random.texi

A C++ class library for scientific computing

@section Overview

These are the basic random number generators (RNGs):

@cindex RNGs
@cindex Random Number Generators
@cindex Random Number Generators overview

@table @code
@item Uniform
Uniform reals on [0,1)

@item Normal
Normal with specified mean and variance

@item Exponential
Exponential with specified mean

@item DiscreteUniform
Integers uniformly distributed over a specified range.

@item Beta
Beta distribution

@item Gamma
Gamma distribution

@item F
F distribution
@end table

To use these generators, you need to include some subset of these headers:

@example
#include <random/uniform.h>
#include <random/normal.h>
#include <random/exponential.h>
#include <random/discrete-uniform.h>
#include <random/beta.h>
#include <random/gamma.h>
#include <random/chisquare.h>
#include <random/F.h>

using namespace ranlib;
@end example

All the generators are inside the namespace @strong{ranlib}, 
@cindex ranlib
so a @strong{using namespace ranlib} directive is required (alternately, you
can write e.g. @code{ranlib::Uniform<>}).

These generators are all class templates.  The first template parameter is
the number type you want to generate: float, double or long double for
continuous distributions, and integer for discrete distributions.  This
parameter defaults to @code{float} for continuous distributions,
and @code{unsigned int} for discrete distributions.

The constructors are:

@example
Uniform();
Normal(T mean, T standardDeviation);
Exponential(T mean);
DiscreteUniform(T n);   // range is 0 .. n-1
Beta(T a, T b);
Gamma(T mean);
ChiSquare(T df);
F(T dfn, T dfd);
@end example

where @code{T} is the first template parameter (@code{float}, @code{double},
or @code{long double}).  To obtain a random number, use the method
@code{random()}.  Here is an example of constructing and using a
@code{Normal} generator:

@example
#include <random/normal.h>

using namespace ranlib;

void foo()
@{
    Normal<double> normalGen;
    double x = normalGen.random();    // x is a normal random number
@}
@end example

@section Note: Parallel random number generators

The generators which Blitz++ provides are not suitable for parallel
programs.  If you need parallel RNGs, you may find
@uref{http://www.ncsa.uiuc.edu/Apps/SPRNG} useful.

@section Seeding a random number generator

@cindex Random Number Generators seeding

You may seed a random number generator using the member function
@code{seed(unsigned int)}.  
@cindex seeding a RNG
By default, all random
number generators share the same underlying integer random number generator.
So seeding one generator will seed them all.  (Note: you can create
generators with their own internal state; see the sections below).  You
should generally only seed a random number generator once, at the beginning
of a program run.  

Here is an example of seeding with the system clock:

@example
#include <random/uniform.h>
#include <time.h>

using namespace ranlib;

int main()
@{
    // At start of program, seed with the system time so we get
    // a different stream of random numbers each run.
    Uniform<float> x;
    x.seed((unsigned int)time(0));

    // Rest of program
    ...
@}
@end example

Note: you may be tempted to seed the random number generator from a static
initializer.  @strong{Don't do it!}  Due to an oddity of C++, there is no
guarantee on the order of static initialization when templates are involved.
Hence, you may seed the RNG before its constructor is invoked, in which case
your program will crash.  If you don't know what a static initializer is,
don't worry -- you're safe!

@section Detailed description of RNGs

@cindex Random Number Generators details

There are really two types of RNGs:

@table @code
@item Integer 
RNGs provide uniformly distributed, unsigned 32 bit integers.
@cindex IRNGs 
@cindex Integer RNGs

@item RNGs
use Integer RNGs to provide other kinds of random numbers. 
@end table

By default, the Integer RNG used is a faithful adaptation of the Mersenne
Twister @code{MT19937} @cindex MersenneTwister due to Matsumoto and
Nishimura (see @cite{ACM Transactions on Modeling and Computer Simulation,
@w{Vol. 8}, @w{No. 1}, @w{January 1998}, @w{pp 3-30}},
@uref{http://www.math.keio.ac.jp/~matumoto/emt.html},
@uref{http://www.acm.org/pubs/citations/journals/tomacs/1998-8-1/p3-matsumoto/}).
This generator has a period of 
@iftex
@math{2^{19937}-1}
@end iftex
@ifinfo
@math{2^{19937}-1}
@end ifinfo
@ifhtml
2^(19937)-1
@end ifhtml
, passed several stringent
statistical tests (including the
@uref{http://stat.fsu.edu/~geo/diehard.html} tests), and has speed
comparable to other modern generators.

