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Subject: Random numbers (again)

[I found the signal-to-noise ratio on this topic rather low,
but decide to post some, I hope, useful information. For
background, I am the Professor of Applied Statistics at 
Oxford, and have written a monograph on simulation.]

Many of the issues (physical vs deterministic, period length and so on)
have been discussed in the literature.  Good books and expository
articles (I won't list the many bad ones) are, in TeX from my course
notes:

References
==========

\ref Bratley, P., Fox, B.L. \& Schrage, L.E. (1987) {\sl A guide to
simulation.  2nd edition} Springer.
\comm{Concentrates on the real problems, lots of challenging exercises.}

\ref Devroye, L. (1986) {\sl Non-uniform random variate generation.}
Springer. \comm{{\sl The} definitive reference on its title.  Very long,
843pp, and lacking recommendations.}

\ref l'Ecuyer, P. (1990) Random numbers for simulation. {\sl Comm.
Assoc. Comput. Mach.} \bl 33 85--97.
\comm{Up-to-date comments on the subject by a computer scientist.}

\ref James, F. (1990) A review of pseudorandom numbers.  {\sl Comp.
Phys. Comm.} \bl 60 329--344.
\comm{Up-to-date comments on the subject by a physicist.}

\ref Knuth, D.E. (1981) {\sl Seminumerical algorithms, 2nd edition.}
Addison-Wesley.\comm{Very mathematical, once the definitve reference.}
 
\ref Morgan, B.J.T. (1984) {\sl Elements of simulation.} Chapman \&
Hall. \comm{A good introduciton, but out-of-date (even when written).}

\ref Ripley, B.D. (1983) The computer generation of random variables
--- a tutorial. \quad{\sl Int. Statist. Rev.} \bl 51 301--319.
\comm{Still a useful tutorial.}

\ref Ripley, B.D. (1987) {\sl Stochastic Simulation.}  Wiley, New York,
237pp. \comm{The most complete reference, rather terse and mathematical.}

\ref Ripley, B.D. (1990) Thoughts on pseudorandom number generators.
{\sl J. Comput. Appl. Math.} \bl 31 153--163.
\comm{Up-to-date comments on the subject by a statistician.}

A revised version of the last is available by anonymous ftp from

markov.stats.ox.ac.uk (192.76.20.1)

as  pub/simul/simul/prng.ps.Z (432Kb)
(One page has a large figure, so prints slowly.)

Brief summary:
==============

Good {31,32}-bit congruential generators are good enough for routine
purposes.  Large simulations (and I have heard of a study wanting 10^9
per second) will do better with GFSR generators or Marsaglia's
add-and-carry.  However, history suggests beware anything new; it takes
10-25 years to find the fatal flaws in new suggestions. For
congruential and GFSRs we know a lot about the defects, and can design
them to be innocuous as possible.

Interesting ideas:
=================

\ref Eichenauer-Herrmann, J. (1992) Inversive congruential pseudorandom 
numbers: a tutorial. {\sl Int. Statist. Rev. } \bl 60 167--176
[More an advertisement than a tutorial.]

\ref Marsaglia, G. \& Zaman, A. (1991) A new class of random number 
generators.  {\sl Ann. Appl. Probab.} \bl 1,3.

.----------------------------------------------------.
|      Prof. Brian D. Ripley                         |
|      Dept. of Statistics, University of Oxford,    |
|      1 South Parks Road, Oxford OX1 3TG, UK        |
|                                                    |
|      +44 865 272861 (self) +44 865 272860 (secr)   |
|      +44 865 272595 (fax)                          |
|      ripley@uk.ac.ox.stats (JANET)                 |
|      ripley@stats.ox.ac.uk (Internet)              |
`----------------------------------------------------'