TOMS738
Niederreiter's Low Discrepancy Sequence

TOMS738 is a FORTRAN90 library which implements ACM TOMS algorithm 738, to compute Niederreiter's low discrepancy sequence.

A low discrepancy or quasirandom sequence, such as the Faure, Halton, Hammersley, Niederreiter, or Sobol sequence, 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 original, true, correct version of ACM TOMS Algorithm 738 is available through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html

The version displayed here has been converted to FORTRAN90, and other internal changes have been made to suit me.

Languages:

TOMS738 is available in 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.

NIEDERREITER, a FORTRAN90 library which is a modification of ACM TOMS algorithm 738, which, among other things, allows the user to start the sequence at any point. This is for the arbitrary base calculations.

NIEDERREITER2, a FORTRAN90 library which is a modification of ACM TOMS algorithm 738, which, among other things, allows the user to start the sequence at any point. This is for the base 2 calculations.

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

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, 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,
   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:

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

• gfarit.f90, 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.f90, the source code;
• gfplys_test.txt, output from a run of GFPLYS;
• gfplys.txt, the data file created by the run;

GENIN can be run to generate a particular Niederreiter sequence. The program is interactive, and requires the user to specify a test integral, the spatial dimension, the base, and the number of "warm up" values to skip.

• genin.f90, the source code;
• genin_test.txt, output from a run of the program on all four test integrals;

GENIN_TWO is a special, very efficient version of GENIN for the special case where the base is 2:

• genin_two.f90, the source code;
• genin_two_test.txt, output from a run of the program on all four test integrals;

SHOW_TWO is a variation of GENIN_TWO which simply writes the quasi-random numbers to a file.

• show_two.f90, the source code;
• show_two_test.txt, output from a run of the program to compute the first 100 values;

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