NIEDERREITER2
The Niederreiter Quasirandom Sequence [Base 2]

NIEDERREITER2 is a C++ library 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.

NIEDERREITER2 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.

Licensing:

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

Languages:

NIEDERREITER2 is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BOX_BEHNKEN, a C++ library which computes a Box-Behnken design, that is, a set of arguments to sample the behavior of a function of multiple parameters;

CVT, a C++ library which computes points in a Centroidal Voronoi Tessellation.

FAURE, a C++ library which computes Faure sequences.

GRID, a C++ library which computes points on a grid.

HALTON, a C++ library which computes Halton sequences.

HAMMERSLEY, a C++ library which computes Hammersley sequences.

HEX_GRID, a C++ library which computes sets of points in a 2D hexagonal grid.

IHS, a C++ library which computes improved Latin Hypercube datasets.

LATIN_CENTER, a C++ library which computes Latin square data choosing the center value.

LATIN_EDGE, a C++ library which computes Latin square data choosing the edge value.

LATIN_RANDOM, a C++ library which computes Latin square data choosing a random value in the square.

NIEDERREITER2_DATASET, a C++ program which creates a Niederreiter quasirandom dataset with base 2;

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

SOBOL, a C++ library which computes Sobol sequences.

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

UNIFORM, a C++ library which computes uniform random values.

VAN_DER_CORPUT, a C++ library which computes van der Corput sequences.

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. R Lidl, Harald Niederreiter,
   Finite Fields,
   Cambridge University Press, 1984, page 553.
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.

Source Code:

• niederreiter2.cpp, the source code;
• niederreiter2.hpp, the include file;
• niederreiter2.sh, commands to compile the source code;

Examples and Tests:

• niederreiter2_prb.cpp, the sample test code;
• niederreiter2_prb.sh, commands to compile the test code;
• niederreiter2_prb_output.txt, the output file;

List of Routines:

• CALCC2 computes values of the constants C(I,J,R).
• CALCV2 calculates the value of the constants V(J,R).
• NIEDERREITER2 returns an element of the Niederreiter sequence base 2.
• NIEDERREITER2_GENERATE generates a set of Niederreiter values.
• PLYMUL2 multiplies two polynomials in the field of order 2
• R8MAT_WRITE writes an R8MAT file.
• SETFLD2 sets up arithmetic tables for the finite field of order 2.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.