23 April, 2006

Copyright (C) 1989, 1995, 1998, 2002: R B Davies

- Overview
- Getting started
- Descriptions of the classes to be accessed by the user
- Descriptions of the supporting classes
- Generating numbers from other distributions
- Other people's code
- Files included in this package
- Class structure
- To do
- History
- To online documentation page

This is a C++ library for generating sequences of random numbers from a wide variety of
distributions. It is particularly appropriate for the situation where one requires
sequences of identically distributed random numbers since the set up time for each type of
distribution is relatively long but it is efficient when generating each new random
number. The library includes *classes* for generating random numbers from a number of
distributions and is easily extended to be able to generate random numbers from almost any
of the standard distributions.

Comments and bug reports to robert** at **statsresearch.co.nz [replace **
at** by you-know-what].

For updates and notes see http://www.robertnz.net.

There are no restrictions on the use of *newran* except that I take no
liability for any problems that may arise from its use.

I welcome its distribution as part of low cost CD-ROM collections.

You can use it in your commercial projects. However, if you distribute the source, please make it clear which parts are mine and that they are available essentially for free over the Internet.

The following are the classes for generating random numbers from particular distributions

Uniform | uniform distribution |

Constant | return a constant |

Exponential | negative exponential distribution |

Cauchy | Cauchy distribution |

Normal | normal distribution |

ChiSq | non-central chi-squared distribution |

Gamma | gamma distribution |

Pareto | Pareto distribution |

Poisson | Poisson distribution |

Binomial | binomial distribution |

NegativeBinomial | negative binomial distribution |

The following classes are available to the user for generating numbers from other distributions

PosGenX | Positive random numbers with a decreasing density |

SymGenX | Random numbers from a symmetric unimodal density |

AsymGenX | Random numbers from an asymmetric unimodal density |

PosGen | Positive random numbers with a decreasing density |

SymGen | Random numbers from a symmetric unimodal density |

AsymGen | Random numbers from an asymmetric unimodal density |

DiscreteGen | Random numbers from a discrete distribution |

SumRandom | Sum and/or product of random numbers |

MixedRandom | Mixture of random numbers |

Each of these classes has the following member functions

Real Next() |
Get a new random number |

char* Name() |
Name of the distribution |

ExtReal Mean() |
Mean of the distribution |

ExtReal Variance() |
Variance of the distribution |

These 4 functions are declared *virtual* so it is easy to write simulation
programs that can be run with different distributions.

*Real* is typedefed to be either *float* or *double*. See customising. Note that `Next()` always returns a *Real*
even for discrete distributions.

ExtReal is a class which is, in effect either a *Real* or
one of the following: *PlusInfinity*, *MinusInfinity*, *Indefinite* or *Missing*.
I use *ExtReal* so that I can return infinite or indefinite values for the mean or
variance of a distribution.

There are two static functions in the class *Random*.

void Random::Set(double)

must be called with an argument between 0 and 1 to set up the base random number
generator before `Next()` is called in any class.

double Random::Get()

returns the current value of the seed.

There are two classes for doing combinations and permutations.

RandomPermutation | Draw numbers without replacement |

RandomCombination | Draw numbers without replacement and sort |

There are three classes for generating random numbers where we want to vary the parameters at each call. It would be inefficient to use the previous classes since you would need to set up a random number object at each call.

VariPoisson | Poisson distribution |

VariBinomial | Binomial distribution |

VariLogNormal | Log normal distribution |

Further details of all these classes including the constructors are given below.

The file include.h sets a variety of options including several compiler dependent options. You may need to edit include.h to get the options you require. If you are using a compiler different from one I have worked with you may have to set up a new section in include.h appropriate for your compiler.

Borland, Gnu, Microsoft and Watcom are recognised automatically. If none of these are recognised a default set of options is used. These are fine for AT&T, HPUX and Sun C++. If you using a compiler I don't know about, you may have to write a new set of options.

There is an option in include.h for selecting whether you use compiler supported exceptions, simulated exceptions, or disable exceptions. Use the option for compiler supported exceptions if and only if you have set the option on your compiler to recognise exceptions. Disabling exceptions sometimes helps with compilers that are incompatible with my exception simulation scheme.

This version of newran does not do memory
clean-up with the simulated exceptions. |

If your (very old) compiler does not recognises *bool* deactivate the
statement `#define bool_LIB`. This will activate my Boolean class.

