The library provides a large collection of random number generators
which can be accessed through a uniform interface. Environment
variables allow you to select different generators and seeds at runtime,
so that you can easily switch between generators without needing to
recompile your program. Each instance of a generator keeps track of its
own state, allowing the generators to be used in multi-threaded
programs. Additional functions are available for transforming uniform
random numbers into samples from continuous or discrete probability
distributions such as the Gaussian, log-normal or Poisson distributions.
These functions are declared in the header file `gsl_rng.h'.
In 1988, Park and Miller wrote a paper entitled "Random number
generators: good ones are hard to find." [Commun. ACM, 31, 1192--1201].
Fortunately, some excellent random number generators are available,
though poor ones are still in common use. You may be happy with the
system-supplied random number generator on your computer, but you should
be aware that as computers get faster, requirements on random number
generators increase. Nowadays, a simulation that calls a random number
generator millions of times can often finish before you can make it down
the hall to the coffee machine and back.
A very nice review of random number generators was written by Pierre
L'Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
ed. (Wiley, 1997). The chapter is available in postscript from from
L'Ecuyer's ftp site (see references). Knuth's volume on Seminumerical
Algorithms (originally published in 1968) devotes 170 pages to random
number generators, and has recently been updated in its 3rd edition
(1997).
It is brilliant, a classic. If you don't own it, you should stop reading
right now, run to the nearest bookstore, and buy it.
A good random number generator will satisfy both theoretical and
statistical properties. Theoretical properties are often hard to obtain
(they require real math!), but one prefers a random number generator
with a long period, low serial correlation, and a tendency not to
"fall mainly on the planes." Statistical tests are performed with
numerical simulations. Generally, a random number generator is used to
estimate some quantity for which the theory of probability provides an
exact answer. Comparison to this exact answer provides a measure of
"randomness".
It is important to remember that a random number generator is not a
"real" function like sine or cosine. Unlike real functions, successive
calls to a random number generator yield different return values. Of
course that is just what you want for a random number generator, but to
achieve this effect, the generator must keep track of some kind of
"state" variable. Sometimes this state is just an integer (sometimes
just the value of the previously generated random number), but often it
is more complicated than that and may involve a whole array of numbers,
possibly with some indices thrown in. To use the random number
generators, you do not need to know the details of what comprises the
state, and besides that varies from algorithm to algorithm.
The random number generator library uses two special structs,
gsl_rng_type which holds static information about each type of
generator and gsl_rng which describes an instance of a generator
created from a given gsl_rng_type.
The functions described in this section are declared in the header file
`gsl_rng.h'.
This function returns a pointer to a newly-created
instance of a random number generator of type T.
For example, the following code creates an instance of the Tausworthe
generator,
gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
If there is insufficient memory to create the generator then the
function returns a null pointer and the error handler is invoked with an
error code of GSL_ENOMEM.
The generator is automatically initialized with the default seed,
gsl_rng_default_seed. This is zero by default but can be changed
either directly or by using the environment variable GSL_RNG_SEED
(see section Random number environment variables).
The details of the available generator types are
described later in this chapter.
Random: void gsl_rng_set(const gsl_rng * r, unsigned long int s)
This function initializes (or `seeds') the random number generator. If
the generator is seeded with the same value of s on two different
runs, the same stream of random numbers will be generated by successive
calls to the routines below. If different values of s are
supplied, then the generated streams of random numbers should be
completely different. If the seed s is zero then the standard seed
from the original implementation is used instead. For example, the
original Fortran source code for the ranlux generator used a seed
of 314159265, and so choosing s equal to zero reproduces this when
using gsl_rng_ranlux.
Random: void gsl_rng_free(gsl_rng * r)
This function frees all the memory associated with the generator
r.
The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers. To obtain
non-uniform distributions see section Random Number Distributions.
Random: unsigned long int gsl_rng_get(const gsl_rng * r)
This function returns a random integer from the generator r. The
minimum and maximum values depend on the algorithm used, but all
integers in the range [min,max] are equally likely. The
values of min and max can determined using the auxiliary
functions gsl_rng_max (r) and gsl_rng_min (r).
