This module provides a free Julia-language Sobol low-discrepancy-sequence (LDS) implementation. This generates "quasi-random" sequences of points in N dimensions which are equally distributed over an N-dimensional hypercube.

The advantage of an LDS over truly random (or pseudo-random) numbers is
that an LDS (which is *not* random) tends to be more evenly
distributed for finite numbers of points. This is used in
quasi-Monte Carlo
methods in
order to accelerate convergence compared to traditional Monte Carlo
methods employing
random sequences.

It can be installed using the Julia package manager via `Pkg.add("Sobol")`

.

This is an independent implementation, originally by Steven G. Johnson, of the algorithm for generation of Sobol sequences in up to 21201 dimensions described in:

- P. Bratley and B. L. Fox, Algorithm 659: Implementing Sobol's quasirandom sequence generator,
*ACM Trans. Math. Soft.***14**(1), pp. 88-100 (1988), doi:10.1145/42288.214372. - S. Joe and F. Y. Kuo, Remark on algorithm 659: Implementing Sobol's quasirandom sequence generator,
*ACM Trans. Math. Soft***29**(1), 49-57 (2003), doi:10.1145/641876.641879.

Originally implemented in C in 2007 as part of the NLopt library for nonlinear optimization, the code was subsequently converted by Ken-B into pure Julia with roughly the same performance.

It is important to emphasize that SGJ's implementation was based on the
mathematical description of the algorithms only, and was done without
reference to the Fortran code from the *Trans. Math. Soft.* (*TOMS*)
papers. The reason is that *TOMS* code is not free/open-source
software (it falls under restrictive ACM copyright
terms).
(SGJ did re-use a table of primitive polynomials and coefficients from
the *TOMS* code, but since this is merely a tabulation of mathematical
facts it is not copyrightable.) SGJ's implementation in NLopt, along
with this Julia translation, is free/open-source software under the MIT
("expat") license.

Direction numbers used were derived from the file http://web.maths.unsw.edu.au/~fkuo/sobol/new-joe-kuo-6.21201

Technically, we implement a 32-bit Sobol sequence. After
2^{32}-1 points, the sequence terminates, and subsequently
our implementation returns pseudo-random numbers generated by the
Mersenne Twister algorithm.
In practical applications, however, this point is rarely reached.

To initialize a Sobol sequence `s`

in `N`

dimensions (`0 < N < 21201`

), use
the `SobolSeq`

constructor:

```
using Sobol
s = SobolSeq(N)
```

Then

`x = next!(s)`

returns the next point (a `Vector{Float64}`

) in the sequence; each point
lies in the hypercube [0,1]^{N}. You can also compute the next
point in-place with

`next!(s, x)`

where `x`

should be a `Vector`

of length `N`

of some floating-point type (e.g. `Float64`

, `Float32`

, or `BigFloat`

).

You can also use a `SobolSeq`

as an iterator in Julia:

```
for x in SobolSeq(N)
...
end
```

Note, however, that the loop will *never terminate* unless you explicitly
call `break`

(or similar) in the loop body at some point of your choosing.

We also provide a different `SobolSeq`

constructor to provide
an `N`

-dimensional Sobol sequence rescaled to an arbitrary hypercube:

`s = SobolSeq(lb, ub)`

where `lb`

and `ub`

are arrays (or other iterables) of length `N`

, giving
the lower and upper bounds of the hypercube, respectively. For example,
`SobolSeq([-1,0,0],[1,3,2])`

generates points in the box [-1,1]×[0,3]×[0,2]. (Although the generated points are `Float64`

by default, in general the precision is determined by that of `lb`

and `ub`

.)

If you know in advance the number `n`

of points that you plan to
generate, some authors suggest that better uniformity can be attained
by first skipping the initial portion of the LDS. In particular,
we skip 2ᵐ−1 for the largest `m`

where `2ᵐ-1 ≤ n`

(see Joe and Kuo, 2003 for a similar suggestion). This
facility is provided by:

`skip(s, n)`

Skipping exactly `n`

elements is also possible:

`skip(s, n, exact=true)`

`skip`

returns `s`

, so you can simply do `s = skip(SobolSeq(N))`

or similar.

Here is a simple example, generating 1024 points in two dimensions and plotting them with the PyPlot package. Note the highly uniform, nonrandom distribution of points in the [0,1]×[0,1] unit square!

```
using Sobol
using PyPlot
s = SobolSeq(2)
p = reduce(hcat, next!(s) for i = 1:1024)'
subplot(111, aspect="equal")
plot(p[:,1], p[:,2], "r.")
```

This module was written by Steven G. Johnson.