# Random Number Generation¶

AMPLGSL uses a global random number generator which is automatically
initialized with the default seed. The seed is zero by default but
can be changed by using the environment variable `GSL_RNG_SEED`

before the library is loaded. Similarly, the random number generator
can be changed by setting the environment variable `GSL_RNG_TYPE`

.

In AMPL version 20120830 and later the standard `randseed`

option
can be used instead of `GSL_RNG_SEED`

to specify the random number
generator seed.

## Random number generator algorithms¶

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.
For the most reliable source of uncorrelated numbers, the second-generation
`ranlux`

generators have the strongest proof of randomness.

Generator: **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. Later versions switched to 5489 as the default seed, you can choose this explicitly via`GSL_RNG_SEED`

instead if you require it.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 3–30The generator

`mt19937`

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 alternative generators`mt19937_1999`

and`mt19937_1998`

.

Generator: **ranlxs0**

Generator: **ranlxs1**

Generator: **ranlxs2**

The generator

`ranlxs0`

is a second-generation version of the`ranlux`

algorithm of Lüscher, which produces “luxury random numbers”. This generator provides single precision output (24 bits) at three luxury levels`ranlxs0`

,`ranlxs1`

and`ranlxs2`

, in increasing order of strength. 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 increased decorrelation between samples as an additional safety margin.Note that the range of allowed seeds for this generator is \([0,2^{31}-1]\). Higher seed values are wrapped modulo \(2^{31}\).

Generator: **ranlxd1**

Generator: **ranlxd2**

These generators produce double precision output (48 bits) from the`ranlxs`

generator. The library provides two luxury levels`ranlxd1`

and`ranlxd2`

, in increasing order of strength.

Generator: **ranlux**

Generator: **ranlux389**

The

`ranlux`

generator is an implementation of the original algorithm developed by Lüscher. 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 Lüscher is provided by`ranlux`

, while`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. Lüscher, “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 Lüscher”,
Computer Physics Communications, 79 (1994) 111–114

Generator: **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,

\[\begin{split}x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) \mod m_1 \\ y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) \mod m_2\end{split}\]with coefficients \(a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098, b_2 = 0, b_3 = -539608\), and moduli \(m_1 = 2^{31} - 1 = 2147483647\) and \(m_2 = 2145483479\).

The period of this generator is \(\operatorname{lcm}(m_1^3-1, m_2^3-1)\), which is approximately \(2^{185}\) (about \(10^{56}\)). 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: **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 Simulation3, 87–98 (1993).

Generator: **taus**

Generator: **taus2**

This is a maximally equidistributed combined Tausworthe generator by L’Ecuyer. The sequence is,

\[x_n = (s^1_n \oplus s^2_n \oplus s^3_n)\]where,

\[\begin{split}s^1_{n+1} = (((s^1_n \& 4294967294) \ll 12) \oplus (((s^1_n \ll 13) \oplus s1_n) \gg 19)) \\ s^2_{n+1} = (((s^2_n \& 4294967288) \ll 4) \oplus (((s^2_n \ll 2) \oplus s2_n) \gg 25)) \\ s^3_{n+1} = (((s^3_n \& 4294967280) \ll 17) \oplus (((s^3_n \ll 3) \oplus s3_n) \gg 11))\end{split}\]computed modulo \(2^{32}\). In the formulas above \(\oplus\) 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

`taus2`

uses the same algorithm as`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), 261–269The generator

`taus2`

should now be used in preference to`taus`

.

Generator: **gfsr4**

The

`gfsr4`

generator is like a lagged-fibonacci generator, and produces each number as an xor’d sum of four previous values.\[r_n = r_{n-A} \oplus r_{n-B} \oplus r_{n-C} \oplus r_{n-D}\]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 example on p392 of Ziff’s article: A=471, B=1586, C=6988, D=9689.

If the offsets are appropriately chosen (such as the one ones in this implementation), then the sequence is said to be maximal; that means that the period is \(2^D - 1\), where D is the longest lag. (It is one less than \(2^D\) because it is not permitted to have all zeros in the ra[] array.) For this implementation with D=9689 that works out to about \(10^{2917}\).

Note that the implementation of this generator using a 32-bit integer amounts to 32 parallel implementations of one-bit generators. One consequence of this is that the period of this 32-bit generator is the same as for the one-bit generator. Moreover, this independence means that all 32-bit patterns are equally likely, and in particular that 0 is an allowed random value. (We are grateful to Heiko Bauke for clarifying for us these properties of GFSR random number generators.)

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.

## Example¶

The following example shows how to select a random number generator and initialize the seed.

```
option GSL_RNG_TYPE 'taus'; # use the "taus" generator
option GSL_RNG_SEED 123; # initialize the seed to 123
include gsl.ampl;
print gsl_ran_gaussian(1);
```