SOBOL
The Sobol Quasirandom Sequence

SOBOL is a MATLAB library which computes elements of the Sobol quasirandom sequence, by Bennett Fox.

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.

SOBOL is an adapation of the INSOBL and GOSOBL routines in ACM TOMS Algorithm 647 and ACM TOMS Algorithm 659. 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.

A remark by Joe and Kuo shows how to extend the algorithm from the original maximum spatial dimension of 40 up to a maximum spatial dimension of 1111. The FORTRAN90 and C++ versions of the code has been updated in this way, but updating the MATLAB code has not been simple, since MATLAB doesn't support 64 bit integers.

The original, true, correct versions of ACM TOMS Algorithm 647 and ACM TOMS Algorithm 659 are 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:

SOBOL is available in a C++ version and a FORTRAN90 version and a MATLAB version and a Python version

Related Data and Programs:

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

FAURE, a MATLAB library which computes Faure sequences.

GRID, a MATLAB library which computes points on a grid.

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

HALTON, a MATLAB library which computes Halton sequences.

HAMMERSLEY, a MATLAB library which computes Hammersley sequences.

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

IHS, a MATLAB library which computes improved Latin Hypercube datasets.

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

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

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

LATTICE_RULE, a MATLAB library which approximates multidimensional integrals using lattice rules.

NIEDERREITER2, a MATLAB library which computes Niederreiter sequences with base 2.

SOBOL_DATASET, a MATLAB program which computes a Sobol quasirandom sequence and writes it to a file.

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

TOMS659, a FORTRAN77 library which evaluates Sobol sequences.

UNIFORM, a MATLAB library which computes uniform random values.

VAN_DER_CORPUT, a MATLAB library which computes van der Corput sequences.

Author:

Original FORTRAN77 version by Bennett Fox; Matlab version by John Burkardt.

Reference:

1. IA Antonov, VM Saleev,
   An Economic Method of Computing LP Tau-Sequences,
   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, March 1988, pages 88-100.
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, July 1992, pages 195-213.
4. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Second Edition,
   Springer, 1987,
   ISBN: 0387964673,
   LC: QA76.9.C65.B73.
5. Bennett Fox,
   Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, December 1986, pages 362-376.
6. Stephen Joe, Frances Kuo,
   Remark on Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator,
   ACM Transactions on Mathematical Software,
   Volume 29, Number 1, March 2003, pages 49-57.
7. Harald Niederreiter,
   Random Number Generation and quasi-Monte Carlo Methods,
   SIAM, 1992,
   ISBN13: 978-0-898712-95-7,
   LC: QA298.N54.
8. William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
   Numerical Recipes in FORTRAN: The Art of Scientific Computing,
   Second Edition,
   Cambridge University Press, 1992,
   ISBN: 0-521-43064-X,
   LC: QA297.N866.
9. Ilya Sobol,
   Uniformly Distributed Sequences with an Additional Uniform Property,
   USSR Computational Mathematics and Mathematical Physics,
   Volume 16, 1977, pages 236-242.
10. Ilya Sobol, YL Levitan,
    The Production of Points Uniformly Distributed in a Multidimensional Cube (in Russian),
    Preprint IPM Akademii Nauk SSSR,
    Number 40, Moscow 1976.

Source Code:

SOBOL is the current version of the program, which is restricted to a maximum spatial dimension of 40.

• i4_bit_hi1.m, returns the position of the high 1 bit base 2 in an integer.
• i4_bit_lo0.m, returns the position of the low 0 bit base 2 in an integer.
• i4_sobol.m, generates a new quasirandom Sobol vector with each call;
• i4_sobol_generate.m, generates a dataset of Sobol vectors.
• i4_uniform.m, returns a random integer in a given range.
• prime_ge.m, finds the smallest prime greater or equal to a given integer.
• r4_uniform_01.m, returns a random real number in [0,1].
• r8mat_write.m, writes an R8MAT file.
• tau_sobol.m defines favorable starting seeds for Sobol sequences.
• timestamp.m, prints the current YMDHMS date as a time stamp;

Examples and Tests:

• sobol_test.m, runs all the tests;
• sobol_test01.m, tests bitxor;
• sobol_test02.m, tests i4_bit_hi1;
• sobol_test03.m, tests i4_bit_lo0;
• sobol_test04.m, tests i4_sobol;
• sobol_test05.m, tests i4_sobol, showing how the value of the seed can be manipulated;
• sobol_test_output.txt, the output file.

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