FAURE
The Faure Quasirandom Sequence

FAURE is a C++ library which computes elements of the Faure 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. Faure sequences, in particular, seem to have become popular in mathematical finance simulations.

FAURE is adapted from code in ACM TOMS Algorithm 647. The original, true, correct version of ACM TOMS Algorithm 647 is 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:

FAURE is available in a C++ version and a FORTRAN90 version and a MATLAB 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 elements of a Centroidal Voronoi Tessellation.

FAURE, a dataset directory which contains files of sample Faure datasets.

FAURE_DATASET, a C++ program which can create a Faure quasirandom dataset.

GRID, a C++ library which computes elements of a grid sequence.

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

HAMMERSLEY, a C++ library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a C++ library which computes elements of a hexagonal grid dataset.

IHS, a C++ library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a C++ library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a C++ library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a C++ library which computes elements of a Latin Hypercube dataset, choosing points at random.

LCVT, a C++ library which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2, a C++ library which computes elements of a Niederreiter quasirandom sequence using base 2.

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

SOBOL, a C++ library which computes Sobol sequences.

TOMS647, a FORTRAN77 library which is a version of ACM TOMS algorithm 647, for evaluating Faure, Halton and Sobol quasirandom sequences.

UNIFORM, a C++ library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a C++ library which computes elements of a 1D van der Corput sequence.

Reference:

1. 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.
2. Henri Faure,
   Discrepance de suites associees a un systeme de numeration (en dimension s),
   Acta Arithmetica,
   Volume 41, 1982, pages 337-351.
3. Henri Faure,
   Good permutations for extreme discrepancy,
   Journal of Number Theory,
   Volume 42, 1992, pages 47-56.
4. 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.

Source Code:

• faure.cpp, the source code;
• faure.hpp, the include file;
• faure.sh, commands to compile the source code;

Examples and Tests:

• faure_prb.cpp, the sample test code;
• faure_prb.sh, commands to compile the test code;
• faure_prb_output.txt, output from a run of the test code;

List of Routines:

• BINOMIAL_TABLE computes a table of bionomial coefficients MOD QS.
• FAURE generates a new quasirandom Faure vector with each call.
• FAURE_GENERATE generates a Faure dataset.
• I4_LOG_I4 returns the logarithm of an I4 to an I4 base.
• I4_MIN returns the minimum of two I4's.
• I4_POWER returns the value of I^J.
• PRIME returns any of the first PRIME_MAX prime numbers.
• PRIME_GE returns the smallest prime greater than or equal to N.
• R8MAT_WRITE writes an R8MAT file.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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