Random: double gsl_rng_uniform(const gsl_rng * r)
This function returns a double precision floating point number uniformly
distributed in the range [0,1). The range includes 0.0 but excludes 1.0.
The value is typically obtained by dividing the result of
gsl_rng_get(r) by gsl_rng_max(r) + 1.0 in double
precision. Some generators compute this ratio internally so that they
can provide floating point numbers with more than 32 bits of randomness
(the maximum number of bits that can be portably represented in a single
unsigned long int).
This function returns a positive double precision floating point number
uniformly distributed in the range (0,1), excluding both 0.0 and 1.0.
The number is obtained by sampling the generator with the algorithm of
gsl_rng_uniform until a non-zero value is obtained. You can use
this function if you need to avoid a singularity at 0.0.
Random: unsigned long int gsl_rng_uniform_int(const gsl_rng * r, unsigned long int n)
This function returns a random integer from 0 to n-1 inclusive.
All integers in the range [0,n-1] are equally likely, regardless
of the generator used. An offset correction is applied so that zero is
always returned with the correct probability, for any minimum value of
the underlying generator.
If n is larger than the range of the generator then the function
calls the error handler with an error code of GSL_EINVAL and
returns zero.
The following functions provide information about an existing
generator. You should use them in preference to hard-coding the generator
parameters into your own code.
This function returns a pointer to the name of the generator.
For example,
printf("r is a '%s' generator\n",
gsl_rng_name (r));
would print something like r is a 'taus' generator.
Random: unsigned long int gsl_rng_max(const gsl_rng * r)
gsl_rng_max returns the largest value that gsl_rng_get
can return.
Random: unsigned long int gsl_rng_min(const gsl_rng * r)
gsl_rng_min returns the smallest value that gsl_rng_get
can return. Usually this value is zero. There are some generators with
algorithms that cannot return zero, and for these generators the minimum
value is 1.
Random: void * gsl_rng_state(const gsl_rng * r)
Random: size_t gsl_rng_size(const gsl_rng * r)
These function return a pointer to the state of generator r and
its size. You can use this information to access the state directly. For
example, the following code will write the state of a generator to a
stream,
void * state = gsl_rng_state (r);
size_t n = gsl_rng_size (r);
fwrite (state, n, 1, stream);
This function returns a pointer to an array of all the available
generator types, terminated by a null pointer. The function should be
called once at the start of the program, if needed. The following code
fragment shows how to iterate over the array of generator types to print
the names of the available algorithms,
The library allows you to choose a default generator and seed from the
environment variables GSL_RNG_TYPE and GSL_RNG_SEED and
the function gsl_rng_env_setup. This makes it easy try out
different generators and seeds without having to recompile your program.
This function reads the environment variables GSL_RNG_TYPE and
GSL_RNG_SEED and uses their values to set the corresponding
library variables gsl_rng_default and
gsl_rng_default_seed. These global variables are defined as
follows,
extern const gsl_rng_type *gsl_rng_default
extern unsigned long int gsl_rng_default_seed
The environment variable GSL_RNG_TYPE should be the name of a
generator, such as taus or mt19937. The environment
variable GSL_RNG_SEED should contain the desired seed value. It
is converted to an unsigned long int using the C library function
strtoul.
If you don't specify a generator for GSL_RNG_TYPE then
gsl_rng_mt19937 is used as the default. The initial value of
gsl_rng_default_seed is zero.
Here is a short program which shows how to create a global
generator using the environment variables GSL_RNG_TYPE and
GSL_RNG_SEED,
#include <stdio.h>
#include <gsl/gsl_rng.h>
gsl_rng * r; /* global generator */
int
main (void)
{
const gsl_rng_type * T;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf("generator type: %s\n", gsl_rng_name (r));
printf("seed = %u\n", gsl_rng_default_seed);
printf("first value = %u\n", gsl_rng_get (r));
return 0;
}
Running the program without any environment variables uses the initial
defaults, an mt19937 generator with a seed of 0,
bash$ ./a.out
generator type: mt19937
seed = 0
first value = 2867219139
By setting the two variables on the command line we can
change the default generator and the seed,
bash$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
GSL_RNG_TYPE=taus
GSL_RNG_SEED=123
generator type: taus
seed = 123
first value = 2720986350
The above methods ignore the random number `state' which changes from
call to call. It is often useful to be able to save and restore the
state. To permit these practices, a few somewhat more advanced
functions are supplied. These include:
Random: int gsl_rng_memcpy(gsl_rng * dest, const gsl_rng * src)
This function copies the random number generator src into the
pre-existing generator dest, making dest into an exact copy
of src. The two generators must be of the same type.
