANLUX INITIALIZED BY RLUXGO FROM SEEDS 11111 31 0
C Error in RESTARTING with RLUXGO:
C The values 11111 31 0 cannot occur at luxury level 4
END

************************************************************************

Darrah Chavey

unread,

Feb 8, 1995, 9:29:02 PM2/8/95

In article <3gtmeq$b...@tribune.usask.ca>, ch...@cs.usask.ca (Henry Choy) wrote:

> I found that the basic rand() in C is not entirely uniformly
> distributed, but it's not too bad.
>
> Over all these years, what is the best random number generator?

There's no real answer to that, since "best" is not really well-defined.
However, the following is one written by one of my students, based on some
of the recent work done on such generators. It was then extensively tested
by a Statistics professor, and certified as strong.

global static variables: seed1, seed2 : integer;

procedure RandomInit (Stream: integer);
begin
{ Get the current day, hour, minute, second, and initialize your seed }
{ along the following lines: }
randseed := (Day + 1) * (Hour + 1) * (Minute + 1) * (Second + 1);
seed1 := HiWord(randseed);
seed1 := LoWord(randseed);
end;

function URand: extended; { "Extended" is Mac-speak for a 16-byte real. }
{ This returns a random real number in the range [0, 1). It can be easily }
{ scaled to any other range you want. }
var
z, k: integer;
begin
{ first linear congruential generator }
k := seed1 div 206;
seed1 := 157 * (seed1 - k * 206) - k * 21;
if seed1 < 0 then
seed1 := seed1 + 32363;

{ second linear congruential generator }
k := seed2 div 217;
seed2 := 146 * (seed2 - k * 217) - k * 45;
if seed2 < 0 then
seed2 := seed2 + 31727;

{ combine the values }
z := seed1 - seed2;
if z > 706 then
z := z - 32362;
if z < 1 then
z := z + 32362;

{ Place Urand in the [0,1) range }
Urand := z * 0.000030899
end;

-- Darrah Chavey Those who do not understand Unix are
Math/CS Dept. condemned to reinvent it, poorly.
Beloit College -- Henry Spencer, Univ. Toronto Unix hack

Douglas Zare (NC)

unread,

Feb 8, 1995, 9:52:37 PM2/8/95

On 3 Feb 1995, Henry Choy wrote:

> I found that the basic rand() in C is not entirely uniformly
> distributed, but it's not too bad.
>
> Over all these years, what is the best random number generator?

Calculus exams?

> --
> Henry Choy "Math class is hard" - Barbie
>
> e-mail: ch...@cs.usask.ca I AGREE!

Douglas Zare
za...@cco.caltech.edu

David Bernier

unread,

Feb 9, 1995, 1:31:13 AM2/9/95

I wrote a fortran subroutine that generates pseudorandom fortran
integers and reals in [0,1] . The method used is a linear feedback
shift-register. See chapter 9 of the book _Random Number Generation
and Quasi-Monte Carlo Methods_ by H. Niederreiter for more
information. I use the primitive trinomial in F2[x]
x^4423 + x^271 + 1 reported by Neal Zierler in _Information and
Control_ (15), 67-69 , 1969 . Note that addition of integers in
fortran is isomorphic to the additive group mod 2^32 ( Z_{2^32} ) .
Because I use a primitive trinomial, the period of the generator
for the random integers is at least 2^4423-1 . For the random
reals, the same is expected to be the case, although I'm not
absolutely sure because many fortran integers result in the same
random real number in [0,1] (see the subroutine part of the source
code below).

I give 3 demo programs using this subroutine below. In addition, one
needs a large file of random integers (named seed5000). A seed file
is included also. If you want to randomize the output, simply replace
(say) the first 10 lines of seed5000 by any fortran integers
of your choice.

Here are some stats about the demo program rndtoss.f below:
On a sparc 20, rndtoss.f ran for 43 seconds.
This gives a rate of about 450000 coin tosses per second.

I hope the demo programs are enough so that someone else can figure
out how to use this subroutine. If you get some strange correlations
or unusual results with this subroutine, I'd be interested in hearing
about it.

