.Modelica.Math.Random.Generators

Information

This package contains various pseudo random number generators. A random number generator is a package that consists of the following elements:

• Integer nState is a constant that defines the length of the internal state vector (in order that an appropriate Integer vector of this length can be declared, depending on the selected random number generator).
• Function initialState(..) is used to initialize the state of the random number generator by providing Integer seeds and calling the random number generator often enough that statistically relevant random numbers are returned by every call of function random(..).
• Function random(..) is used to return a random number of type Real in the range 0.0 < random ≤ 1.0 for every call. Furthermore, the updated (internal) state of the random number generator is returned as well.

The Generators package contains the xorshift suite of random number generators from Sebastiano Vigna (from 2014; based on work of George Marsaglia). The properties of these random number generators are summarized below and compared with the often used Mersenne Twister (MT19937-64) generator. The table is based on http://xorshift.di.unimi.it/ and on the articles:

Sebastiano Vigna: An experimental exploration of Marsaglia's xorshift generators, scrambled, 2014.
Sebastiano Vigna: Further scramblings of Marsaglia's xorshift generators, 2014.

Summary of the properties of the random number generators:

Property	xorshift64*	xorshift128+	xorshift1024*	MT19937-64
Period	2^64	2^128	2^1024	2^19937
Length of state (# 32 bit integer)	2	4	33	624
Statistic failures (Big Crush)	363	64	51	516
Systematic failures (Big Crush)	yes	no	no	yes
Worst case startup	> 1 call	> 20 calls	> 100 calls	> 100000 calls
Run time (MT=1.0)	0.39	0.27	0.33	1.0

Further explanations of the properties above:

• The period defines the number of random numbers generated before the sequence begins to repeat itself. According to "A long period does not imply high quality" a period of 2^1024 is by far large enough for even massively parallel simulations with huge number of random number computations per simulation. A period of 2^128 might be not enough for massively parallel simulations.
• Length of state (# 32 bit integer) defines the number of "int" (that is Modelica Integer) elements used for the internal state vector.
• Big Crush is part of TestU01 a huge framework for testing random number generators. According to these tests, the statistical properties of the xorshift random number generators are better than the ones of the Mersenne Twister random number generator.
• Worst case startup means how many calls are needed until getting from a bad seed to random numbers with appropriate statistical properties. Here, the xorshift random number suite has much better properties than the Mersenne Twister. When initializing a random number generator, the above property is taken into account and appropriate random numbers are generated, so that a subsequent call of random(..) will generate statistically relevant random numbers, even if the user provides a bad initial seed (such as localSeed=1). This means, any Integer number can be given as initial seed without influencing the quality of the generated random numbers.
• Run time shows that the xorshift random number generators are all much faster than the Mersenne Twister random number generator, although this is not really relevant for most simulations, because the execution time of the other parts of the simulations is usually much larger.

The xorshift random number generators are used in the following way in the Blocks.Noise package:

1. Xorshift64star (xorshift64*) is used to generate the initial internal state vectors of the other generators from two Integer values, due to the very good startup properties.
2. Xorshift128plus (xorshift128+) is the random number generator used by the blocks in Blocks.Noise. Since these blocks hold the internal state vector for every block instance, and the internal state vector is copied whenever a new random number is drawn, it is important that the internal state vector is short (and still has good statistical properties as shown in the table above).
3. Xorshift1024star (xorshift1024*) is the basis of the impure function Math.Random.Utilities.impureRandom which in turn is used with Blocks.Noise.GlobalSeed. The internal state vector is not exposed. It is updated internally, whenever a new random number is drawn.

Note, the generators produce 64 bit random numbers. These numbers are mapped to the 52 bit mantissa of double numbers in the range 0.0 .. 1.0.

Contents

Revisions

Date	Description
June 22, 2015	Initial version implemented by A. Klöckner, F. v.d. Linden, D. Zimmer, M. Otter. DLR Institute of System Dynamics and Control