HALTON_ADVANCED is a C++ 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
For more convenience, there are two related routines with almost no input arguments:
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:
The NDIM-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 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.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
HALTON_ADVANCED is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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You can go up one level to the C++ source codes.