FAURE
The Faure Quasirandom Sequence

FAURE is a MATLAB implementation 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.

Related Data and Programs:

CVT is a MATLAB library of routines which computes elements of a Centroidal Voronoi Tessellation.

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

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

FAURE_DATASET is an interactive FORTRAN90 program which can create a Faure dataset.

GRID is a MATLAB library of routines which computes elements of a grid dataset.

HALTON is a MATLAB library of routines which computes elements of a Halton quasirandom sequence.

HAMMERSLEY is a MATLAB library of routines which computes elements of a Hammersley quasirandom sequence.

HEX_GRID is a MATLAB library of routines which computes elements of a hexagonal grid dataset.

HEX_GRID_ANGLE is a FORTRAN90 library of routines which computes elements of an angled hexagonal grid dataset.

IHS is a MATLAB library of routines which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER is a MATLAB library of routines which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE is a MATLAB library of routines which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM is a MATLAB library of routines which computes elements of a Latin Hypercube dataset, choosing points at random.

LCVT is a MATLAB library of routines which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2 is a MATLAB library of routines which computes elements of a Niederreiter quasirandom sequence with base 2.

SOBOL is a MATLAB library of routines which computes elements of a Sobol quasirandom sequence.

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

UNIFORM is a MATLAB library of routines which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT is a MATLAB library of routines which computes elements of a van der Corput quasirandom 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:

• binomial_table.m, returns a table of binomial coefficients mod QS.
• i4_log_i4.m, returns the logarithm of an I4 to an I4 base.
• faure.m, computes the SEED-th element of the Faure sequence with the given BASE vector.
• prime_ge.m, returns the smallest prime number greater than or equal to the input argument.

Examples and Tests:

• faure_test.m, runs all the tests.
• faure_test005.m, tests binomial_table.
• faure_test006.m, tests i4_log_i4.
• faure_test01.m, tests faure.
• faure_test02.m, demonstrates how the seed parameter is used.
• faure_test03.m, calls FAURE with some high values of the dimension parameter.
• timestamp.m, used to put a timestamp on the tests.
• faure_test.out, output from the tests.

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