NIEDERREITER
The Niederreiter Quasirandom Sequence [Arbitrary base]

NIEDERREITER is a C++ library 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 GNU LGPL license.

Languages:

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

Related Data and Programs:

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

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

GRID, a C++ library which computes elements of a grid dataset.

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

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

HEX_GRID, a C++ library which computes elements of a hexagonal grid dataset.

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

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

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

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

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

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

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

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

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

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

VAN_DER_CORPUT, a C++ library 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_output.txt, output from a run of GFARIT;
• gfarit.txt, the data file created by the run;

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

• gfplys.cpp, the source code;
• gfplys_output.txt, output from a run of GFPLYS;
• gfplys.txt, the data file created by the run;

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

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

Examples and Tests:

• niederreiter_test.cpp, the sample test code;
• niederreiter_test.txt, output from a run of the test code;

List of Routines:

• CALCC calculates the value of the constants C(I,J,R).
• CALCV calculates the constants V(J,R).
• GOLO generates a new quasi-random vector on each call.
• I4_CHARACTERISTIC gives the characteristic for an integer.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_POWER returns the value of I^J.
• INLO calculates the values of C(I,J,R).
• NIEDERREITER returns an element of a Niederreiter sequence for base BASE.
• NIEDERREITER_GENERATE generates a set of Niederreiter values.
• NIEDERREITER_WRITE writes a set of Niederreiter values to a file.
• PLYMUL multiplies one polynomial by another.
• R8_EPSILON returns the R8 roundoff unit.
• SETFLD sets up the arithmetic tables for a finite field.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.

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