The functions described above make no reference to the actual algorithm
used. This is deliberate so that you can switch algorithms without
having to change any of your application source code. The library
provides a large number of generators of different types, including
simulation quality generators, generators provided for compatibility
with other libraries and historical generators from the past.
The following generators are recommended for use in simulation. They
have extremely long periods, low correlation and pass most statistical
tests.
Generator:gsl_rng_mt19937
The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
variant of the twisted generalized feedback shift-register algorithm,
and is known as the "Mersenne Twister" generator. It has a Mersenne
prime period of
2^19937 - 1 (about
10^6000) and is
equi-distributed in 623 dimensions. It has passed the DIEHARD
statistical tests. It uses 624 words of state per generator and is
comparable in speed to the other generators. The original generator used
a default seed of 4357 and choosing s equal to zero in
gsl_rng_set reproduces this.
For more information see,
Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number
generator". ACM Transactions on Modeling and Computer
Simulation, Vol. 8, No. 1 (Jan. 1998), Pages 3-30
The generator gsl_rng_19937 uses the second revision of the
seeding procedure published by the two authors above in 2002. The
original seeding procedures could cause spurious artifacts for some seed
values. They are still available through the alternate generators
gsl_rng_mt19937_1999 and gsl_rng_mt19937_1998.
Generator:gsl_rng_ranlxs0
Generator:gsl_rng_ranlxs1
Generator:gsl_rng_ranlxs2
The generator ranlxs0 is a second-generation version of the
RANLUX algorithm of L@"uscher, which produces "luxury random
numbers". This generator provides single precision output (24 bits) at
three luxury levels ranlxs0, ranlxs1 and ranlxs2.
It uses double-precision floating point arithmetic internally and can be
significantly faster than the integer version of ranlux,
particularly on 64-bit architectures. The period of the generator is
about
10^171. The algorithm has mathematically proven properties and
can provide truly decorrelated numbers at a known level of randomness.
The higher luxury levels provide additional decorrelation between samples
as an additional safety margin.
Generator:gsl_rng_ranlxd1
Generator:gsl_rng_ranlxd2
These generators produce double precision output (48 bits) from the
RANLXS generator. The library provides two luxury levels
ranlxd1 and ranlxd2.
Generator:gsl_rng_ranlux
Generator:gsl_rng_ranlux389
The ranlux generator is an implementation of the original
algorithm developed by L@"uscher. It uses a
lagged-fibonacci-with-skipping algorithm to produce "luxury random
numbers". It is a 24-bit generator, originally designed for
single-precision IEEE floating point numbers. This implementation is
based on integer arithmetic, while the second-generation versions
RANLXS and RANLXD described above provide floating-point
implementations which will be faster on many platforms.
The period of the generator is about
10^171. The algorithm has mathematically proven properties and
it can provide truly decorrelated numbers at a known level of
randomness. The default level of decorrelation recommended by L@"uscher
is provided by gsl_rng_ranlux, while gsl_rng_ranlux389
gives the highest level of randomness, with all 24 bits decorrelated.
Both types of generator use 24 words of state per generator.
For more information see,
M. L@"uscher, "A portable high-quality random number generator for
lattice field theory calculations", Computer Physics
Communications, 79 (1994) 100-110.
F. James, "RANLUX: A Fortran implementation of the high-quality
pseudo-random number generator of L@"uscher", Computer Physics
Communications, 79 (1994) 111-114
Generator:gsl_rng_cmrg
This is a combined multiple recursive generator by L'Ecuyer.
Its sequence is,
z_n = (x_n - y_n) mod m_1
where the two underlying generators x_n and y_n are,
The period of this generator is
2^205
(about
10^61). It uses
6 words of state per generator. For more information see,
P. L'Ecuyer, "Combined Multiple Recursive Random Number
Generators," Operations Research, 44, 5 (1996), 816--822.