@section Template parameters

RNGs take three template parameters, all of which have default values.
Using the @code{Uniform} RNG as an example, the template parameters of
@code{Uniform<T, IRNG, stateTag>} are:

@table @code
@item T
is the type of random number to generate (one of @code{float},
@code{double}, or @code{long double} for continuous distributions; an
integer type for discrete distributions).  Note that generating double and
long double RNGs takes longer, because filling the entire mantissa with
random bits requires several random integers.  The default parameter for
most generators is @code{float}.

@item IRNG
is the underlying Integer RNG to use.  The default is MersenneTwister.

@item stateTag
@cindex @code{stateTag} (RNGs) 
is either @code{sharedState} or @code{independentState}.  If
@code{sharedState}, the IRNG is shared with other generators.  If
@code{independentState}, the RNG contains its own IRNG.  The default is
sharedState.  
@end table

@section Member functions

@cindex Random Number Generators member functions

RNGs have these methods:

@example
T random();
@end example
@findex random()
Returns a random number.

@example
void seed(unsigned int);
@end example
@findex seed()
Seeds the underlying IRNG.  See above for an example of seeding
with the system timer.

@section Detailed listing of RNGs
@cindex Random Number Generators list of

To save space in the below list, template parameters have been omitted and
only constructors are listed.  The notation [a,b] means an interval which
includes the endpoints a and b; (a,b) is an interval which does not include
the endpoints.

@subsection @file{random/uniform.h}
@findex random/uniform.h

@example
Uniform<>()
@end example
@cindex Uniform RNG
Continuous uniform distribution on [0,1). 

@example
UniformClosedOpen<>()
@end example
@cindex UniformClosedOpen RNG
Continuous uniform distribution on [0,1).  Same as @code{Uniform<>}.

@example
UniformClosed<>()
@end example
@cindex UniformClosed RNG
Continuous uniform distribution on [0,1].

@example
UniformOpen<>()
@end example
@cindex UniformOpen RNG
Continuous uniform distribution on (0,1).

@example
UniformOpenClosed<>()
@end example
@cindex UniformOpenClosed RNG
Continuous uniform distribution on (0,1].

@subsection @file{random/normal.h}

@example
NormalUnit<>()
@end example
@cindex NormalUnit RNG
Continuous normal distribution with mean 0 and variance 1.

@example
Normal<>(T mean, T standardDeviation)
@end example
@cindex Normal RNG
Continuous normal distribution with specified mean and standard 
deviation.  

@subsection @file{random/exponential.h}

@example
ExponentialUnit<>()
@end example
@cindex ExponentialUnit RNG
Continuous exponential distribution with mean 1.

@example
Exponential<>(T mean)
@end example
@cindex Exponential RNG
Continuous exponential distribution with specified mean.

@subsection @file{random/beta.h}

@example
Beta<>(T a, T b)
@end example
@cindex Beta RNG
Beta distribution with parameters a and b.  The mean of the distribution is
@math{a/(a+b)} and its variance is @math{ab/((a+b)^2(a+b+1))}.  Use the
method @code{setParameters(T a, T b)} to change the parameters.

@subsection @file{random/chisquare.h}

@example
ChiSquare<>(T df)
@end example
@cindex ChiSquare RNG
@math{\chi^2} distribution with @code{df} degrees of freedom.  The parameter
df must be positive.  Use the method @code{setDF(T df)} to change the
degrees of freedom.

@subsection @file{random/gamma.h}

@example
Gamma<>(T mean)
@end example
@cindex Gamma RNG
Gamma distribution with specified mean.  The mean must
be positive.  Use the method @code{setMean(T mean)} to
change the mean.

@subsection @file{random/F.h}

@example
F<>(T numeratorDF, T denominatorDF)
@end example
@cindex F distribution RNG
F distribution with numerator and denominator degrees
of freedom specified.  Both these parameters must be
positive.  Use @code{setDF(T dfn, T dfd)} to change the
degrees of freedom. 

@subsection @file{random/discrete-uniform.h}

@example
DiscreteUniform<>(T n)
@end example
@cindex DiscreteUniform RNG
Discrete uniform distribution over @math{0, 1, \ldots, n-1}.