niederreiter

niederreiter, a C++ code which implements the Niederreiter quasirandom sequence, using an "arbitrary" base; more correctly, the code is not restricted to using a base of 2, but can instead use a base that is a prime or a power of a prime.

A quasirandom or low discrepancy sequence, such as the Faure, Halton, Hammersley, Niederreiter or Sobol sequences, is "less random" than a pseudorandom number sequence, but more useful for such tasks as approximation of integrals in higher dimensions, and in global optimization. This is because low discrepancy sequences tend to sample space "more uniformly" than random numbers. Algorithms that use such sequences may have superior convergence.

NIEDERREITER is an adaptation of the INLO and GOLO routines in ACM TOMS Algorithm 738. The original code can only compute the "next" element of the sequence. The revised code allows the user to specify the index of the desired element.

The original, true, correct version of ACM TOMS Algorithm 738 is available in the TOMS subdirectory of the NETLIB web site. The version displayed here has been converted to FORTRAN90, and other internal changes have been made to suit me.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

niederreiter is available in a C++ version and a FORTRAN90 version.

Related Data and Programs:

CVT, a C++ code which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a C++ code which computes elements of a Faure quasirandom sequence.

HALTON, a C++ code which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a C++ code which computes elements of a Hammersley quasirandom sequence.

IHS, a C++ code which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a C++ code which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a C++ code which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a C++ code which computes elements of a Latin Hypercube dataset, choosing points at random.

LCVT, a C++ code which computes a latinized Centroidal Voronoi Tessellation.

niederreiter_test

NIEDERREITER2, a C++ code which computes a Niederreiter sequence for a base of 2.

NORMAL, a C++ code which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a C++ code which computes elements of a Sobol quasirandom sequence.

UNIFORM, a C++ code which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a C++ code which computes elements of a van der Corput pseudorandom sequence.

Reference:

1. Paul Bratley, Bennett Fox,
   Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator,
   ACM Transactions on Mathematical Software,
   Volume 14, Number 1, 1988, pages 88-100.
2. Paul Bratley, Bennett Fox, Harald Niederreiter,
   Algorithm 738: Programs to Generate Niederreiter's Low-Discrepancy Sequences,
   ACM Transactions on Mathematical Software,
   Volume 20, Number 4, 1994, pages 494-495.
3. Paul Bratley, Bennett Fox, Harald Niederreiter,
   Implementation and Tests of Low Discrepancy Sequences,
   ACM Transactions on Modeling and Computer Simulation,
   Volume 2, Number 3, 1992, pages 195-213.
4. Bennett Fox,
   Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, 1986, pages 362-376.
5. Rudolf Lidl, Harald Niederreiter,
   Finite Fields,
   Second Edition,
   Cambridge University Press, 1997,
   ISBN: 0521392314,
   LC: QA247.3.L53
6. Harald Niederreiter,
   Low-discrepancy and low-dispersion sequences,
   Journal of Number Theory,
   Volume 30, 1988, pages 51-70.
7. Harald Niederreiter,
   Random Number Generation and quasi-Monte Carlo Methods,
   SIAM, 1992,
   ISBN13: 978-0-898712-95-7.

Source Code:

GFARIT must be run first, to set up a tables of addition and multiplication.

• gfarit.cpp, the source code;
• gfarit.sh, compiles the source code;

GFPLYS must be run second, to set up a table of irreducible polynomials.

• gfplys.cpp, the source code;
• gfplys.sh, compiles the source code;

Once GFARIT and GFPLYS have been run to set up the tables, the NIEDERREITER routines can be used.

• niederreiter.cpp, the source code;
• niederreiter.sh, compiles the source code;
• niederreiter.hpp, the include file;