Activate the appropriate statement to make the element type *Real* to mean *float*
or *double*.

Activate the *namespace* option if your want to use namespaces and have a compiler
that *really* does support them.

You will need to compile newran.cpp, myexcept.cpp and extreal.cpp and link the
resulting object files to your programs. Your source files which access *newran* will
need to have newran.h as an include file.

I have tested newran02 with the following compilers (all PC ones in 32 bit console mode)

Borland 5.0, 5.5, 5.6 | OK |

Microsoft 5.0, 6.0, 7.0 | OK |

Watcom 10A | OK |

Sun CC 6 | OK |

Gnu G++ 2.96 (Linux), 2.95 (Sun) | OK |

Intel 5 for Windows or Linux | OK |

I have included make files for Borland 5.5, CC and Gnu G++. See files section. You can use my genmake program to generate make files for other compilers.

The files tryrand.cpp, tryrand1.cpp, tryrand2.cpp, tryrand3.cpp, tryrand4.cpp, tryrand5.cpp, hist.cpp run the generators in the library and print histograms of the resulting distributions. Sample means and variances are also calculated and can be compared with the population values. The results from Borland 5.5 are in tryrand.txt. Other compilers may give different but still correct results. In particular, Microsoft C++ compilers give different results when global optimisation is set. This is most likely due to different round-off error but may also be due to different order of evaluation of expressions. The appearance of the histograms in the output should be similar to that in tryrand.txt and the statistical tests should still be passed. There are some notes in tryrand.txt for assessing the output.

If you are compiling on a PC with a 16 bit compiler you will need to set n_large
to be 8000 rather than 1000000 and n to 8000 rather than 200000 in tryrand.cpp.
This will change the output from the test program from that given in tryrand.txt although
the general appearance should be the same. |

The test program tryrand.cpp includes a simple test for memory leaks. This is valid for only some compilers. It seems to work for Borland C++ in console mode but not for Gnu G++ or Microsoft C++, where it almost always (presumably incorrectly) suggests an error.

This is the basic uniform random number generator, used to drive all the others. The
Lewis-Goodman-Miller algorithm is used with Marsaglia mixing. While not perfect, and now
superseded, the LGM generator has given me acceptable results in a wide variety of
simulations. See *Numerical Recipes in C* by Press, Flannery, Teukolsky, Vetterling
published by the Cambridge University Press for details. The LGM generator does pass the
Marsaglia *diehard* tests when
you include the mixing. (It doesn't pass without the mixing). Nevertheless it should be
upgraded. It is very dubious if you are going to call it more than 100 million
times in a simulation. Ideally the basic generator should be recoded in assembly language to give the
maximum speed to all the generators in this package. You can access the numbers directly
using `Next()` but I suggest you use class Uniform for
uniform random numbers and reserve Random for setting the starting seed and as the base
class for the random number generators.

Remember that you need to call `Random::Set(double seed)` at the beginning of
your program. See the overview and tryrand.cpp.

Return a uniform random number from the range (0, 1). The constructor has no parameters. For example

Uniform U; for (int i=0; i<100; i++) cout << U.Next() << "\n";

prints a column of 100 numbers drawn from a uniform distribution.

This returns a constant. The constructor takes one *Real* parameter; the value of
the constant to be returned. So

Constant C(5.5); cout << C.Next() << "\n";

prints 5.5.

This generates random numbers with density `exp(-x)` for `x>=0`. The
constructor takes no arguments.

Exponential E; for (int i=0; i<100; i++) cout << E.Next() << "\n";

Generates random numbers from a standard Cauchy distribution. The constructor takes no parameters.

Cauchy C; for (int i=0; i<100; i++) cout << C.Next() << "\n";

Generates standard normal random numbers. The constructor has no arguments. This class has been augmented to ensure only one copy of the arrays generated by the constructor exist at any given time. That is, if the constructor is called twice (before the destructor is called) only one copy of the arrays is generated.

Normal Z; for (int i=0; i<100; i++) cout << Z.Next() << "\n";

Non-Central chi-squared distribution. The method uses ChiSq1 to generate the
non-central part and Gamma2 or Exponential to generate the central part. The constructor
takes as arguments the number of degrees of freedom `(>=1)` and the
non-centrality parameter (omit if zero).

int df = 10; Real noncen = 2.0; ChiSq CS(df, noncen); for (int i=0; i<100; i++) cout << CS.Next() << "\n";

Gamma distribution. The constructor takes the shape parameter as argument. Uses Gamma1, Gamma2 or Exponential.

