# Generalized Halton Number Generator

This library allows to generate quasi-random numbers according to the generalized Halton sequence. For more information on Generalized Halton Sequences, their properties, and limits see Braaten and Weller (1979), Faure and Lemieux (2009), and De Rainville et al. (2012) and reference therein.

The library is compatible Python 2 and Python 3.

##
`pip`

Install with Simply type in

```
$ pip install ghalton
```

## Building the Code

To build the code you'll need a working C++ compiler.

```
$ python setup.py install
```

## Using the Library

The library contains two generators one producing the standard Halton sequence and the other a generalized version of it. The former constructor takes a single argument, which is the dimensionalty of the sequence.

```
import ghalton
sequencer = ghalton.Halton(5)
```

The last code will produce a sequence in five dimension. To get the points use

```
points = sequencer.get(100)
```

A list of 100 lists will be produced, each sub list will containt 5 points

```
print(points[0])
# [0.5, 0.3333, 0.2, 0.1429, 0.0909]
```

The halton sequence produce points in sequence, to reset it call
`sequencer.reset()`

.

The generalised Halton sequence constructor takes at least one argument, either the dimensionality, or a configuration. When the dimensionality is given, an optional argument can be used to seed for the random permutations created.

```
import ghalton
sequencer = ghalton.GeneralizedHalton(5, 68)
points = sequencer.get(100)
print(points[0])
# [0.5, 0.6667, 0.4, 0.8571, 0.7273]
```

A configuration is a series of permutations each of *n_i* numbers,
where *n_i* is the *n_i*'th prime number. The dimensionality is infered from
the number of sublists given.

```
import ghalton
perms = ((0, 1),
(0, 2, 1),
(0, 4, 2, 3, 1),
(0, 6, 5, 4, 3, 2, 1),
(0, 8, 2, 10, 4, 9, 5, 6, 1, 7, 3))
sequencer = ghalton.GeneralizedHalton(perms)
points = sequencer.get(100)
print(points[0])
# [0.5, 0.6667, 0.8, 0.8571, 0.7273]
```

The configuration presented in De Rainville et al. (2012) is available in the
ghalton module. Use the first *dim* dimensions of the `EA_PERMS`

constant.
The maximum dimensionality provided is 100.

```
import ghalton
dim = 5
sequencer = ghalton.GeneralizedHalton(ghalton.EA_PERMS[:dim])
points = sequencer.get(100)
print(points[0])
# [0.5, 0.6667, 0.8, 0.8571, 0.7273]
```

The complete API is presented on read the docs.

## Building the SWIG wrapper

In the main directory use command

```
swig -Wall -c++ -python -outdir ghalton src/Halton.i
```

## Configuration Repository

See the Quasi Random Sequences Repository for more configurations.

## References

E. Braaten and G. Weller. An improved low-discrepancy sequence for multidi-
mensional quasi-Monte Carlo integration. *J. of Comput. Phys.*,
33(2):249-258, 1979.

F.-M. De Rainville, C. Gagné, O. Teytaud, D. Laurendeau. Evolutionary
optimization of low-discrepancy sequences. *ACM Trans. Model. Comput. Simul.*,
22(2):1-25, 2012.

H. Faure and C. Lemieux. Generalized Halton sequences in 2008: A comparative
study. *ACM Trans. Model. Comput. Simul.*, 19(4):1-43, 2009.