HALTON
The Halton Quasirandom Sequence

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

HALTON includes routines to make it easy to manipulate this computation, to compute the next N entries, to compute a particular entry, to restart the sequence at a particular point, or to compute DIM_NUM-dimensional versions of the sequence.

For the most straightforward use, try either

• I4_TO_HALTON, for one element of a sequence;
• I4_TO_HALTON_SEQUENCE, for N elements of a sequence;

For more convenience, there are two related routines with almost no input arguments:

• HALTON, for one element of a sequence;
• HALTON_SEQUENCE, for N elements of a sequence;

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

Routines in this library 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:DIM_NUM) + STEP * LEAP(1:DIM_NUM).
      

The arguments that the user may set include:

• DIM_NUM, the spatial dimension,
  default: DIM_NUM = 1,
  required: 1 <= DIM_NUM;
• STEP, the subsequence index.
  default: STEP = 0,
  required: 0 <= STEP.
• SEED(1:DIM_NUM), the Halton sequence index corresponding to STEP = 0.
  default: SEED(1:DIM_NUM) = (0, 0, ... 0),
  required: 0 <= SEED(1:DIM_NUM);
• LEAP(1:DIM_NUM), the succesive jumps in the Halton sequence.
  default: LEAP(1:DIM_NUM) = (1, 1, ..., 1).
  required: 1 <= LEAP(1:DIM_NUM).
• BASE(1:DIM_NUM), the Halton bases.
  default: BASE(1:DIM_NUM) = (2, 3, 5, 7, 11... ),
  required: 1 < BASE(1:DIM_NUM).

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

The DIM_NUM-dimensional Halton sequence is related to the DIM_NUM+1 dimensional Hammersley sequence of length NMAX. An DIM_NUM+1 dimensional Hammersley sequence of length NMAX becomes an DIM_NUM-dimensional Halton sequence by deleting the first dimension. An DIM_NUM dimensional Halton sequence of NMAX points becomes an DIM_NUM+1 dimensional Hammersley sequence of length NMAX by prefixing a first coordinate, and setting the value of this first coordinate to I/NMAX for the I-th entry of the sequence.

While the Hammersley sequence has better dispersion properties in technical measures such as the discrepancy, it suffers from the problem that you must know, beforehand, the number of points you are going to generate. Thus, if you have computed a Hammersley sequence of length 100, and you want to compute a Hammersley sequence of length 200, you must discard your current values and start over. By contrast, you can compute 100 points of a Halton sequence, and then 100 more, and this will be the same as computing the first 200 points of the Halton sequence in one calculation.

In low dimensions, the multidimensional Halton sequence quickly "fills up" the space in a well-distributed pattern. However, for higher dimensions (such as DIM_NUM = 40) for instance, the initial elements of the Halton sequence can be very poorly distributed; it is only when N, the number of sequence elements, is large enough relative to the spatial dimension, that the sequence is properly behaved. Remedies for this problem were suggested by Kocis and Whiten.

As an example of the use of Halton sequences, we also use them to compute "random" points on or in the the unit circle in 2D, and on the unit sphere in 3D.

Licensing:

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

Languages:

HALTON is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

FAURE, a MATLAB program which computes elements of a Faure sequence.

GRID, a MATLAB program which computes elements of a grid sequence.

HALTON_DATASET, a MATLAB program which creates a Halton sequence and write it to a file.

HAMMERSLEY, a MATLAB program which computes elements of a Hammersley sequence.

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

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

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

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

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

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

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

NIEDERREITER2, a MATLAB program which computes elements of a Niederreiter sequence using base 2.

SOBOL, a MATLAB program which computes elements of a Sobol sequence.

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

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

VAN_DER_CORPUT, a MATLAB program which computes elements of a (1 dimensional) van der Corput sequence.

