HAMMERSLEY
The Hammersley Quasirandom Sequence

HAMMERSLEY is a C++ library which computes the Hammersley quasirandom sequence.

HAMMERSLEY includes several subroutines 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_HAMMERSLEY, for one element of a sequence;
• I4_TO_HAMMERSLEY_SEQUENCE, for N elements of a sequence;

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

• HAMMERSLEY, for one element of a sequence;
• HAMMERSLEY_SEQUENCE, for N elements of a sequence;

Routines in this library select elements of a "leaped" subsequence of the Hammersley sequence. The subsequence elements are indexed by a quantity called STEP, which starts at 0. The STEP-th subsequence element is simply the Hammersley 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 Hammersley 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 Hammersley sequence.
  default: LEAP(1:NDIM) = (1, 1, ..., 1).
  required: 1 <= LEAP(1:NDIM).
• BASE(1:NDIM), the Hammersley bases.
  default: BASE(1:NDIM) = (2, 3, 5, 7, 11... ), or
  (-N, 2, 3, 5, 7, 11,...) if N is known,
  required: 1 < BASE(I) for any van der Corput dimension I, or BASE(I) < 0 to generate the fractional sequence J/|BASE(I)|.

In the standard NDIM-dimensional Hammersley sequence, it is assumed that N, the number of values to be generated, is known beforehand. The first dimension of entries in the sequence will have the form J/N for J from 1 to N. The remaining dimensions are computed using the 1-dimensional van der Corput sequence, using successive primes as bases.

In a generalized Hammersley sequence, each coordinate is allowed to be a fractional or van der Corput sequence. For any fractional sequence, the denominator is arbitrary. However, it is extremely desirable that the values in all coordinates stay between 0 and 1. This happens automatically for any van der Corput sequence, but for fractional sequences, this criterion is enforced using an appropriate modulus function. The consequence is that if you specify a small "base" for a fractional sequence, your sequence will soon wrap around and you will get repeated values.

If you drop the first dimension of the standard NDIM-dimensional Hammersley sequence, you get the standard Halton sequence of dimension NDIM-1.

The standard Hammersley sequence has slightly better dispersion properties than the standard Halton sequence. However, 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 N = 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 Hammersley 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 Hammersley 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.

Licensing:

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

Languages:

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

Related Data and Programs:

BOX_BEHNKEN, a C++ library which computes a Box-Behnken design, that is, a set of arguments to sample the behavior of a function of multiple parameters;

CVT, a C++ library which computes points in a Centroidal Voronoi Tessellation.

FAURE, a C++ library which computes Faure sequences.

GRID, a C++ library which computes points on a grid.

HALTON, a C++ library which computes Halton sequences.

HAMMERSLEY_DATASET, a C++ program which computes Hammersley datasets.

HEX_GRID, a C++ library which computes sets of points in a 2D hexagonal grid.

IHS, a C++ library which computes improved Latin Hypercube datasets.

LATIN_CENTER, a C++ library which computes Latin square data choosing the center value.

LATIN_EDGE, a C++ library which computes Latin square data choosing the edge value.

LATIN_RANDOM, a C++ library which computes Latin square data choosing a random value in the square.

NIEDERREITER2, a C++ library which computes Niederreiter sequences with base 2.

SOBOL, a C++ library which computes Sobol sequences.

UNIFORM, a C++ library which computes uniform random values.

VAN_DER_CORPUT, a C++ library which computes van der Corput sequences.

Reference:

1. John Hammersley,
   Monte Carlo methods for solving multivariable problems,
   Proceedings of the New York Academy of Science,
   Volume 86, 1960, pages 844-874.
2. 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:

• hammersley.cpp, the source code.
• hammersley.hpp, the include file.
• hammersley.sh, commands to compile the source code.

Examples and Tests:

• hammersley_prb.cpp, a sample problem.
• hammersley_prb.sh, commands to compile, link and run the sample problem.
• hammersley_prb_output.txt, the output file.
• hammersley_04_00010.txt, a Hammersley datafile created by the sample problem.

List of Routines:

• ARC_COSINE computes the arc cosine function, with argument truncation.
• ATAN4 computes the inverse tangent of the ratio Y / X.
• DIGIT_TO_CH returns the base 10 digit character corresponding to a digit.
• GET_SEED returns a random seed for the random number generator.
• 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 dataset to a file.
• HAMMERSLEY computes the next element in a leaped Hammersley subsequence.
• HAMMERSLEY_BASE_CHECK checks BASE for a Hammersley sequence.
• HAMMERSLEY_BASE_GET gets the base vector for a leaped Hammersley subsequence.
• HAMMERSLEY_BASE_SET sets the base vector for a leaped Hammersley subsequence.
• HAMMERSLEY_DIM_NUM_GET gets the spatial dimension for a leaped Hammersley subsequence.
• HAMMERSLEY_DIM_NUM_SET sets the spatial dimension for a leaped Hammersley subsequence.
• HAMMERSLEY_LEAP_GET gets the leap vector for a leaped Hammersley subsequence.
• HAMMERSLEY_LEAP_SET sets the leap vector for a leaped Hammersley subsequence.
• HAMMERSLEY_SEED_GET gets the seed vector for a leaped Hammersley subsequence.
• HAMMERSLEY_SEED_SET sets the seed vector for a leaped Hammersley subsequence.
• HAMMERSLEY_SEQUENCE computes N elements in an DIM_NUM-dimensional Hammersley sequence.
• HAMMERSLEY_STEP_GET gets the step for the leaped Hammersley subsequence.
• HAMMERSLEY_STEP_SET sets the step for a leaped Hammersley subsequence.
• I4_LOG_10 returns the whole part of the logarithm base 10 of an integer.
• I4_MIN returns the smaller of two I4's.
• I4_TO_HAMMERSLEY computes one element of a leaped Hammersley subsequence.
• I4_TO_HAMMERSLEY_SEQUENCE computes N elements of a leaped Hammersley subsequence.
• I4_TO_S converts an integer to a string.
• I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed".
• PRIME returns any of the first PRIME_MAX prime numbers.
• R8_EPSILON returns the R8 roundoff unit.
• R8VEC_DOT_PRODUCT returns the dot product of two R8VEC's.
• R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• U1_TO_SPHERE_UNIT_2D maps a point in the unit interval onto the circle in 2D.
• 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 to the unit sphere in 3D.
• U3_TO_BALL_UNIT_3D maps points from the unit box to the unit ball in 3D.

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