# SOBOL The Sobol Quasirandom Sequence

SOBOL, a Python 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 adaptation 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. These changes have been implemented in the FORTRAN90 and C++ versions of the program.

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 Python, and other internal changes have been made.

### Languages:

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

### Related Data and Programs:

HALTON, a Python library which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HAMMERSLEY, a Python library which computes elements of a Hammersley Quasi Monte Carlo (QMC) sequence, using a simple interface.

NORMAL, a Python library which contains random number generators (RNG's) for normally distributed values.

VAN_DER_CORPUT, a Python library which computes elements of a 1D van der Corput Quasi Monte Carlo (QMC) sequence using a simple interface.

### Author:

This Python implementation was written by Corrado Chisari.

### 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),