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 "Random Number Generation in ",
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 " 6"
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Cell["Andrzej Kozlowski", "Text",
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Cell["\<\
Tokyo Denki University, Chiba, Japan and Warsaw University, Warsaw, Poland\
\>", "Text",
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Cell["Abstract", "Subsubsection",
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Cell[TextData[{
 "We describe the main aspects of the new mechanism of random number \
generation in ",
 StyleBox["Mathematica",
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 " 6.  After discussing some problems in the design of the random number \
generation in earlier versions of ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ", we show how they were overcome in  versions 6, and explain why the \
flexibility and extensibility of the new design should prevent such problems \
arising again in the future. We illustrate the new flexibility with an \
example from mathematical finance, by computing the value of a \"european \
call option\" by Monte-Carlo simulation using different built-in generators \
with variable parameters, and the new \"extensibility\" by  defining a  \
random number generator for a a finite non-uniform distribution. \n"
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Cell["\<\
(Pseudo-) Random Number Generators
\
\>", "Section",
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Cell[TextData[{
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 " 6 comes with a completely new set of tools for generating and using \
pseudo-random sequences of numbers. These may not be the most celebrated or \
spectacular of the new features but their importance is difficult to  \
overstate.  Pseudo-random numbers (\"pseudo\" -to distinguish them from \
genuine \"random numbers\", which can, at best, be generated only by certain \
physical processes), to which from now on we shall refer to as simply as \
\"random numbers\"  or RNs, have played  a huge role in many areas of \
science, mathematics and computer science ever since the invention by Enrico \
Fermi, Stanislaw Ulam, Nicholas Metropolis and John von Neumann  of what is \
now known as the \"Monte-Carlo\" method of integration. In order to apply \
this technique John Von Neumann invented  the first RNG (Random Number \
Generator) - the \"mid square method\". By today's  standards it was of poor \
quality  but it served its purpose and gave rise to a whole new area of \
study.\n\nAs von Neumann pointed out in an often quoted remark: \"there are \
no such things as random numbers - there are only methods to generate random \
numbers\".  Each such method generates random numbers according to some \
statistical distribution. Usually one particular distribution, the Uniform \
Distribution,  is considered as basic; once we have a RNG which approximates \
the Uniform Distribution we can define RNGs which approximate any other \
distribution by means o several well known techniques. \n\nSince a RNG \
approximates a statistical distribution we can always ask how close is the \
approximation to the  statistical distribution it is supposed to approximate. \
A good uniform RNG is one that produces sequences without any \"apparent \
patterns\", or more exactly: the patterns that do occur do so not \
significantly more  or less often than one would expect in the case of \
genuine uniform distribution. There are a variety of well known statistical \
tests that are used to assess the quality of RNGs. Many are discussed in \
Knuth's book [Kn].  One of the most popular  test suites is  George \
Marsaglia's \"Diehard\" battery of tests (the pun is deliberate, see [Ma]).  \
It should always be remembered that even the best RNG  cannot be expected to \
pass all possible tests.  As Donald Knuth wrote in 1969: \"Every random \
number generator will fail in at least one application\". To what extend this \
is true about some of the modern RNGs may be argued about (in particular see \
the discussion of RNGs in [Wo2]) but it is clear that rather than aiming for \
an ideal RNG it is more important to have a number of RNGs that perform well \
under well understood conditions, and which offer us enough flexibility so \
that we can choose the best one for the task at hand. Mathematica 6 \
represents a big advance in this direction over the previous versions.\n"
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Cell[TextData[{
 " RNGs in ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 "  prior to version 6.\n"
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Cell[TextData[{
 "\nRandom Number generation has from the beginning played a somewhat special \
role in ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ".  The very first version of ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " used two RNGs. One of them  was, at that time, a  new and remarkable \
