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<H1> 9. Random Number Generators </H1>
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<H2> 9.1 Overview </H2>
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<P>

These are the basic random number generators (RNGs):
</P><P>

<A NAME="IDX339"></A>
<A NAME="IDX340"></A>
<A NAME="IDX341"></A>
</P><P>

<DL COMPACT>
<DT><CODE>Uniform</CODE>
<DD>Uniform reals on [0,1)
<P>

<DT><CODE>Normal</CODE>
<DD>Normal with specified mean and variance
<P>

<DT><CODE>Exponential</CODE>
<DD>Exponential with specified mean
<P>

<DT><CODE>DiscreteUniform</CODE>
<DD>Integers uniformly distributed over a specified range.
<P>

<DT><CODE>Beta</CODE>
<DD>Beta distribution
<P>

<DT><CODE>Gamma</CODE>
<DD>Gamma distribution
<P>

<DT><CODE>F</CODE>
<DD>F distribution
</DL>
<P>

To use these generators, you need to include some subset of these headers:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>#include &#60;random/uniform.h&#62;
#include &#60;random/normal.h&#62;
#include &#60;random/exponential.h&#62;
#include &#60;random/discrete-uniform.h&#62;
#include &#60;random/beta.h&#62;
#include &#60;random/gamma.h&#62;
#include &#60;random/chisquare.h&#62;
#include &#60;random/F.h&#62;

using namespace ranlib;
</pre></td></tr></table></P><P>

All the generators are inside the namespace <STRONG>ranlib</STRONG>, 
<A NAME="IDX342"></A>
so a <STRONG>using namespace ranlib</STRONG> directive is required (alternately, you
can write e.g. <CODE>ranlib::Uniform&#60;&#62;</CODE>).
</P><P>

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</CODE> for continuous distributions,
and <CODE>unsigned int</CODE> for discrete distributions.
</P><P>

The constructors are:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>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);
</pre></td></tr></table></P><P>

where <CODE>T</CODE> is the first template parameter (<CODE>float</CODE>, <CODE>double</CODE>,
or <CODE>long double</CODE>).  To obtain a random number, use the method
<CODE>random()</CODE>.  Here is an example of constructing and using a
<CODE>Normal</CODE> generator:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>#include &#60;random/normal.h&#62;

using namespace ranlib;

void foo()
{
    Normal&#60;double&#62; normalGen;
    double x = normalGen.random();    // x is a normal random number
}
</pre></td></tr></table></P><P>

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<H2> 9.2 Note: Parallel random number generators </H2>
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<P>

The generators which Blitz++ provides are not suitable for parallel
programs.  If you need parallel RNGs, you may find
<A HREF="http://www.ncsa.uiuc.edu/Apps/SPRNG">http://www.ncsa.uiuc.edu/Apps/SPRNG</A> useful.
</P><P>

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<H2> 9.3 Seeding a random number generator </H2>
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<P>

<A NAME="IDX343"></A>
</P><P>

You may seed a random number generator using the member function
<CODE>seed(unsigned int)</CODE>.  
<A NAME="IDX344"></A>
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.  
</P><P>

Here is an example of seeding with the system clock:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>#include &#60;random/uniform.h&#62;
#include &#60;time.h&#62;

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&#60;float&#62; x;
    x.seed((unsigned int)time(0));

    // Rest of program
    ...
}
</pre></td></tr></table></P><P>

Note: you may be tempted to seed the random number generator from a static
initializer.  <STRONG>Don't do it!</STRONG>  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!
</P><P>

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<H2> 9.4 Detailed description of RNGs </H2>
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<P>

<A NAME="IDX345"></A>
</P><P>

There are really two types of RNGs:
</P><P>

<DL COMPACT>
<DT><CODE>Integer</CODE>
<DD>RNGs provide uniformly distributed, unsigned 32 bit integers.
<A NAME="IDX346"></A>
<A NAME="IDX347"></A>
<P>

<DT><CODE>RNGs</CODE>
<DD>use Integer RNGs to provide other kinds of random numbers. 
</DL>
<P>

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

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<H2> 9.5 Template parameters </H2>
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<P>

RNGs take three template parameters, all of which have default values.
Using the <CODE>Uniform</CODE> RNG as an example, the template parameters of
<CODE>Uniform&#60;T, IRNG, stateTag&#62;</CODE> are:
</P><P>

<DL COMPACT>
<DT><CODE>T</CODE>
<DD>is the type of random number to generate (one of <CODE>float</CODE>,
<CODE>double</CODE>, or <CODE>long double</CODE> 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</CODE>.
<P>

<DT><CODE>IRNG</CODE>
<DD>is the underlying Integer RNG to use.  The default is MersenneTwister.
<P>