Generator:gsl_rng_mrg
This is a fifth-order multiple recursive generator by L'Ecuyer, Blouin
and Coutre. Its sequence is,
x_n = (a_1 x_{n-1} + a_5 x_{n-5}) mod m
with
a_1 = 107374182,
a_2 = a_3 = a_4 = 0,
a_5 = 104480
and
m = 2^31 - 1.
The period of this generator is about
10^46. It uses 5 words
of state per generator. More information can be found in the following
paper,
P. L'Ecuyer, F. Blouin, and R. Coutre, "A search for good multiple
recursive random number generators", ACM Transactions on Modeling and
Computer Simulation 3, 87-98 (1993).
Generator:gsl_rng_taus
Generator:gsl_rng_taus2
This is a maximally equidistributed combined Tausworthe generator by
L'Ecuyer. The sequence is,
computed modulo
2^32. In the formulas above
^^
denotes "exclusive-or". Note that the algorithm relies on the properties
of 32-bit unsigned integers and has been implemented using a bitmask
of 0xFFFFFFFF to make it work on 64 bit machines.
The period of this generator is
2^88 (about
10^26). It uses 3 words of state per generator. For more
information see,
P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe
Generators", Mathematics of Computation, 65, 213 (1996), 203--213.
The generator gsl_rng_taus2 uses the same algorithm as
gsl_rng_taus but with an improved seeding procedure described in
the paper,
P. L'Ecuyer, "Tables of Maximally Equidistributed Combined LFSR
Generators", Mathematics of Computation, 68, 225 (1999), 261--269
The generator gsl_rng_taus2 should now be used in preference to
gsl_rng_taus.
Generator:gsl_rng_gfsr4
The gfsr4 generator is like a lagged-fibonacci generator, and
produces each number as an xor'd sum of four previous values.
r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}
Ziff (ref below) notes that "it is now widely known" that two-tap
registers (such as R250, which is described below)
have serious flaws, the most obvious one being the three-point
correlation that comes from the definition of the generator. Nice
mathematical properties can be derived for GFSR's, and numerics bears
out the claim that 4-tap GFSR's with appropriately chosen offsets are as
random as can be measured, using the author's test.
This implementation uses the values suggested the the example on p392 of
Ziff's article: A=471, B=1586, C=6988, D=9689.
If the offsets are appropriately chosen (such the one ones in
this implementation), then the sequence is said to be maximal.
I'm not sure what that means, but I would guess that means all
states are part of the same cycle, which would mean that the
period for this generator is astronomical; it is
(2^K)^D \approx 10^{93334}
where K=32 is the number of bits in the word, and D is the longest
lag. This would also mean that any one random number could
easily be zero; ie
0 <= r < 2^32.
Ziff doesn't say so, but it seems to me that the bits are
completely independent here, so one could use this as an efficient
bit generator; each number supplying 32 random bits. The quality of the
generated bits depends on the underlying seeding procedure, which
may need to be improved in some circumstances.
For more information see,
Robert M. Ziff, "Four-tap shift-register-sequence random-number
generators", Computers in Physics, 12(4), Jul/Aug
1998, pp 385-392.
The standard Unix random number generators rand, random
and rand48 are provided as part of GSL. Although these
generators are widely available individually often they aren't all
available on the same platform. This makes it difficult to write
portable code using them and so we have included the complete set of
Unix generators in GSL for convenience. Note that these generators
don't produce high-quality randomness and aren't suitable for work
requiring accurate statistics. However, if you won't be measuring
statistical quantities and just want to introduce some variation into
your program then these generators are quite acceptable.
Generator:gsl_rng_rand
This is the BSD rand() generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
with
a = 1103515245,
c = 12345 and
m = 2^31.
The seed specifies the initial value,
x_1. The period of this
generator is
2^31, and it uses 1 word of storage per
generator.
