Quasirandom numbers

These are sequences of numbers to be used in Monte Carlo calculations ( Random Numbers), optimized not to appear highly random, but rather to give the fastest convergence in the computation. They are applicable mainly to multidimensional integration, where the theory is based on that of uniformity of distribution (Kuipers74).

Because the way of generating and using them is quite different, one must distinguish between finite and infinite quasirandom sequences:

• A finite quasirandom sequence is optimized for a particular number of points in a particular dimensionality of space. However, the complexity of this optimization is so horrendous that exact solutions are known only for very small point sets (Kuipers74, Zaremba72) The most widely used sequences in practice are the Korobov sequences.
• An infinite quasirandom sequence is an algorithm which allows the generation of sequences of an arbitrary number of vectors of arbitrary length (p-dimensional points). The properties of these sequences are generally known only asymptotically, where they perform considerably better than truly random or pseudorandom sequences, since they give 1/N convergence for Monte Carlo integration instead of 1/ File:Hepa img879.gif . The short-term distribution may, however, be rather poor, and generators should be examined carefully before being used in sensitive calculations. Major improvements are possible by shuffling, or changing the order in which the numbers are used. An effective shuffling technique is given in Braaten79.