TOMS659
The Sobol Quasirandom Sequence

TOMS659 is a FORTRAN90 library which computes elements of the Sobol quasirandom sequence.

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 original, true, correct version of ACM TOMS Algorithm 659 is available through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html.

Languages:

TOMS659 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.

GSL, a C++ library which includes routines to compute elements of the Sobol sequence.

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 elements of a Niederreiter quasirandom sequence with base 2.

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

SOBOL_DATASET, a FORTRAN90 program which specifies a Sobol dataset and writing it to a file.

TOMS647, a FORTRAN90 library which is a version of ACM TOMS algorithm 647, for evaluating Faure, Halton and Sobol 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 quasirandom sequence.

Reference:

1. IA Antonov, VM Saleev,
   USSR Computational Mathematics and Mathematical Physics,
   Volume 19, 1980, pages 252-256.
2. 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.
3. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Springer Verlag, pages 201-202, 1983.
4. 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.
5. 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.
6. Stephen Joe, Frances Kuo
   Remark on Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator,
   ACM Transactions on Mathematical Software,
   Volume 29, Number 1, pages 49-57, March 2003.
7. Harald Niederreiter,
   Random Number Generation and quasi-Monte Carlo Methods,
   SIAM, 1992,
   ISBN13: 978-0-898712-95-7.
8. William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
   Numerical Recipes in FORTRAN: The Art of Scientific Computing,
   Second Edition,
   Cambridge University Press, 1992,
   LC: QA297.N866,
   ISBN: 0-521-43064-X.
9. Ilya Sobol,
   USSR Computational Mathematics and Mathematical Physics,
   Volume 16, pages 236-242, 1977.
10. Ilya Sobol, YL Levitan,
    The Production of Points Uniformly Distributed in a Multidimensional Cube (in Russian),
    Preprint IPM Akad. Nauk SSSR,
    Number 40, Moscow 1976.

Source Code:

• toms659.f90, the source code;

Examples and Tests:

• toms659_test.f90, a sample calling program;
• toms659_test_input.txt, an input file;
• toms659_test.txt, output from a run of the test program;
• toms659_test_expected.txt, the expected output from the run;

List of Routines:

• BDSOBL initializes labelled common block /SOBDAT/.
• INSOBL initializes the Sobol sequence computation.
• GOSOBL generates a new quasirandom vector with each call.
• EXOR calculates the exclusive OR of its two input parameters.

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