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:

• arc_cosine.m, computes the arc cosine function, with argument truncation.
• atan4.m, computes the inverse tangent of the ratio Y / X.
• get_seed.m, returns a random seed for the random number generator.
• halham_dim_num_check.m, checks DIM_NUM for a Halton or Hammersley sequence.
• halham_leap_check.m, checks LEAP for a Halton or Hammersley sequence.
• halham_n_check.m, checks N for a Halton or Hammersley sequence.
• halham_seed_check.m, checks SEED for a Halton or Hammersley sequence.
• halham_step_check.m, checks STEP for a Halton or Hammersley sequence.
• halham_write.m, writes a Halton or Hammersley sequence to a file.
• halton.m, computes the next element of the Halton sequence in the current base.
• halton_base_check.m, checks BASE for a Halton sequence.
• halton_base_get.m, returns the current base.
• halton_base_set.m, sets the current base.
• halton_dim_num_get.m, returns the current dimension.
• halton_dim_num_set.m, sets the current dimension.
• halton_leap_get.m, returns the leap vector.
• halton_leap_set.m, sets the leap vector.
• halton_seed_get.m, returns the current seed.
• halton_seed_set.m, sets the current seed.
• halton_sequence.m, computes the next N elements of the Halton sequence, starting at the current seed, in the current base.
• halton_step_get.m, returns the step of the leaped Halton subsequence.
• halton_step_set.m, sets the step of the leaped Halton subsequence.
• i4_to_halton.m, computes the SEED-th element of the Halton sequence with the given BASE vector.
• i4_to_halton_sequence.m, computes the next N elements, starting at SEED, of the Halton sequence with the given BASE vector.
• i4vec_transpose_print.m, prints an integer vector "transposed".
• prime.m, returns any of the first PRIME_MAX prime numbers.
• r8_epsilon.m, returns the R8 roundoff unit.
• s_len_trim.m, returns the length of a string to the last nonblank.
• timestamp.m, prints the current YMDHMS date as a timestamp.
• u1_to_sphere_unit_2d.m, maps a point in the unit interval onto the unit sphere in 2D.
• u2_to_ball_unit_2d.m, maps a point in the unit box to the unit ball in 2D.
• u2_to_sphere_unit_3d.m, maps a point in the unit box onto the unit sphere in 3D.
• u3_to_ball_unit_3d.m, maps a point in the unit box to the unit ball in 3D.

Examples and Tests:

• halton_test.m, runs all the tests.
• halton_test_output.txt, output from the tests.
• halton_test01.m, tests HALTON and HALTON_SEED_SET.
• halton_test0125.m, tests I4_TO_HALTON.
• halton_test0126.m, tests I4_TO_HALTON.
• halton_test02.m, tests HALTON_SEQUENCE.
• halton_test025.m, tests I4_TO_HALTON_SEQUENCE.
• halton_test03.m, tests HALTON_SEQUENCE, HALTON_SEED_GET and HALTON_SEED_GET.
• halton_test04.m, tests HALTON, HALTON_BASE_GET, and HALTON_BASE_SET.
• halton_test045.m, tests HALTON_SEQUENCE.
• halton_test05.m, tests HALTON.
• halton_test06.m, tests HALTON_SEQUENCE.
• halton_test07.m, tests HALTON_SEQUENCE, HALTON_SEED_GET, HALTON_SEED_SET.
• halton_test08.m, tests HALTON_SEQUENCE, HALTON_BASE_GET, HALTON_BASE_SET.
• halton_test09.m, tests U1_TO_SPHERE_UNIT_2D.
• halton_test10.m, tests U2_TO_BALL_UNIT_2D.
• halton_test11.m, tests U2_TO_SPHERE_UNIT_3D.
• halton_test12.m, tests U3_TO_BALL_UNIT_3D.
• halton_test13.m, tests HALTON_WRITE.
• halton_02_00010.txt, a sample Halton dataset created by HALHAM_WRITE.
• halton_02_00010.png, a PNG image of the dataset.
• halton_02_00100.txt, a sample Halton dataset created by HALHAM_WRITE.
• halton_02_00100.png, a PNG image of the dataset.

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