HAMMERSLEY_ADVANCED is a C++ library which computes the Hammersley Quasi Monte Carlo (QMC) sequence.
HAMMERSLEY_ADVANCED 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
For more convenience, there are two related routines with almost no input arguments:
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:
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.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
HAMMERSLEY_ADVANCED is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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