RUNRANDOM Features

• Provides quasi-random generators based on versions of FAURE-NIEDERREITER, SOBOL', TEZUKA-FAURE-NIEDERREITER, and RICHTMEYER
• Provides pseudo-random generators based on versions of LECUYER, and KNUTH
• Provides Microsoft Excel utilities to view random vectors
• Provides a fast transformation to generate normal variates
• As much as 5,000 dimensions

Background: Computer-Generated Random Numbers

Computer generated numbers are not really random but only appear so when subjected to certain statistical tests.  Quasi-random numbers are based on properties of prime numbers and prime polynomial fields (an application of number theory). They are also called low discrepancy sequences because when they are generated, they fill in rectangular regions very evenly and uniformly. Pseudo random numbers do not have this property: their strength is that when they are generated, sequences of numbers have minimal correlation with each other. Simple pseudo random number generators are the ones that are included in many systems (e.g., via a rand() function).  They probably should not be trusted unless their autocorrelation properties are known.

Many theoretical and empirical results indicate that simulations based on quasi random numbers converge an order of magnitude faster than simulations based on pseudo random numbers. 

(More formally, for N simulations, the error associated with pseudo random simulations is proportional to .  For quasi random numbers, this error is proportional to .  This means that a simulation based on 10,000 pseudo random numbers may be as accurate as a simulation based on only 100 quasi random numbers: consequently, we can may be able to get the same answer with less work.)

The RUNRANDOM library provides an easy way to incorporate different types of quasi and pseudo random generators into your simulation applications.  RUNRANDOM generates multivariate uniform random variables and multivariate zero-mean unit variance normal random variables (via a very fast approximation to the inverse normal distribution).  These numbers are accessed in memory (via a vector object) or by reading a file.

RUNRANDOM provides 6 different generators and comes with a tool to display samples of these numbers and for timing analysis.  Here are some extracts of the RUNRANDOM tool output.  For example, the first 6 vectors of a Uniform and Normal 3-dimensional RICHTMEYER sequence are

This sequence generates random vectors based on prime numbers. Note that the first vector is the zero vector.  (RUNRANDOM can skip several vectors of numbers use by changing the SKIP parameter; in this and the following examples, SKIP is set to zero.)

The first 6 vectors of a Uniform and Normal 3-dimensional SOBOL' sequence look like this:

Note that these numbers appear "less random" than the other sequence, and that they fill in the space "more evenly." 

A related method is the FAURE-NIEDERREITER sequence.  The first 6 vectors of a Uniform and Normal 3-dimensional sequence look like this: