FAURE
The Faure Quasirandom Sequence

FAURE is a MATLAB 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:

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

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

faure_test

FAURE_DATASET, a MATLAB program which creates a Faure quasirandom dataset.

GRID, a MATLAB library which computes elements of a grid dataset.

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

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

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

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

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

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

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

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

LCVT, a MATLAB library which computes a latinized Centroidal Voronoi Tessellation.

LATTICE_RULE, a MATLAB library which approximates multidimensional integrals using lattice rules.

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

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

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

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

VAN_DER_CORPUT, a MATLAB library which computes elements of a van der Corput quasirandom sequence.

Reference:

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.
Henri Faure,
Discrepance de suites associees a un systeme de numeration (en dimension s),
Acta Arithmetica,
Volume 41, 1982, pages 337-351.
Henri Faure,
Good permutations for extreme discrepancy,
Journal of Number Theory,
Volume 42, 1992, pages 47-56.
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.
faure_generate.m generates a Faure dataset.
prime_ge.m, returns the smallest prime number greater than or equal to the input argument.
r8mat_write.m, writes an R8MAT file.
timestamp.m, used to put a timestamp on the tests.