Real shape = 0.75; Gamma G(shape); for (int i=0; i<100; i++) cout << G.Next() << "\n";

Pareto distribution. The constructor takes the shape parameter as argument. I follow
the definition of Kotz and Johnson's *Continuous univariate distributions 1*, chapter
19, page 234, with *k* = 1. The generator uses a power transform of a uniform random
number.

Real shape = 0.75; Pareto P(shape); for (int i=0; i<100; i++) cout << P.Next() << "\n";

Poisson distribution: uses Poisson1 or Poisson2. Constructor takes the mean as its argument.

Real mean = 5.0; Poisson P(mean); for (int i=0; i<100; i++) cout << (int)P.Next() << "\n";

Binomial distribution: uses Binomial1 or Binomial2. Constructor takes *n* and *p*
as its arguments.

int n = 50; Real p = 0.25; Binomial B(n, p); for (int i=0; i<100; i++) cout << (int)B.Next() << "\n";

Negative binomial distribution. Constructor takes *N* and *P* as its
arguments. I use the notation of Kotz and Johnson's *Discrete distributions*. Some
people use *p* = 1/(*P*+1) in place of the second parameter.

Real N = 12.5; Real P = 3.0; NegativeBinomial NB(N, P); for (int i=0; i<100; i++) cout << (int)NB.Next() << "\n";

This uses an arbitrary density satisfying the previous conditions to generate random
numbers from that density. Suppose `Real pdf(Real)` is the density. Then use `pdf`
as the argument of the constructor. For example

PosGenX P(pdf); for (int i=0; i<100; i++) cout << P.Next() << "\n";

Note that the probability density pdf must drop to exactly 0 for
the argument large enough. For example, include a statement in the program for pdf
that, if the value is less than 1.0E-15, then return 0. |

This corresponds to PosGenX for symmetric distributions.

Note that the probability density pdf must drop to exactly 0 for
the argument large enough. For example, include a statement in the program for pdf
that, if the value is less than 1.0E-15, then return 0. |

Corresponds to PosGenX. The arguments of the constructor are the name of the density function and the location of the mode.

Real pdf(Real); Real mode; ..... AsymGenX X(pdf, mode); for (int i=0; i<100; i++) cout << X.Next() << "\n";

Note that the probability density pdf must drop to exactly 0 for
the argument large (large positive and large negative) enough. For example, include a
statement in the program for pdf that, if the value is less than 1.0E-15, then
return 0. |

PosGen is not used directly. It is used as a base class for generating a random number
from an arbitrary probability density `p(x)`. `p(x)` must be non-zero only
for `x>=0`, be monotonically decreasing for `x>=0`, and be finite. For
example, `p(x)` could be `exp(-x)` for `x>=0`.

The method is to cover the density in a set of rectangles of equal area as in the
diagram (indicated by `---`).

|x|xx------|xx||xxx||.......xxx---------||xxxx|||xxxx|||.........xxxxx------------|| |xxxxx||| |xxxxxx||| |..............xxxxxx----------------------|| | |xxxxxxx||| | |xxxxxxx||| | |xxxxxxxx|+===========================================================================

The numbers are generated by generating a pair of numbers uniformly distributed over
these rectangles and then accepting the *X* coordinate as the next random number if
the pair corresponds to a point below the density function. The acceptance can be done in
two stages, the first being whether the number is below the dotted line. This means that
the density function need be checked only very occasionally and on the average only just
over 3 uniform random numbers are required for each of the random numbers produced by this
generator.

See PosGenX or Exponential for the method of deriving a class to generate random
numbers from a given distribution.

Note that the probability density p(x) must drop to exactly 0 for
the argument, x, large enough. For example, include a statement in the program for p(x)
that, if the value is less than 1.0E-15, then return 0. |

SymGen is a modification of PosGen for unimodal distributions symmetric about the origin, such as the standard normal.

A general random number generator for unimodal distributions following the method used by PosGen. The constructor takes one argument: the location of the mode of the distribution.

