A Random Number Generator Test Suite for the C++ ... - ETH Zürich

2. What are random numbers?Random numbers are characterised by the fact that their value can not be predicted. Or, inother words, if one constructs a sequence of random numbers, the probability distribution ofthe following random numbers have to be completely independent of all the other generatednumbers.A more sophisticated mathematical definition and discussion can be found in [6].2.1. Types of random numbersThere are three types of random numbers, quasi-, pseudo- and true- random numbers. Thesedifferent types of random numbers have different applications. (It is philosophical questionwhat we can call random or not, but here, we use the following descriptions, its simpler. . . )True Random Number The most often used example for “truly” random numbers is the decay ofa radioactive material. If a Geiger counter is put in front of such a radioactive source, theintervals between the decay events are truly random. True random numbers are gained fromphysical processes like radioactive decay or also rolling a dice. But rolling a dice is difficult,perhaps someone could control the dice so well to determine the outcome.Pseudo Random Number These numbers are generated by a computer or that is to say, by an algorithmand because of this not truly random. Every new number is generated from the previousones by an algorithm. This means that the new value is fully determined by the previous ones.But, depending on the algorithm, they often have properties making them very suitable forsimulations.Quasi Random Number A good description quoted from [25], Chapter 7.7Sequences of n-tuples that fill n-space more uniformly than uncorrelated randompoints are called quasi-random sequences. That term is somewhat of a misnomer,since there is nothing random about quasi-random sequences: They arecleverly crafted to be, in fact, sub-random. The sample points in a quasi-randomsequence are, in a precise sense, maximally avoiding each other.Quasi random numbers are not designed to appear random, rather to be uniformly distributed.One aim of such numbers is to reduce and control errors in Monte Carlo simulations.A picture is always a good way to illustrate the difference between this two types. In figure2.1 1 and 2.2 2 we have plots with different numbers of pseudo- and quasi-random numbers.This is a good demonstration to show the structure of quasi-random numbers, but it is also1 This plot was generated with the Matlab 6 rand generator, a combination of a lagged Fibonacci generator,with a cache of 32 floating point numbers and a shift register random integer generator.2 This plot was generated with the sobol.m routine for Matlab from http://www.csit.fsu.edu/~burkardt/m_src/sobol/sobol.html. This web-site includes also a variety of references for Sobolsequences and some implementations in different programming languages.2