SOBOL
The Sobol Quasirandom Sequence

SOBOL is a C++ 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 any 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 this program have been updated in this way. In particular, the extra data in the C++ version of the program was kindly formatted and supplied by Steffan Berridge.

The routines with a prefix of I8_ use 64 bit integers, and use the long int to get this. On some systems, a long int is simply 32 bits. In that case, try using the long long int datatype instead.

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. 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:

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

Related Data and Programs:

BOX_BEHNKEN, a C++ library which computes a Box-Behnken design, that is, a set of arguments to sample the behavior of a function of multiple parameters;

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

FAURE, a C++ library which computes Faure sequences.

GRID, a C++ library which computes points on a grid.

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

HALTON, a C++ library which computes Halton sequences.

HAMMERSLEY, a C++ library which computes Hammersley sequences.

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

IHS, a C++ library which computes improved Latin Hypercube datasets.

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

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

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

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

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

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

SOBOL_OLD, a C++ library which implements the ACM TOMS algorithm, and is restricted to a maximal spatial dimension of 40.

TOMS647 a FORTRAN90 library which evaluates Faure, Halton and Sobol sequences.

TOMS659 a FORTRAN77 library which evaluates Sobol sequences.

UNIFORM, a C++ library which computes uniform random values.

VAN_DER_CORPUT, a C++ library which computes a van der Corput sequences.

Author:

Original FORTRAN77 version by Bennett Fox; C++ 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.cpp, the source code;
• sobol.hpp, the include file;
• sobol.sh, commands to compile the source code;

Examples and Tests:

• sobol_prb.cpp, the sample test code;
• sobol_prb.sh, commands to compile the test code;
• sobol_prb_output.txt, the output file.

List of Routines:

• I4_BIT_HI1 returns the position of the high 1 bit base 2 in an integer.
• I4_BIT_LO0 returns the position of the low 0 bit base 2 in an integer.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4_SOBOL generates a new quasirandom Sobol vector with each call.
• I4_SOBOL_GENERATE generates a Sobol dataset.
• I4_UNIFORM returns a scaled pseudorandom I4.
• I8_BIT_HI1 returns the position of the high 1 bit base 2 in an integer.
• I8_BIT_LO0 returns the position of the low 0 bit base 2 in an integer.
• I8_SOBOL generates a new quasirandom Sobol vector with each call.
• I8_SOBOL_GENERATE generates a Sobol dataset.
• I8_MAX returns the maximum of two I8's.
• I8_MIN returns the smaller of two I8's.
• I8_UNIFORM returns a scaled pseudorandom I8.
• R4_ABS returns the absolute value of an R4.
• R4_NINT returns the nearest integer to an R4.
• R4_UNIFORM_01 returns a unit pseudorandom R4.
• R8_ABS returns the absolute value of an R8.
• R8_NINT returns the nearest integer to an R8.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_WRITE writes an R8MAT file.
• TAU_SOBOL defines favorable starting seeds for Sobol sequences.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.