niederreiter2

niederreiter2, a FORTRAN90 code which implements the Niederreiter quasirandom sequence, using a base of 2.

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.

The code is an adapation of the INLO2 and GOLO2 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:

niederreiter2 is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CVT, a FORTRAN90 code which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a FORTRAN90 code which computes elements of a Faure quasirandom sequence.

HALTON, a FORTRAN90 code which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HAMMERSLEY, a FORTRAN90 code which computes elements of a Hammersley Quasi Monte Carlo (QMC) sequence, using a simple interface.

IEEE_UNIFORM_SAMPLE, a FORTRAN90 code which tries to uniformly sample the discrete set of values that represent the legal IEEE real numbers;

LATTICE_RULE, a FORTRAN90 code which approximates multidimensional integrals using lattice rules.

LCVT, a FORTRAN90 code which computes a latinized Centroidal Voronoi Tessellation.

niederreiter2_test

NORMAL, a FORTRAN90 code which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a FORTRAN90 code which computes elements of a Sobol quasirandom sequence.

TOMS738, a FORTRAN90 code which is a version of ACM TOMS algorithm 738, for evaluating Niederreiter sequences.

UNIFORM, a FORTRAN90 code which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a FORTRAN90 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, pages 88-100, 1988.
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, pages 494-495, 1994.
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, pages 195-213, 1992.
4. Bennett Fox,
   Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, pages 362-376, 1986.
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:

• niederreiter2.f90, the source code;
• niederreiter2.sh, compiles the source code;