This is for generating random numbers taking just a finite number of values. There are two alternative forms of the constructor:

DiscreteGen D(n,prob); DiscreteGen D(n,prob,val);

where `n` is an integer giving the number of values, `prob` a Real array
of length `n` giving the probabilities and `val` a Real array of length `n`
giving the set of values that are generated. If `val` is omitted the values are `0,1,...,n-1`.

The method requires two uniform random numbers for each number it produces. This method
is described by Kronmal and Peterson, *American Statistician*, 1979, Vol 33, No 4,
pp214-218.

This is for building a random number generator as a linear or multiplicative
combination of existing random number generators. Suppose `RV1`, `RV2`, `RV3`,
`RV4` are random number generators defined with constructors given above and `r1`,
`r2`, `r0` are Reals and `i1`, `i3` are integers.

Then the generator `S` defined by something like

SumRandom S = RV1(i1)*r1 - RV2*r2 + RV3(i3)*RV4 + r0;

has the obvious meaning. `RV1(i1)` means that the sum of `i1` independent
values from `RV1` should be used. Note that `RV1*RV1` means the product of
two independent numbers generated from `RV1`. Remember that `SumRandom` is
slow if the number of terms or copies is large. I support the four arithmetic operators `+`,
`-`, `*` and `/` but cannot calculate the means and variances if you
divide by a random variable.

Use `SumRandom` to quickly set up simple combinations of the existing
generators. But if the combination is going to be used extensively, then it is probably
better to write a new class to do this.

Example: *normal* with mean = 10, standard deviation = 5:

Normal N; SumRandom Z = 10 + 5 * N; for (int i=0; i<100; i++) cout << Z.Next() << "\n";

Example: *F* distribution with *m* and *n* degrees of freedom:

int m, n; ... put values in m and n ChiSq Num(m); ChiSq Den(n); SumRandom F = (double)n/(double)m * Num / Den; for (int i=0; i<100; i++) cout << F.Next() << "\n";

This is for mixtures of distributions. Suppose `rv1`, `rv2`, `rv3`
are random number generators and `p1`, `p2`, `p3` are Reals summing
to 1. Then the generator `M` defined by

MixedRandom M = rv1(p1) + rv2(p2) + rv3(p3);

produces a random number generator with selects its next random number from `rv1`
with probability `p1`, `rv2` with probability `p2`, `rv3` with
probability `p3`.

Alternatively one can use the constructor

MixedRandom M(n, prob, rv);

where `n` is the number of distributions in the mixture, `prob` the Real
array of probabilities, `rv` an array of pointers to random variables.

Normal with outliers:

Normal N; Cauchy C; MixedRandom Z = N(0.9) + C(0.1); for (int i=0; i<100; i++) cout << Z.Next() << "\n";

or:

Normal N; MixedRandom Z = N(0.9) + (10*N)(0.1); for (int i=0; i<100; i++) cout << Z.Next() << "\n";

To draw `M` numbers without replacement from `start, start+1, ..., start+N-1`
use

RandomPermutation RP; RP.Next(N, M, p, start);

where `p` is an `int*` pointing to an array of length `M` or
longer. Results are returned to that array.

RP.Next(N, p, start);

assumes `M = N`. The parameter, `start`
has a default value of 0.

The method is rather inefficient if `N` is very large and `M` is much
smaller.

To draw `M` numbers without replacement from `start, start+1, ..., start+N-1`
and then sort use

RandomCombination RC; RC.Next(N, M, p, start);

where `p` is an `int*` pointing to an array of length `M` or
longer. Results are returned to that array.

RC.Next(N, p, start);

assumes `M = N`. The parameter, `start`
has a default value of 0.

The method is rather inefficient if `N` is large. A better approach for large `N`
would be to generate the sorted combination directly. This would also provide a better way
of doing permutations with large `N`, small `M`.

Use this class if you want to generate a Poisson random variable but you want to change the parameter frequently, so using the Poisson class would be inefficient. There is one member function

int VariPoisson::iNext(Real mu);

which returns a new Poisson random number with mean `mu`. To generate
100 Poisson random numbers with means 1,2,...,100 use the following program

VariPoisson VP; for (int i = 1; i <= 100; ++i) { Real mu = i; cout << VP.iNext(mu) << end; }

The constructor is slow so put it outside any loop. The individual
calls to `iNext` should be quite fast. The method is approximate for `mu >= 300.` The
constructor is not in any class hierarchy and `iNext` is not virtual. This class is somewhat beta-ish and may change in a future release of *
newran*.