David Bernier dber...@mat.ulaval.ca

demo program rnd.f
------------------------------------------------------------------

dimension ifrandom(3),iseedrandom(4423),iscrandom(4423)
open(95,file='seed5000')
do 1997 i=1,4423
read(95,*)iseedrandom(i)
iscrandom(i)=1+mod(i,4423)
1997 continue
ifrandom(1)=1
ifrandom(2)=272
ifrandom(3)=4423
* the preceding lines need to be executed before calling
* the random subroutine .
* the variable names ifrandom iseedrandom iscrandom
* and the subroutine name random need to be reserved for the
* use of the random subroutine. the output of random is put
* in the integer variable integer and the real variable real .
* integer is a pseudorandom fortran integer in the range
* -2147483648 to 2147483647 . real is a pseudorandom real
* number in the range [0,1] following a uniform distribution.
* there is no problem in changing the names of the two
* output variables of random, ie integer and random.
* this demo program computes and sends to stdout 20 values
* of random integers and random reals.
* this subroutine is for free use and is public domain
do 50 j=1,20
call random(iseedrandom,ifrandom,iscrandom,integer,real)
write(6,*)integer,real
50 continue
end
* the MAIN program ends on the preceding line
* the random subroutine begins on the following line
subroutine random(iseed,if,isc,int,real)
dimension iseed(1),if(1),isc(1)
int=iseed(if(1))+iseed(if(2))
if(1)=isc(if(1))
if(2)=isc(if(2))
if(3)=isc(if(3))
iseed(if(3))=int
real=abs(int/2147483648.)
return
end
--------------------------------------------------------------------

demo program rndtoss.f
----------------------------------------------------------------------

dimension ifrandom(3),iseedrandom(4423),iscrandom(4423)
integer distrib(21)
open(95,file='seed5000')
do 1997 i=1,4423
read(95,*)iseedrandom(i)
iscrandom(i)=1+mod(i,4423)
1997 continue
ifrandom(1)=1
ifrandom(2)=272
ifrandom(3)=4423
* the preceding lines need to be executed before calling
* the random subroutine .
* the variable names ifrandom iseedrandom iscrandom
* and the subroutine name random need to be reserved for the
* use of the random subroutine. the output of random is put
* in the integer variable integer and the real variable real .
* integer is a pseudorandom fortran integer in the range
* -2147483648 to 2147483647 . real is a pseudorandom real
* number in the range [0,1] following a uniform distribution.
* there is no problem in changing the names of the two
* output variables of random, ie integer and random.
* this demo program simulates the following experiment:
* toss a fair coin 20 times and note the number of tails;
* repeat 1000000 times and write how many times you got
* 0 , 1, ... , 20 tails .
* this subroutine is for free use and is public domain
do 50 j=1,21
distrib(j)=0
50 continue
do 60 i=1,1000000
ntails=0
do 65 j=1,20
call random(iseedrandom,ifrandom,iscrandom,integer,real)
if(real.lt..5)then
ntails=ntails+1
endif
65 continue
distrib(ntails+1)=distrib(ntails+1)+1
60 continue
do 80 i=0,20
write(6,*)i,distrib(i+1)
80 continue
end
* the MAIN program ends on the preceding line
* the random subroutine begins on the following line
subroutine random(iseed,if,isc,int,real)
dimension iseed(1),if(1),isc(1)
int=iseed(if(1))+iseed(if(2))
if(1)=isc(if(1))
if(2)=isc(if(2))
if(3)=isc(if(3))
iseed(if(3))=int
real=abs(int/2147483648.)
return
end
----------------------------------------------------------------------

demo program rnddeck.f
-----------------------------------------------------------------------

