NIEDERREITER
The Niederreiter Quasirandom Sequence [Arbitrary base]

NIEDERREITER is a FORTRAN90 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 FORTRAN90 library which computes elements of a Centroidal Voronoi Tessellation.

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

GRID, a FORTRAN90 library which computes elements of a grid dataset.

HALTON, a FORTRAN90 library which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a FORTRAN90 library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a FORTRAN90 library which computes elements of a hexagonal grid dataset.

HEX_GRID_ANGLE, a FORTRAN90 library which computes elements of an angled hexagonal grid dataset.

IHS, a FORTRAN90 library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing points at random.

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

NIEDERREITER2, a FORTRAN90 library which computes a Niederreiter sequence for a base of 2.

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

SOBOL, a FORTRAN90 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 FORTRAN90 library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a FORTRAN90 library which computes elements of a van der Corput pseudorandom sequence.

Reference:

1. Paul Bratley and 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:

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

• gfarit.f90, the source code;
• gfarit.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.f90, the source code;
• gfplys.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.f90, the source code;

Examples and Tests:

• niederreiter_test.f90, the sample test code;
• niederreiter_test.txt, the output file;

List of Routines:

• CALCC computes the constants C(I,J,R).
• CALCV calculates the constants V(J,R).
• GET_UNIT returns a free FORTRAN unit number.
• NIEDERREITER returns an element of the Niederreiter sequence.
• NIEDERREITER_GENERATE generates a set of Niederreiter values.
• NIEDERREITER_WRITE writes a set of Niederreiter values to a file.
• PLYMUL multiplies two polynomials in the field.
• SETFLD sets up arithmetic tables for the finite field.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING writes the current YMDHMS date into a string.

You can go up one level to the FORTRAN90 source codes.