Use this class if you want to generate a Binomial random variable but you want to change the parameters of the frequently, so using the Binomial class would be inefficient. There is one member function

int VariBinomial::iNext(int n, Real p);

which returns a new Binomial random number with number of trials `n` and
probability of success `p`. To generate
100 Binomial random numbers with `n` = 1,2,...,100 and `p` = 0.5 use the following program

VariBinomial VB; for (int n = 1; n <= 100; ++n) { Real p = 0.5; cout << VB.iNext(n, p) << end; }

The constructor is slow so put it outside any loop. The individual
calls to `iNext` should be quite fast. The method is approximate if both `
n*p > 200` and `n*(1-p) > 200.` The
constructor is not in any class hierarchy and `iNext` is not virtual. This class is somewhat beta-ish and may change in a future release of *
newran*.

Use this class if you want to generate a log normal random variable and you want to change the parameters of the frequently. There is one member function

Real VariLogNormal::Next(Real mean, Real sd);

which returns a new log normal random number with mean `mean` and
standard deviation `sd`. Note that `mean` and
`sd` are the mean and standard deviation of the log normal distribution and not of
the underlying normal distribution. To generate
100 log normal random numbers with `mean` = 1,2,...,100 and `sd` = 1.0 use the following program

VariLogNormal VLN; for (int i = 1; i <= 100; ++i) { Real mean = i; Real sd = 1.0; cout << VLN.Next(mean, sd) << end; }

The
constructor is not in any class hierarchy and `Next` is not virtual. This class is somewhat beta-ish and may change in a future release of *
newran*.

A class consisting of a Real and an enumeration, `EXT_REAL_CODE`, taking the
following values:

- Finite
- PlusInfinity
- MinusInfinity
- Indefinite
- Missing

The arithmetic functions `+`, `-`, `*`, `/` are defined in
the obvious ways, as is `<<` for printing. The constructor can take either a
Real or a value of `EXT_REAL_CODE` as an argument. If there is no argument the
object is given the value *Missing*. Member function `IsReal()` returns *true*
if the enumeration value is *Finite* and in this case value of the Real can be found
with `Value()`. The enumeration value can be found with member function `Code()`.

*ExtReal* is used at the type for values returned from the *Mean* and *Variance*
member functions since these values may be infinite, indefinite or missing.

Non-central chi-squared with one degree of freedom. Used as part of ChiSq.

This generates random numbers from a gamma distribution with shape parameter `alpha
< 1`. Because the density is infinite at *x* = 0 a power transform is
required. The constructor takes `alpha` as an argument.

Gamma distribution for the shape parameter, `alpha`, greater than 1. The
constructor takes `alpha` as the argument.

Poisson distribution; derived from AsymGen. The constructor takes the mean as the argument. Used by Poisson for values of the mean greater than 10.

Poisson distribution with mean less than or equal to 10. Uses DiscreteGen. Constructor takes the mean as its argument.

Binomial distribution; derived from AsymGen. Used by Binomial for `n >= 40`.
Constructor takes *n* and *p* as arguments.

Binomial distribution with `n < 40`. Uses DiscreteGen. Constructor takes *n*
and *p* as arguments.

These are used by SumRandom and MixedRandom.

Distribution type |
Method |
Example |
---|---|---|

Continuous finite unimodal density (no parameters, can calculate density) | Use PosGenX, SymGenX or AsymGenX. | |

Continuous finite unimodal density (with parameters, can calculate density) | Derive a new class from PosGen, SymGen
or AsymGen, over-ride Density. |
Gamma2 |

Can calculate inverse of distribution | Transform uniform random number. | Pareto |

Transformation of supported random number | Derive a new class from the existing class | ChiSq1 |

Transformation of several random numbers | Derive new class from Random; generate the new random number from the existing generators. | ChiSq |

Density with infinite singularity | Transform a random variable generated by PosGen, SymGen or AsymGen. | Gamma1 |

Distribution with several modes | Breakdown into a mixture of unimodal distributions. | |

Linear or quadratic combination of supported random numbers | Use SumRandom. | |

Mixture of supported random numbers | Use MixedRandom. | |

Discrete distribution (< 100 possible values) | Use DiscreteGen. | Poisson2 |

Discrete distribution (many possible values) | Use PosGen, SymGen or AsymGen. | Poisson1 |

