Quasi-Random Sequences¶ ↑

This chapter describes the quasi-random sequence generator GSL::QRng of arbitrary dimensions. A quasi-random sequence progressively covers a d-dimensional space with a set of points that are uniformly distributed. Quasi-random sequences are also known as low-discrepancy sequences. The quasi-random sequence generators use an interface that is similar to the interface for random number generators.

Contents:

1. Quasi-random number generator initialization

2. Sampling from a quasi-random number generator

3. Auxiliary quasi-random number generator functions

4. Saving and resorting quasi-random number generator state

5. Quasi-random number generator algorithms

Quasi-random number generator initialization¶ ↑

• GSL::QRng.alloc(T, d)

  This returns a GSL::QRng object, a quasi-random sequence generator of type T and dimension d.

• GSL::QRng::init

  This reinitializes the generator to its starting point.

Sampling from a quasi-random number generator¶ ↑

• GSL::QRng::get(x)

  This calculate the next point x from the sequence generator. Here x is an instance of the GSL::Vector class. The space available for x must match the dimension of the generator. The point x will lie in the range 0 < x_i < 1 for each x_i.

  This is used as

  q = QRng.alloc(QRng::SOBOL, dim)
  v = Vector.alloc(dim)
  for i in 0..1024 do
    q.get(v)
    printf("%.5f %.5f\n", v[0], v[1])
  end
  

Auxiliary quasi-random number generator functions¶ ↑

• GSL::QRng::name

  Returns the name of the generator self.

• GSL::QRng::size

Saving and resorting quasi-random number generator state¶ ↑

• GSL::QRng::clone

• GSL::QRng::duplicate

  Return a newly created generator which is an exact copy of the generator self.

Quasi-random number generator algorithms¶ ↑

In creating a generator by the method GSL::QRng.alloc(T, d), the algorithm type T is given by a String or a Fixnum constant. The following quasi-random sequence algorithms are available,

• “niederreiter_2” (String)

• GSL::QRng::NIEDERREITER_2 (Fixnum)

  The generator of this type uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.

• “sobol” (String)

• GSL::QRng::SOBOL (Fixnum)

  This generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys. 19, 252 (1980). It is valid up to 40 dimensions.

• “halton” (String)

• GSL::QRng::HALTON (Fixnum)

• “reversehalton” (String)

• GSL::QRng::REVERSEHALTON (Fixnum)

  (GSL-1.11 or later) These generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics 189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.

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