Generator:gsl_rng_random_bsd
Generator:gsl_rng_random_libc5
Generator:gsl_rng_random_glibc2
These generators implement the random() family of functions, a
set of linear feedback shift register generators originally used in BSD
Unix. There are several versions of random() in use today: the
original BSD version (e.g. on SunOS4), a libc5 version (found on
older GNU/Linux systems) and a glibc2 version. Each version uses a
different seeding procedure, and thus produces different sequences.
The original BSD routines accepted a variable length buffer for the
generator state, with longer buffers providing higher-quality
randomness. The random() function implemented algorithms for
buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm with
the largest length that would fit into the user-supplied buffer was
used. To support these algorithms additional generators are available
with the following names,
where the numeric suffix indicates the buffer length. The original BSD
random function used a 128-byte default buffer and so
gsl_rng_random_bsd has been made equivalent to
gsl_rng_random128_bsd. Corresponding versions of the libc5
and glibc2 generators are also available, with the names
gsl_rng_random8_libc5, gsl_rng_random8_glibc2, etc.
Generator:gsl_rng_rand48
This is the Unix rand48 generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
defined on 48-bit unsigned integers with
a = 25214903917,
c = 11 and
m = 2^48.
The seed specifies the upper 32 bits of the initial value, x_1,
with the lower 16 bits set to 0x330E. The function
gsl_rng_get returns the upper 32 bits from each term of the
sequence. This does not have a direct parallel in the original
rand48 functions, but forcing the result to type long int
reproduces the output of mrand48. The function
gsl_rng_uniform uses the full 48 bits of internal state to return
the double precision number x_n/m, which is equivalent to the
function drand48. Note that some versions of the GNU C Library
contained a bug in mrand48 function which caused it to produce
different results (only the lower 16-bits of the return value were set).
The following generators are provided for compatibility with
Numerical Recipes. Note that the original Numerical Recipes
functions used single precision while we use double precision. This will
lead to minor discrepancies, but only at the level of single-precision
rounding error. If necessary you can force the returned values to single
precision by storing them in a volatile float, which prevents the
value being held in a register with double or extended precision. Apart
from this difference the underlying algorithms for the integer part of
the generators are the same.
Generator:gsl_rng_ran0
Numerical recipes ran0 implements Park and Miller's MINSTD
algorithm with a modified seeding procedure.
Generator:gsl_rng_ran1
Numerical recipes ran1 implements Park and Miller's MINSTD
algorithm with a 32-element Bayes-Durham shuffle box.
Generator:gsl_rng_ran2
Numerical recipes ran2 implements a L'Ecuyer combined recursive
generator with a 32-element Bayes-Durham shuffle-box.
The generators in this section are provided for compatibility with
existing libraries. If you are converting an existing program to use GSL
then you can select these generators to check your new implementation
against the original one, using the same random number generator. After
verifying that your new program reproduces the original results you can
then switch to a higher-quality generator.
Note that most of the generators in this section are based on single
linear congruence relations, which are the least sophisticated type of
generator. In particular, linear congruences have poor properties when
used with a non-prime modulus, as several of these routines do (e.g.
with a power of two modulus,
2^31 or
2^32). This
leads to periodicity in the least significant bits of each number,
with only the higher bits having any randomness. Thus if you want to
produce a random bitstream it is best to avoid using the least
significant bits.
Generator:gsl_rng_ranf
This is the CRAY random number generator RANF. Its sequence is
x_{n+1} = (a x_n) mod m
defined on 48-bit unsigned integers with a = 44485709377909 and
m = 2^48. The seed specifies the lower
32 bits of the initial value,
x_1, with the lowest bit set to
prevent the seed taking an even value. The upper 16 bits of
x_1
are set to 0. A consequence of this procedure is that the pairs of seeds
2 and 3, 4 and 5, etc produce the same sequences.
The generator compatibile with the CRAY MATHLIB routine RANF. It
produces double precision floating point numbers which should be
identical to those from the original RANF.
There is a subtlety in the implementation of the seeding. The initial
state is reversed through one step, by multiplying by the modular
inverse of a mod m. This is done for compatibility with
the original CRAY implementation.
Note that you can only seed the generator with integers up to
2^32, while the original CRAY implementation uses
non-portable wide integers which can cover all
2^48 states of the generator.