The gamma function is adapted from *Numerical Recipes in C*
by Press, Flannery, Teukolsky, Vetterling published by the Cambridge University Press.
The Shell sort and quick sort are adapted from *Algorithms in C++* by
Sedgewick published by Addison Wesley.

readme.txt | readme file |

nr02doc.htm | this file |

rbd.css | style sheet for newran02.htm |

newran.h | header file for newran |

newran.cpp | main code file |

extreal.h | header file for extended reals |

extreal.cpp | code file for extended reals |

boolean.h | definition of bool |

include.h | option file |

tryrand.h | header file for tryrand |

tryrand.cpp | test file |

tryrand1.cpp | called by tryrand - histograms of simple examples |

tryrand2.cpp | called by tryrand - histograms of advanced examples |

tryrand3.cpp | called by tryrand - statistical tests |

tryrand4.cpp | called by tryrand - test permutations |

tryrand5.cpp | called by tryrand - test "vari" versions of generators |

hist.cpp | called by tryrand - draw histogram |

tryrand.txt | output from tryrand |

nr_cc.mak | make file for CC compiler |

nr_gnu.mak | make file for gnu G++ compiler |

nr_b55.mak | make file for Borland 5.5 compiler |

The following diagram gives the class hierarchy of the package.

ExtReal..........................Extended real numbersRandom........................... Uniform random number generator | +---Constant.................... Return a constant | +---PosGen...................... Used by PosGenX etc | | | +---PosGenX................ Positive random #s from decreasing density | | | +---Exponential............ Negative exponential rng | | | +---Gamma1................. Used by Gamma (shape parameter < 1) | | | +---SymGen................. Used by SymGenX etc | | | +---SymGenX........... Random numbers from symmetric density | | | +---Cauchy............ Cauchy random number generator | | | +---Normal............ Standard normal random number generator | | | +---ChiSq1....... Used by ChiSq (one df) | +---AsymGen..................... Used by AsymGenX etc | | | +---AsymGenX............... Random numbers from asymmetric density | | | +---Poisson1............... Used by Poisson (mean > 8) | | | +---Binomial1.............. Used by Binomial (n >= 40) | | | +---NegativeBinomial....... Negative binomial random number generator | | | +---Gamma2................. Used by Gamma (shape parameter > 1) | +---ChiSq....................... Non-central chi-squared rng | +---Gamma....................... Gamma random number generator | +---Pareto...................... Pareto random number generator | +---DiscreteGen................. Discrete random number generator | +---Poisson2.................... Used by Poisson (mean <= 8) | +---Binomial2................... Used by Binomial (n < 40) | +---Poisson..................... Poisson random number generator | +---Binomial.................... Binomial random number generator | +---SumRandom................... Sum of random numbers | +---MixedRandom................. Mixture of random numbers | +---MultipliedRandom............ Used by SumRandom | | | +---AddedRandom............ Used by SumRandom | | | +---SubtractedRandom....... Used by SumRandom | +---ShiftedRandom............... Used by SumRandom | | | +---ReverseShiftedRandom... Used by SumRandom | | | +---ScaledRandom........... Used by SumRandom | +---NegatedRandom.......... .... Used by SumRandom | +---RepeatedRandom.............. Used by SumRandom | +---AddedSelectedRandom......... Used by MixedRandom | +---SelectedRandom.............. Used by MixedRandomRandomPermutation................ Random permutation | +---RandomCombination........... Sorted random permutation

VariPoisson...................... Poisson generator

VariBinomial..................... Binomial generator

VariLogNormal.................... Log normal generator

- More modern alternatives to the LGM generator;
- Additional generator classes;
- Better methods for combinations and permutations with large
`N`and small`M`; - Faster method for normal distribution
- Improve test program

April, 2006 - make compatible with G++ 4.1 and VC++8 (you should also set the standard option in include.h with these compilers).

July, 2002 - bring into line with my other libraries; VariPoisson, VariBinomial, VariLogNormal classes; change to Sedgewick's Shell sort.

August, 1998 - update exception package; work around problem with MS VC++ 5

January, 1998 - version compatible with newmat09

1995 - *newran* version, additional distributions

1989 - initial version