1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
|
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/**
* Random number and random data generators.
*
* <p>Commons-math provides a few pseudo random number generators. The top level interface is
* RandomGenerator. It is implemented by three classes:
*
* <ul>
* <li>{@link org.apache.commons.math3.random.JDKRandomGenerator JDKRandomGenerator} that extends
* the JDK provided generator
* <li>AbstractRandomGenerator as a helper for users generators
* <li>BitStreamGenerator which is an abstract class for several generators and which in turn is
* extended by:
* <ul>
* <li>{@link org.apache.commons.math3.random.MersenneTwister MersenneTwister}
* <li>{@link org.apache.commons.math3.random.Well512a Well512a}
* <li>{@link org.apache.commons.math3.random.Well1024a Well1024a}
* <li>{@link org.apache.commons.math3.random.Well19937a Well19937a}
* <li>{@link org.apache.commons.math3.random.Well19937c Well19937c}
* <li>{@link org.apache.commons.math3.random.Well44497a Well44497a}
* <li>{@link org.apache.commons.math3.random.Well44497b Well44497b}
* </ul>
* </ul>
*
* <p>The JDK provided generator is a simple one that can be used only for very simple needs. The
* Mersenne Twister is a fast generator with very good properties well suited for Monte-Carlo
* simulation. It is equidistributed for generating vectors up to dimension 623 and has a huge
* period: 2<sup>19937</sup> - 1 (which is a Mersenne prime). This generator is described in a paper
* by Makoto Matsumoto and Takuji Nishimura in 1998: <a
* href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf">Mersenne Twister: A
* 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator</a>, ACM Transactions on
* Modeling and Computer Simulation, Vol. 8, No. 1, January 1998, pp 3--30. The WELL generators are
* a family of generators with period ranging from 2<sup>512</sup> - 1 to 2<sup>44497</sup> - 1
* (this last one is also a Mersenne prime) with even better properties than Mersenne Twister. These
* generators are described in a paper by François Panneton, Pierre L'Ecuyer and Makoto
* Matsumoto <a href="http://www.iro.umontreal.ca/~lecuyer/myftp/papers/wellrng.pdf">Improved
* Long-Period Generators Based on Linear Recurrences Modulo 2</a> ACM Transactions on Mathematical
* Software, 32, 1 (2006). The errata for the paper are in <a
* href="http://www.iro.umontreal.ca/~lecuyer/myftp/papers/wellrng-errata.txt">wellrng-errata.txt</a>.
*
* <p>For simple sampling, any of these generators is sufficient. For Monte-Carlo simulations the
* JDK generator does not have any of the good mathematical properties of the other generators, so
* it should be avoided. The Mersenne twister and WELL generators have equidistribution properties
* proven according to their bits pool size which is directly linked to their period (all of them
* have maximal period, i.e. a generator with size n pool has a period 2<sup>n</sup>-1). They also
* have equidistribution properties for 32 bits blocks up to s/32 dimension where s is their pool
* size. So WELL19937c for exemple is equidistributed up to dimension 623 (19937/32). This means a
* Monte-Carlo simulation generating a vector of n variables at each iteration has some guarantees
* on the properties of the vector as long as its dimension does not exceed the limit. However,
* since we use bits from two successive 32 bits generated integers to create one double, this limit
* is smaller when the variables are of type double. so for Monte-Carlo simulation where less the 16
* doubles are generated at each round, WELL1024 may be sufficient. If a larger number of doubles
* are needed a generator with a larger pool would be useful.
*
* <p>The WELL generators are more modern then MersenneTwister (the paper describing than has been
* published in 2006 instead of 1998) and fix some of its (few) drawbacks. If initialization array
* contains many zero bits, MersenneTwister may take a very long time (several hundreds of thousands
* of iterations to reach a steady state with a balanced number of zero and one in its bits pool).
* So the WELL generators are better to <i>escape zeroland</i> as explained by the WELL generators
* creators. The Well19937a and Well44497a generator are not maximally equidistributed (i.e. there
* are some dimensions or bits blocks size for which they are not equidistributed). The Well512a,
* Well1024a, Well19937c and Well44497b are maximally equidistributed for blocks size up to 32 bits
* (they should behave correctly also for double based on more than 32 bits blocks, but
* equidistribution is not proven at these blocks sizes).
*
* <p>The MersenneTwister generator uses a 624 elements integer array, so it consumes less than 2.5
* kilobytes. The WELL generators use 6 integer arrays with a size equal to the pool size, so for
* example the WELL44497b generator uses about 33 kilobytes. This may be important if a very large
* number of generator instances were used at the same time.
*
* <p>All generators are quite fast. As an example, here are some comparisons, obtained on a 64 bits
* JVM on a linux computer with a 2008 processor (AMD phenom Quad 9550 at 2.2 GHz). The generation
* rate for MersenneTwister was about 27 millions doubles per second (remember we generate two 32
* bits integers for each double). Generation rates for other PRNG, relative to MersenneTwister:
*
* <p>
*
* <table border="1" align="center">
* <tr BGCOLOR="#CCCCFF"><td colspan="2"><font size="+2">Example of performances</font></td></tr>
* <tr BGCOLOR="#EEEEFF"><font size="+1"><td>Name</td><td>generation rate (relative to MersenneTwister)</td></font></tr>
* <tr><td>{@link org.apache.commons.math3.random.MersenneTwister MersenneTwister}</td><td>1</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.JDKRandomGenerator JDKRandomGenerator}</td><td>between 0.96 and 1.16</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.Well512a Well512a}</td><td>between 0.85 and 0.88</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.Well1024a Well1024a}</td><td>between 0.63 and 0.73</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.Well19937a Well19937a}</td><td>between 0.70 and 0.71</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.Well19937c Well19937c}</td><td>between 0.57 and 0.71</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.Well44497a Well44497a}</td><td>between 0.69 and 0.71</td></tr>
* <tr><td>{@link org.apache.commons.math3.random.Well44497b Well44497b}</td><td>between 0.65 and 0.71</td></tr>
* </table>
*
* <p>So for most simulation problems, the better generators like {@link
* org.apache.commons.math3.random.Well19937c Well19937c} and {@link
* org.apache.commons.math3.random.Well44497b Well44497b} are probably very good choices.
*
* <p>Note that <em>none</em> of these generators are suitable for cryptography. They are devoted to
* simulation, and to generate very long series with strong properties on the series as a whole
* (equidistribution, no correlation ...). They do not attempt to create small series but with very
* strong properties of unpredictability as needed in cryptography.
*/
package org.apache.commons.math3.random;
|