non-linear  RNG - which was the first important practical application of \
Stephen Wolfram's work on Cellular Automata (nowadays usually referred to as \
NKS [Wo2]). In [Wo1] it was first observed that a simple 1-dimensional CA \
with neighbourhood rule of size 3, CA30 (so named  because of the decimal \
value of its eight-row truth table) makes it possible to construct a \
non-linear RNG (for the uniform distribution) that performed better on tests \
then  linear feedback shift-register RNGs popular at the time. In fact, the \
Wolfram CA30 RNG was able to pass all the tests in the  Marsaglia Diehard \
test suite. The other RNG also used by ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " was a well known and, at that time, highly regarded linear congruential \
RNG  of Marsaglia and Zaman,  which used a \"lagged\" version of the \
\"subtract with borrow\" (SWB) algorithm.  Unfortunately, in versions of \
Mathematica prior to 6, these generators were not available as two separate, \
user selectable, methods of generating RNs  but rather ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " used one  or the other RNG or a combination of the two depending on what \
sort of random numbers were required. For example, to generate small integers \
the CA30 generator was used, to generate machine reals the SWB was used and \
to generate large integers both were used together. (The precise description \
of the approach was given by Daniel Lichtblau in [Li] ). \n   As computers \
became faster more RNs were needed for sophisticated applications and RNGs \
that were once regarded as satisfactory were found wanting. This is what \
happened to the Marsaglia Zaman SWB RNG. In 1995 this RNG was found to fail \
Marsaglia's gap test (one of Marsaglia's Diehard suite), when the gap exceeds \
the \"long lag\".  Unfortunately the lags in the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " implementation of the SWB algorithm were rather small and from around the \
year 2000 posts began to appear on the ",
 StyleBox["MathGroup",
  FontSlant->"Italic"],
 " from people who had discovered something \"fishy\" in the results of \
simulations they run which depended on the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " Random[] function. In response to this Daniel Lichtblau posted a partial \
fix on the ",
 StyleBox["MathGroup",
  FontSlant->"Italic"],
 " [Li]. The fix used the fact that all known problems were traced to the SWB \
RNG while the CA30 RNG, which had passed all the Diehard tests with flying \
colors. Hence by replacing the use of the SWB RNG whenever possible by the \
CA30 one could bypass the problem. \n  The fix, however, was only a stop-gap \
measure and a far from satisfactory one. Firstly, it resulted in a very \
substantial loss in performance. Secondly, it fixed only one (albeit the most \
common) problem; the one that occurred when machine random reals where \
generated. Another  problem, which occurred with large random integers, \
remained. Thirdly, there was no guarantee that, as computer speeds progressed \
further, new kind of problems affecting perhaps even the CA30 would not be \
found or at least new RNGs found that perform better than the built-in ones. \
In fact, there are now known CA based RNGs  that perform better than the  \
CA30. Thus, only by allowing the user a choice of an RNG to fit his purpose \
and perhaps the ability to add  new RNGs could one solve the existing \
problems with enough flexibility to cope with future ones. Such a solution \
obviously required a fundamentally new approach, which is why it took longer \
to fix the problems  than many expected it to take. Thanks to that, however, \
",
 StyleBox["Mathematica",
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 " 6 now offers practically everything that one could wish for in this \
respect. \n  "
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 "  \nAdopting a radically new approach meant, however, that it was \
impossible to preserve at the same time the old function names and \
compatibility with earlier versions of ",
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 ". Thus it was decided to keep the old function Random[] with its \
problematic legacy behaviour and add a new collection of functions with names \
identifying the kind of random number that they generated. Thus,  the old \
single \"form\" Random[type, range], where type could be Integer, Real or \
Complex,  has been superseded by three new functions RandomReal[range], \
RandomInteger[range], RandomComplex[range] . Furthermore, these new functions \
can now be used to generate random arrays very efficiently :"
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This use of RandomInteger, RandomReal, etc, replaces the function RandomArray \
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RNGs work with built in distributions other than the Uniform distribution. \
Here we use this to simulate 20 paths of a random walk approximation to the \
standard Wiener Process (Brownian motion). \
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