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# Quasi-Random Sequences

This chapter describes functions for generating quasi-random sequences in 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.

The functions described in this section are declared in the header file ``gsl_qrng.h'`.

## Quasi-random number generator initialization

Function: gsl_qrng * gsl_qrng_alloc (const gsl_qrng_type * T, unsigned int d)
  This function returns a pointer to a newly-created instance of a quasi-random sequence generator of type T and dimension d. If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of `GSL_ENOMEM`.

Function: void gsl_qrng_free (gsl_qrng * q)
  This function frees all the memory associated with the generator q.

Function: void gsl_qrng_init (gsl_qrng * q)
  This function reinitializes the generator q to its starting point.

## Sampling from a quasi-random number generator

Function: int gsl_qrng_get (const gsl_qrng * q, double x[])
  This function returns the next point x from the sequence generator q. 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.

## Auxiliary quasi-random number generator functions

Function: const char * gsl_qrng_name (const gsl_qrng * q)
  This function returns a pointer to the name of the generator.

Function: size_t gsl_qrng_size (const gsl_qrng * q)
Function: void * gsl_qrng_state (const gsl_qrng * q)
  These function return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,

      void * state = gsl_qrng_state (q);
      size_t n = gsl_qrng_size (q);
      fwrite (state, n, 1, stream);

## Saving and resorting quasi-random number generator state

Function: int gsl_qrng_memcpy (gsl_qrng * dest, const gsl_qrng * src)
  This function copies the quasi-random sequence generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.

Function: gsl_qrng * gsl_qrng_clone (const gsl_qrng * q)
  This function returns a pointer to a newly created generator which is an exact copy of the generator r.

## Quasi-random number generator algorithms

The following quasi-random sequence algorithms are available,

Generator: gsl_qrng_niederreiter_2
  This generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.

Generator: gsl_qrng_sobol
  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.

## Examples

The following program prints the first 1024 points of the 2-dimensional Sobol sequence.

    #include <stdio.h>
    #include <gsl/gsl_qrng.h>

    int
    main (void)
    {
      int i;
      gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);

      for (i = 0; i < 1024; i++)
        {
          double v[2];
          gsl_qrng_get(q, v);
          printf("%.5f %.5f\n", v[0], v[1]);
        }

      gsl_qrng_free(q);
      return 0;
    }

Here is the output from the program,

    $ ./a.out
    0.50000 0.50000
    0.75000 0.25000
    0.25000 0.75000
    0.37500 0.37500
    0.87500 0.87500
    0.62500 0.12500
    0.12500 0.62500
    ....

It can be seen that successive points progressively fill-in the spaces between previous points.

## References

The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,

P. Bratley and B.L. Fox and H. Niederreiter, "Algorithm 738: Programs to Generate Niederreiter's Low-discrepancy Sequences", Transactions on Mathematical Software, Vol. 20, No. 4, December, 1994, p. 494-495.

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