HALTON_ADVANCED
The Halton Quasi Monte Carlo (QMC) Sequence

HALTON_ADVANCED is a FORTRAN90 library which computes elements of a Halton Quasi Monte Carlo (QMC) sequence.

HALTON_ADVANCED 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 NDIM-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 NDIM-dimensional Halton sequence is really NDIM 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:NDIM) + STEP * LEAP(1:NDIM).
      

The arguments that the user may set include:

• NDIM, the spatial dimension,
  default: NDIM = 1,
  required: 1 <= NDIM;
• STEP, the subsequence index.
  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).

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 NDIM-dimensional Halton sequence is related to the NDIM+1 dimensional Hammersley sequence of length NMAX. An NDIM+1 dimensional Hammersley sequence of length NMAX becomes an NDIM-dimensional Halton sequence by deleting the first dimension. An NDIM dimensional Halton sequence of NMAX points becomes an NDIM+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 NDIM = 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 unit circle in 2D, and 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_ADVANCED is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CVT, a FORTRAN90 library which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a FORTRAN90 library which computes elements of a Faure quasirandom sequence.

HALTON, a FORTRAN90 library which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HALTON_DATASET, a FORTRAN90 program which creates a Halton sequence and writes it to a file.

HAMMERSLEY, a FORTRAN90 library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a FORTRAN90 library which computes elements of a hexagonal grid dataset.

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

IEEE_UNIFORM_SAMPLE, a FORTRAN90 library which tries to uniformly sample the discrete set of values that represent the legal IEEE real numbers;

IHS, a FORTRAN90 library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing points at random.

LATTICE_RULE, a FORTRAN90 library which approximates multidimensional integrals using lattice rules.

LCVT, a FORTRAN90 library which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2, a FORTRAN90 library which computes elements of a Niederreiter sequence using base 2.

NORMAL, a FORTRAN90 library which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a FORTRAN90 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 quasirandom sequences.

UNIFORM, a FORTRAN90 library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a FORTRAN90 library which computes elements of a 1D 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:

• halton_advanced.f90, the source code.

Examples and Tests:

• halton_advanced_test.f90, a sample problem.
• halton_advanced_test.txt, the output file.
• halton_02_00010.txt, a Halton datafile created by the sample problem.
• halton_02_00010.png, a PNG image of the dataset.
• halton_02_00100.txt, a Halton datafile created by the sample problem.
• halton_02_00100.png, a PNG image of the dataset.

List of Routines:

• ARC_COSINE computes the arc cosine function, with argument truncation.
• ATAN4 computes the inverse tangent of the ratio Y / X.
• GET_SEED returns a seed for the random number generator.
• GET_UNIT returns a free FORTRAN unit number.
• HALHAM_LEAP_CHECK checks LEAP for a Halton or Hammersley sequence.
• HALHAM_N_CHECK checks N for a Halton or Hammersley sequence.
• HALHAM_DIM_NUM_CHECK checks DIM_NUM for a Halton or Hammersley sequence.
• HALHAM_SEED_CHECK checks SEED for a Halton or Hammersley sequence.
• HALHAM_STEP_CHECK checks STEP for a Halton or Hammersley sequence.
• HALHAM_WRITE writes a Halton or Hammersley subsequence to a file.
• HALTON computes the next element in a leaped Halton subsequence.
• HALTON_BASE_GET gets the base vector for a leaped Halton subsequence.
• HALTON_BASE_CHECK checks BASE for a Halton sequence.
• HALTON_BASE_SET sets the base vector for a leaped Halton subsequence.
• HALTON_LEAP_GET gets the leap vector for a leaped Halton subsequence.
• HALTON_LEAP_SET sets the leap vector for a leaped Halton subsequence.
• HALTON_MEMORY holds data associated with a leaped Halton subsequence.
• HALTON_DIM_NUM_GET gets the spatial dimension for a leaped Halton subsequence.
• HALTON_DIM_NUM_SET sets the spatial dimension for a leaped Halton subsequence.
• HALTON_SEED_GET gets the seed vector for a leaped Halton subsequence.
• HALTON_SEED_SET sets the seed vector for a leaped Halton subsequence.
• HALTON_SEQUENCE computes N elements of a leaped Halton subsequence.
• HALTON_STEP_GET gets the "step" for a leaped Halton subsequence.
• HALTON_STEP_SET sets the "step" for a leaped Halton subsequence.
• I4_TO_HALTON computes one element of a leaped Halton subsequence.
• I4_TO_HALTON_SEQUENCE computes N elements of a leaped Halton subsequence.
• I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed".
• PRIME returns any of the first PRIME_MAX prime numbers.
• U1_TO_SPHERE_UNIT_2D maps a point in the unit interval to the unit circle.
• U2_TO_BALL_UNIT_2D maps points from the unit box to the unit ball in 2D.
• U2_TO_SPHERE_UNIT_3D maps a point in the unit box onto the unit sphere in 3D.
• U3_TO_BALL_UNIT_3D maps points from the unit box to the unit ball in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.