The function gsl_rng_get returns the upper 32 bits from each term
of the sequence. The function gsl_rng_uniform uses the full 48
bits to return the double precision number x_n/m.
The period of this generator is
2^46.
Generator:gsl_rng_ranmar
This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and
Tsang. It is a 24-bit generator, originally designed for
single-precision IEEE floating point numbers. It was included in the
CERNLIB high-energy physics library.
Generator:gsl_rng_r250
This is the shift-register generator of Kirkpatrick and Stoll. The
sequence is
x_n = x_{n-103} ^^ x_{n-250}
where
^^ denote "exclusive-or", defined on
32-bit words. The period of this generator is about
2^250 and it
uses 250 words of state per generator.
For more information see,
S. Kirkpatrick and E. Stoll, "A very fast shift-register sequence random
number generator", Journal of Computational Physics, 40, 517-526
(1981)
Generator:gsl_rng_tt800
This is an earlier version of the twisted generalized feedback
shift-register generator, and has been superseded by the development of
MT19937. However, it is still an acceptable generator in its own
right. It has a period of
2^800 and uses 33 words of storage
per generator.
For more information see,
Makoto Matsumoto and Yoshiharu Kurita, "Twisted GFSR Generators
II", ACM Transactions on Modelling and Computer Simulation,
Vol. 4, No. 3, 1994, pages 254-266.
Generator:gsl_rng_vax
This is the VAX generator MTH$RANDOM. Its sequence is,
x_{n+1} = (a x_n + c) mod m
with
a = 69069, c = 1 and
m = 2^32. The seed specifies the initial value,
x_1. The
period of this generator is
2^32 and it uses 1 word of storage per
generator.
Generator:gsl_rng_transputer
This is the random number generator from the INMOS Transputer
Development system. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 1664525 and
m = 2^32.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_randu
This is the IBM RANDU generator. Its sequence is
x_{n+1} = (a x_n) mod m
with a = 65539 and
m = 2^31. The
seed specifies the initial value,
x_1. The period of this
generator was only
2^29. It has become a textbook example of a
poor generator.
Generator:gsl_rng_minstd
This is Park and Miller's "minimal standard" MINSTD generator, a
simple linear congruence which takes care to avoid the major pitfalls of
such algorithms. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 16807 and
m = 2^31 - 1 = 2147483647.
The seed specifies the initial value,
x_1. The period of this
generator is about
2^31.
This generator is used in the IMSL Library (subroutine RNUN) and in
MATLAB (the RAND function). It is also sometimes known by the acronym
"GGL" (I'm not sure what that stands for).
For more information see,
Park and Miller, "Random Number Generators: Good ones are hard to find",
Communications of the ACM, October 1988, Volume 31, No 10, pages
1192-1201.
Generator:gsl_rng_uni
Generator:gsl_rng_uni32
This is a reimplementation of the 16-bit SLATEC random number generator
RUNIF. A generalization of the generator to 32 bits is provided by
gsl_rng_uni32. The original source code is available from NETLIB.
Generator:gsl_rng_slatec
This is the SLATEC random number generator RAND. It is ancient. The
original source code is available from NETLIB.
Generator:gsl_rng_zuf
This is the ZUFALL lagged Fibonacci series generator of Peterson. Its
sequence is,
t = u_{n-273} + u_{n-607}
u_n = t - floor(t)
The original source code is available from NETLIB. For more information
see,
W. Petersen, "Lagged Fibonacci Random Number Generators for the NEC
SX-3", International Journal of High Speed Computing (1994).
Generator:gsl_rng_borosh13
This is the Borosh, Niederreiter random number generator. It is taken
from Knuth's Seminumerical Algorithms, 3rd Ed., pages
106-108. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 1812433253 and
m = 2^32.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_coveyou
This is the Coveyou random number generator. It is taken from Knuth's
Seminumerical Algorithms, 3rd Ed., Section 3.2.2. Its sequence
is,
x_{n+1} = (x_n (x_n + 1)) mod m
with
m = 2^32.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_fishman18
This is the Fishman, Moore III random number generator. It is taken from
Knuth's Seminumerical Algorithms, 3rd Ed., pages 106-108. Its
sequence is,
x_{n+1} = (a x_n) mod m
with a = 62089911 and
m = 2^31 - 1.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_fishman20
This is the Fishman random number generator. It is taken from Knuth's
Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 48271 and
m = 2^31 - 1.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_fishman2x
This is the L'Ecuyer - Fishman random number generator. It is taken from
Knuth's Seminumerical Algorithms, 3rd Ed., page 108. Its sequence
is,
z_{n+1} = (x_n - y_n) mod m
with
m = 2^31 - 1.
