# HALTON_ADVANCED The Halton Quasi Monte Carlo (QMC) Sequence

HALTON_ADVANCED is a MATLAB 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 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;
Both of these routines require explicit input values for all parameters.

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;
These routines allow the user to either rely on the default values of parameters, or to change a few of them by calling appropriate routines.

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.

### Languages:

HALTON_ADVANCED 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.

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

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.