Resources - Gallery
Available Versions
Quick Order - Free Demo

Short Product Description

The construction and application of low-discrepancy sequences recieved a lot of attention over the last decades in various fields of numerical computation, as well as in theoretical mathematics. QR Streams, a joint development of CSK (Switzerland) and MathConsult, provides some of the most important types of low-discrepancy sequences.
QR Streams is a set of optimized Mathematica packages to produce high quality low-discrepancy sequences.  It is based on an object-oriented design and provides a stream interface as a unified framework for the various sequences. Moreover, all the available statistical distributions available in Mathematica can be used in conjunction with QR Streams.
A C++ template implementation, included into Mathematica via MathLink, increases performance for demanding applications and runs on Windows NT/9x and various Unix platforms (Solaris, Irix, AIX, Linux, others on request).
All the calculation can  be done at full precision and in case of the Niederreiter sequence even formal calculation in any finite field is possible. QR Streams is packaged with extensive documentation on low-discrepancy sequences and Quasi-Monte Carlo methods.  This is of great additional value for educational purpose. Moreover, various applications, in particular in the area of finance, are shown.

Low-discrepancy sequences

The explicit construction of point sets which fill out the -dimensional unit cube as uniformly as possible is an important task for many numerical applications. Pseudo-random numbers are often used for this purpose. However, it is well-known that pseudo-random numbers are only a subsitute for true random numbers and tend to show clustering effects. This led to the search for more evenly spread sequences. The general study of uniformly distributed sequences was initiated by Herman Weyl in 1916 with a famous paper "Über die Gleichverteilung von Zahlen mod. Eins". He defined the notion of discrepancy to quantify the quality of uniformity of a finite point set. Now, a sequence is called a low-discrepancy sequence or quasi-random sequence if the discrepancy of the first points decays asymptotically as . There are various explicit constructions, based on combinatiorial methods, algebraic number theory and ergodic theory.

Monte Carlo and Quasi-Monte Carlo Methods

Monte Carlo techniques are methods to calculate a numerical quantity by casting it into a stochastic framework and applying repeated simulation using pseudorandom numbers. The expectation E[X] of a random variable X is approximated by a sample mean. In practice, the uniform distribution is sampled by pseudo-random sequences which are then transformed to the law of .
Quasi-Monte Carlo techniques replace pseudo-random sequences with low-discrepancy sequences which have a more uniform behaviour. In a certain sense, Quasi-Monte Carlo methods combine the advantages of Monte Carlo and uniform lattice methods. In particular, fewer quasi-random samples are needed to achieve a similar level of accuracy as obtained by using pseudo-random sequeces.

Quick Ordering

Free Demonstration Package

To dowload the free demonstration package, register (if you do not have your username yet), then go to your personal home page (available after registration) and select the QR Streams Demo package in our store. You'll go through order processing in the same way you would with any of our other software, but the price of the package will be 0 USD. After your order has been processed you will be taken to the download area.

© 1998 MathConsult Dr. R. Mäder
http://www.mathdirect.com/products/qrn/index.html
Last update: 22.01.2000