x_n and y_n are given by the fishman20
and lecuyer21 algorithms.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_knuthran2
This is a second-order multiple recursive generator described by Knuth
in Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m
with
a_1 = 271828183,
a_2 = 314159269,
and
m = 2^31 - 1.
Generator:gsl_rng_knuthran
This is a second-order multiple recursive generator described by Knuth
in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth
provides its C code.
Generator:gsl_rng_lecuyer21
This is the L'Ecuyer random number generator. It is taken from Knuth's
Seminumerical Algorithms, 3rd Ed., page 106-108. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 40692 and
m = 2^31 - 249.
The seed specifies the initial value,
x_1.
Generator:gsl_rng_waterman14
This is the Waterman random number generator. It is taken from Knuth's
Seminumerical Algorithms, 3rd Ed., page 106-108. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 1566083941 and
m = 2^32.
The seed specifies the initial value,
x_1.
The following table shows the relative performance of a selection the
available random number generators. The simulation quality generators
which offer the best performance are taus, gfsr4 and
mt19937.
1754 k ints/sec, 870 k doubles/sec, taus
1613 k ints/sec, 855 k doubles/sec, gfsr4
1370 k ints/sec, 769 k doubles/sec, mt19937
565 k ints/sec, 571 k doubles/sec, ranlxs0
400 k ints/sec, 405 k doubles/sec, ranlxs1
490 k ints/sec, 389 k doubles/sec, mrg
407 k ints/sec, 297 k doubles/sec, ranlux
243 k ints/sec, 254 k doubles/sec, ranlxd1
251 k ints/sec, 253 k doubles/sec, ranlxs2
238 k ints/sec, 215 k doubles/sec, cmrg
247 k ints/sec, 198 k doubles/sec, ranlux389
141 k ints/sec, 140 k doubles/sec, ranlxd2
1852 k ints/sec, 935 k doubles/sec, ran3
813 k ints/sec, 575 k doubles/sec, ran0
787 k ints/sec, 476 k doubles/sec, ran1
379 k ints/sec, 292 k doubles/sec, ran2
The following program demonstrates the use of a random number generator
to produce uniform random numbers in range [0.0, 1.0),
#include <stdio.h>
#include <gsl/gsl_rng.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < n; i++)
{
double u = gsl_rng_uniform (r);
printf("%.5f\n", u);
}
gsl_rng_free (r);
return 0;
}
The numbers depend on the seed used by the generator. The default seed
can be changed with the GSL_RNG_SEED environment variable to
produce a different stream of numbers. The generator itself can be
changed using the environment variable GSL_RNG_TYPE. Here is the
output of the program using a seed value of 123 and the
mutiple-recursive generator mrg,
The subject of random number generation and testing is reviewed
extensively in Knuth's Seminumerical Algorithms.
Donald E. Knuth, The Art of Computer Programming: Seminumerical
Algorithms (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
Further information is available in the review paper written by Pierre
L'Ecuyer,
P. L'Ecuyer, "Random Number Generation", Chapter 4 of the
Handbook on Simulation, Jerry Banks Ed., Wiley, 1998, 93--137.
http://www.iro.umontreal.ca/~lecuyer/papers.html
in the file `handsim.ps'.
On the World Wide Web, see the pLab home page
(http://random.mat.sbg.ac.at/) for a lot of information on the
state-of-the-art in random number generation, and for numerous links to
various "random" WWW sites.
The source code for the DIEHARD random number generator tests is also
available online.
Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
making the source code to their generators (MT19937, MM&TN; TT800,
MM&YK) available under the GNU General Public License. Thanks to Martin
L@"uscher for providing notes and source code for the RANLXS and
RANLXD generators.