HALTON_DATASET
Generate Halton Datasets

HALTON_DATASET is a MATLAB program which creates a Halton sequence dataset.

The program is interactive, and allows the user to choose the parameters that define the sequence.

The NDIM-dimensional Halton sequence is really NDIM separate sequences, each generated by a particular base.

The program applies the user's data to 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 data that the user may set include:

• NDIM, the spatial dimension,
  default: NDIM = 1,
  required: 1 <= NDIM;
• N, the number of entries of the subsequence to compute,
  required: 1 <= N;
• STEP, the subsequence index of the first entry to compute,
  default: STEP = 0,
  required: 0 <= STEP.
• SEED(1:NDIM), the Halton sequence index corresponding to STEP = 0.
  default: SEED(1:NDIM) = (0, 0, ... 0),
  required: 0 <= SEED(1:NDIM);
• LEAP(1:NDIM), the succesive jumps in the Halton sequence.
  default: LEAP(1:NDIM) = (1, 1, ..., 1).
  required: 1 <= LEAP(1:NDIM).
• BASE(1:NDIM), the Halton bases.
  default: BASE(1:NDIM) = (2, 3, 5, 7, 11... ),
  required: 1 < BASE(1:NDIM).

Once these parameters are set, the program generates the data, and writes it to a file. The user may then specify another set of data, or terminate the program.

Related Data and Programs:

HALTON is a MATLAB library which contains many routines needed in order to run HALTON_DATASET.

HALTON is a dataset directory which contains sample datasets created by HALTON_DATASET.

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

TABLE is the format used for the output files.

TABLE_DISCREPANCY is an executable C++ program which can read a TABLE file of points (presumed to lie in the unit hypercube) and compute bounds on the star discrepancy, a measure of dispersion.

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, 1997, pages 266-294.

Source Code:

• halton_dataset.m, the main program.

Examples and Tests:

• halton_dataset.inp, is a simple input file for creation of sequences of dimension 2, 3 and 7.
• halton_dataset.out, is the printed output from a run of the program with the input file.
• halton_2_10.txt, a leaped Halton subsequence dataset with NDIM = 2 and N = 10.
• halton_02_00010.png, a PNG image of the dataset.
• halton_3_100.txt, a leaped Halton subsequence dataset with NDIM = 3 and N = 100.
• halton_7_1000.txt, a leaped Halton subsequence dataset with NDIM = 7 and N = 1000.

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