SOBOL
The Sobol Quasirandom Sequence

SOBOL is a MATLAB 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.

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 version 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.

Related Data and Programs:

CVT is a MATLAB library of routines for computing points in a Centroidal Voronoi Tessellation.

FAURE is a MATLAB library of routines for computing Faure sequences.

GRID is a MATLAB library of routines for computing points on a grid.

GSL is the Gnu Scientific Library which includes routines to compute elements of the Sobol sequence.

HALTON is a MATLAB library of routines for computing Halton sequences.

HAMMERSLEY is a MATLAB library of routines for computing Hammersley sequences.

HEX_GRID is a MATLAB library of routines for computing sets of points in a 2D hexagonal grid.

IHS is a MATLAB library of routines for computing improved Latin Hypercube datasets.

LATIN_CENTER is a MATLAB library of routines for computing Latin square data choosing the center value.

LATIN_EDGE is a MATLAB library of routines for computing Latin square data choosing the edge value.

LATIN_RANDOM is a MATLAB library of routines for computing Latin square data choosing a random value in the square.

NIEDERREITER2 is a MATLAB library of routines for computing Niederreiter sequences with base 2.

SOBOL is also available in a C++ version and a FORTRAN90 version.

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

TOMS659 is the FORTRAN77 source of ACM TOMS algorithm 659 for evaluating Sobol sequences.

UNIFORM is a MATLAB library of routines for computing uniform random values.

VAN_DER_CORPUT is a MATLAB library of routines for computing van der Corput sequences.

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_uniform.m, returns a random integer in a given range.
• i4_xor.m, computes the bitwise exclusive OR of two integers.
• 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].
• 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_test07.m, tests i4_xor;
• sobol_test_output.txt, the output file.

Junk:

SOBOL_NOT_WORKING is a version of the program which is designed to work up to dimension 1111. However, this code is not working, partly because MATLAB integers are not really 64 bit but 53 bit. I expect that fixing this code will be a major headache. At the moment, MATLAB does not support some necessary operations on 64 bit integers.

• i4_sobol_not_working.m, generates a new quasirandom Sobol vector with each call; in case you weren't paying attention, this code is NOT WORKING.
• sobol_test04_not_working.m, tests i4_sobol;
• sobol_test05_not_working.m, tests i4_sobol, showing how the value of the seed can be manipulated;
• sobol_test06_not_working.m, tests i4_sobol, computing a few values in high dimensional spaces;
• sobol_test_not_working.out, output from a run of the tests;

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