dimension ifrandom(3),iseedrandom(4423),iscrandom(4423)
dimension im(52),int(52)
logical ct
open(95,file='seed5000')
do 1997 i=1,4423
read(95,*)iseedrandom(i)
iscrandom(i)=1+mod(i,4423)
1997 continue
ifrandom(1)=1
ifrandom(2)=272
ifrandom(3)=4423
* the preceding lines need to be executed before calling
* the random subroutine .
* the variable names ifrandom iseedrandom iscrandom
* and the subroutine name random need to be reserved for the
* use of the random subroutine. the output of random is put
* in the integer variable integer (which is replaced below by
* isum) and the real variable real (replaced below by r) .
* integer is a pseudorandom fortran integer in the range
* -2147483648 to 2147483647 . real is a pseudorandom real
* number in the range [0,1] following a uniform distribution.
* there is no problem in changing the names of the two
* output variables of random, ie integer and random.
* this demo program tries to answer the question:
* if two people each shuffle a 52 card deck, and then each
* turns over a card and announces the kind (A,2,3,... 10,J,Q,K),
* what is the probability that there will be no match in the
* 52 announcements? this problem was discussed in 1994 in sci.math
* and the exact probability was reported to be .016232727...
* (the name of the thread was:
* Probability problem, no match in 52 card deck )
* this subroutine is for free use and is public domain
do 40 i=1,52
im(i)=mod(i,13)
40 continue
do 70 j=1,100
do 75 l=1,50000
ct=.true.
do 85 ix=1,52
call random(iseedrandom,ifrandom,iscrandom,isum,r)
int(ix)=isum
85 continue
do 95 ix=1,52
imin=2147483647
do 105 iy=1,52
if(int(iy).lt.imin) then
imin=int(iy)
isa=iy
endif
105 continue
ct=ct.and.(im(ix).ne.im(isa))
int(isa)=2147483647
95 continue
ic=ic+1
if(ct)then
isuc=isuc+1
endif
75 continue
r=isuc/(ic+0.)
write(6,*)ic,isuc,r
70 continue
end
* the MAIN program ends on the preceding line
* the random subroutine begins on the following line
subroutine random(iseed,if,isc,int,real)
dimension iseed(1),if(1),isc(1)
int=iseed(if(1))+iseed(if(2))
if(1)=isc(if(1))
if(2)=isc(if(2))
if(3)=isc(if(3))
iseed(if(3))=int
real=abs(int/2147483648.)
return
end
------------------------------------------------------------------------

filename=seed5000 (5000 lines)
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--------------------------------------------------------------------------

Matthias Fischmann

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Feb 6, 1995, 1:37:00 PM2/6/95

reply to msg [What's the best random number generator?] in newsgroup [/sci/math]
from [ch...@cs.usask.ca]
date: [03.02.95] time: [16:43]

> I found that the basic rand() in C is not entirely uniformly
> distributed, but it's not too bad.
>
> Over all these years, what is the best random number generator?
>

1) get any keystrokes from the user and calculate the time passing
between them in an arbitrary way (you can add them all and then take the
first two digits from the result, so you have a real random number between
zero and 99). realized in pgp by philip zimmermann and many others.

2) there are also radioactive random number generators (they base
on the completely arbitrary decay of atoms in radioactive substances), but
has anybody heard something about any suppliers or prices?

matze

ma...@phil15.uni-sb.de

--
matthias fischmann bismarckstr.6 66111 saarbruecken
phone +49 (681) 37392-4 fax +49 (681) 37392-5
email: ma...@phil.uni-sb.de f...@redlite.saar.de

Ken Smith

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Feb 9, 1995, 9:53:06 AM2/9/95

In article <5fKU5...@redlite.saar.de>,

Matthias Fischmann <ma...@redlite.saar.de> wrote:
> 2) there are also radioactive random number generators (they base
>on the completely arbitrary decay of atoms in radioactive substances), but
>has anybody heard something about any suppliers or prices?

All you really need is a noise source and a counter. The noise generated
by the reverse break down of a diode junction amplified then connected to
a counter is just as random as the noise of a radioactive device. If you
really need random bits that a truely random you may want to have one of
you EE friends build you one of these.

--
Ken Smith <kens...@rahul.net>

Ǝ

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Aug 4, 2023, 7:53:53 AM8/4/23

Your hand.

markus...@gmail.com

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Aug 4, 2023, 3:50:29 PM8/4/23

fredag 3 februari 1995 kl. 17:43:06 UTC+1 skrev Henry Choy:

random.org

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