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* <tr BGCOLOR="#EEEEFF"> * <tr> * <tr> * <tr> * <tr> * <tr> * <tr> * <tr> * <tr> * </table> * </p> * * <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> * * <p> * Note that <em>none 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. * </p> * * */ package org.apache.commons.math3.random;

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 * 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,
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *      <p>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> * <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> * </li> * </ul> * </p> * * <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 - 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 - 1 * to 2<sup>44497 - 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> * * <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-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> * * <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 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> * * <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> * * <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> * * <p> * <table border="1" align="center"> * <tr BGCOLOR="#CCCCFF">
Example of performances
Namegeneration rate (relative to MersenneTwister)
{@link org.apache.commons.math3.random.MersenneTwister MersenneTwister}1
{@link org.apache.commons.math3.random.JDKRandomGenerator JDKRandomGenerator}between 0.96 and 1.16
{@link org.apache.commons.math3.random.Well512a Well512a}between 0.85 and 0.88
{@link org.apache.commons.math3.random.Well1024a Well1024a}between 0.63 and 0.73
{@link org.apache.commons.math3.random.Well19937a Well19937a}between 0.70 and 0.71
{@link org.apache.commons.math3.random.Well19937c Well19937c}between 0.57 and 0.71
{@link org.apache.commons.math3.random.Well44497a Well44497a}between 0.69 and 0.71
{@link org.apache.commons.math3.random.Well44497b Well44497b}between 0.65 and 0.71

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