<DT><CODE>stateTag</CODE>
<DD><A NAME="IDX348"></A>
is either <CODE>sharedState</CODE> or <CODE>independentState</CODE>.  If
<CODE>sharedState</CODE>, the IRNG is shared with other generators.  If
<CODE>independentState</CODE>, the RNG contains its own IRNG.  The default is
sharedState.  
</DL>
<P>

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<H2> 9.6 Member functions </H2>
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<P>

<A NAME="IDX349"></A>
</P><P>

RNGs have these methods:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>T random();
</pre></td></tr></table><A NAME="IDX350"></A>
Returns a random number.
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>void seed(unsigned int);
</pre></td></tr></table><A NAME="IDX351"></A>
Seeds the underlying IRNG.  See above for an example of seeding
with the system timer.
</P><P>

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<H2> 9.7 Detailed listing of RNGs </H2>
<!--docid::SEC149::-->
<P>

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.
</P><P>

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<H3> 9.7.1 <TT>`random/uniform.h'</TT> </H3>
<!--docid::SEC150::-->
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>Uniform&#60;&#62;()
</pre></td></tr></table><A NAME="IDX352"></A>
Continuous uniform distribution on [0,1). 
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>UniformClosedOpen&#60;&#62;()
</pre></td></tr></table><A NAME="IDX353"></A>
Continuous uniform distribution on [0,1).  Same as <CODE>Uniform&#60;&#62;</CODE>.
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>UniformClosed&#60;&#62;()
</pre></td></tr></table><A NAME="IDX354"></A>
Continuous uniform distribution on [0,1].
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>UniformOpen&#60;&#62;()
</pre></td></tr></table><A NAME="IDX355"></A>
Continuous uniform distribution on (0,1).
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>UniformOpenClosed&#60;&#62;()
</pre></td></tr></table><A NAME="IDX356"></A>
Continuous uniform distribution on (0,1].
</P><P>

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<H3> 9.7.2 <TT>`random/normal.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>NormalUnit&#60;&#62;()
</pre></td></tr></table><A NAME="IDX357"></A>
Continuous normal distribution with mean 0 and variance 1.
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>Normal&#60;&#62;(T mean, T standardDeviation)
</pre></td></tr></table><A NAME="IDX358"></A>
Continuous normal distribution with specified mean and standard 
deviation.  
</P><P>

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<H3> 9.7.3 <TT>`random/exponential.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>ExponentialUnit&#60;&#62;()
</pre></td></tr></table><A NAME="IDX359"></A>
Continuous exponential distribution with mean 1.
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>Exponential&#60;&#62;(T mean)
</pre></td></tr></table><A NAME="IDX360"></A>
Continuous exponential distribution with specified mean.
</P><P>

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<H3> 9.7.4 <TT>`random/beta.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>Beta&#60;&#62;(T a, T b)
</pre></td></tr></table><A NAME="IDX361"></A>
Beta distribution with parameters a and b.  The mean of the distribution is
<EM>a/(a+b)</EM> and its variance is <EM>ab/((a+b)^2(a+b+1))</EM>.  Use the
method <CODE>setParameters(T a, T b)</CODE> to change the parameters.
</P><P>

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<H3> 9.7.5 <TT>`random/chisquare.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>ChiSquare&#60;&#62;(T df)
</pre></td></tr></table><A NAME="IDX362"></A>
<EM>\chi^2</EM> distribution with <CODE>df</CODE> degrees of freedom.  The parameter
df must be positive.  Use the method <CODE>setDF(T df)</CODE> to change the
degrees of freedom.
</P><P>

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<H3> 9.7.6 <TT>`random/gamma.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>Gamma&#60;&#62;(T mean)
</pre></td></tr></table><A NAME="IDX363"></A>
Gamma distribution with specified mean.  The mean must
be positive.  Use the method <CODE>setMean(T mean)</CODE> to
change the mean.
</P><P>

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<H3> 9.7.7 <TT>`random/F.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>F&#60;&#62;(T numeratorDF, T denominatorDF)
</pre></td></tr></table><A NAME="IDX364"></A>
F distribution with numerator and denominator degrees
of freedom specified.  Both these parameters must be
positive.  Use <CODE>setDF(T dfn, T dfd)</CODE> to change the
degrees of freedom. 
</P><P>

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<H3> 9.7.8 <TT>`random/discrete-uniform.h'</TT> </H3>
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<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>DiscreteUniform&#60;&#62;(T n)
</pre></td></tr></table><A NAME="IDX365"></A>
Discrete uniform distribution over <EM>0, 1, \ldots, n-1</EM>.
</P><P>

<A NAME="Numeric properties"></A>
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