HALTON
Halton Datasets

HALTON is a dataset directory which contains points generated by the M-dimensional Halton sequence.

The NDIM-dimensional Halton sequence is derived from the 1-dimensional van der Corput sequence. Each dimension typically uses a different prime number as the base of the calculation.

The HALTON_DATASET programs select elements of a "leaped" subsequence of the Halton sequence. The subsequence elements are indexed by a quantity called STEP, which starts at 0. The STEP-th subsequence element is simply the Halton sequence element with index 

        SEED(1:NDIM) + STEP * LEAP(1:NDIM).

The arguments that the user may set include:

In some cases, it is recommended that the initial portion of the sequence be skipped over. A general suggestion is to let STEP be the first power of 2 that is equal to or greater than N, the number of points to generate.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

HALTON_DATASET, a C++ program which creates a Halton sequence and writes it to a file.

PLOT_POINTS, a FORTRAN90 program which can create Encapsulated PostScript images (EPS) of some of the two dimensional datasets.

TABLE, a file format which is used to store the data.

TABLE_TOP a FORTRAN90 program which can display pairwise coordinate plots of higher dimensional datasets.

Example dataset:

A typical (but small) Halton dataset looks like this: 

#  halton_02_00010.txt
#  created by HALTON_WRITE.F90
#
#  File generated on July 11 2004  12:58:28.788 PM
#
#  NDIM =            2
#  N =              10
#  STEP =            0
#  SEED =            0           0
#  LEAP =            1           1
#  BASE =            2           3
#  EPSILON (unit roundoff ) =   0.222045E-15
#
  0.000000  0.000000
  0.500000  0.333333
  0.250000  0.666667
  0.750000  0.111111
  0.125000  0.444444
  0.625000  0.777778
  0.375000  0.222222
  0.875000  0.555556
  0.062500  0.888889
  0.562500  0.037037

Reference:

  1. John Halton,
    On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,
    Numerische Mathematik,
    Volume 2, 1960, pages 84-90.
  2. John Halton, GB Smith,
    Algorithm 247: Radical-Inverse Quasi-Random Point Sequence,
    Communications of the ACM,
    Volume 7, 1964, pages 701-702.
  3. Ladislav Kocis, William Whiten,
    Computational Investigations of Low-Discrepancy Sequences,
    ACM Transactions on Mathematical Software,
    Volume 23, Number 2, June 1997, pages 266-294.

Datasets:

Datasets in M = 2 dimensions, with no skipping, include:

Datasets in M = 2 dimensions, with power of 2 skipping, include:

Datasets in M = 3 dimensions, with no skipping, include:

Datasets in M = 4 dimensions, with no skipping, include:

Datasets in M = 5 dimensions, with no skipping, include:

Datasets in M = 6 dimensions, with no skipping: (number of points in datasets were chosen to match the growth in size of a 6D sparse grid based on Clenshaw Curtis points)

Datasets in M = 7 dimensions, with no skipping include:

Datasets in M = 7 dimensions, with power of 2 skipping, include:

Datasets in M = 10 dimensions, with no skipping: (number of points in datasets were chosen to match the growth in size of a 10D sparse grid based on Clenshaw Curtis points)

Datasets in M = 16 dimensions, with no skipping include:

Datasets in M = 16 dimensions, with power of 2 skipping, include:

Datasets in M = 40 dimensions, with no skipping, include:

Datasets in M = 40 dimensions, a nonunit LEAP, include:

You can go up one level to the DATASETS directory.

Last revised on 01 August 2007.