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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'.
- 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.
- 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.
- 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);
- 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.
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.
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.
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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