VAN_DER_CORPUT
The van der Corput
Quasi Monte Carlo (QMC) sequence

VAN_DER_CORPUT is a C library which computes the van der Corput Quasi Monte Carlo (QMC) sequence, using a simple interface.

A more sophisticated library is available in VAN_DER_CORPUT_ADVANCED, but I find this simple version to be preferable for everyday use!

The van der Corput sequence generates a sequence of points in [0,1] which never repeats. For positive index I, the elements of the van der Corput sequence are strictly between 0 and 1.

The I-th element of the van der Corput sequence is computed by writing I in the base B (usually 2) and then reflecting its digits about the decimal point. For example, if we start with I = 11, its binary expansion is 1011, and so its reflected binary expansion is 0.1101 which is 1/2+1/4+1/16=0.8125.

The generation is quite simple. Given an index I, the expansion of I in base B is generated. Then, essentially, the result R is generated by writing a decimal point followed by the digits of the expansion of I, in reverse order. This decimal value is actually still in base B, so it must be properly interpreted to generate a usable value.

Here is an example in base 2:

I (decimal)	I (binary)	R (binary)	R (decimal)
0	0	.0	0.0
1	1	.1	0.5
2	10	.01	0.25
3	11	.11	0.75
4	100	.001	0.125
5	101	.101	0.625
6	110	.011	0.375
7	111	.111	0.875
8	1000	.0001	0.0625

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

VAN_DER_CORPUT is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

HALTON, a C library which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HAMMERSLEY, a C library which computes elements of a Hammersley Quasi Monte Carlo (QMC) sequence, using a simple interface.

LATIN_RANDOM, a C library which computes elements of a Latin Hypercube dataset, choosing points at random.

NORMAL, a C library which computes elements of a sequence of pseudorandom normally distributed values.

UNIFORM, a C library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a dataset directory which contains datasets of van der Corput sequences.

Reference:

1. J G van der Corput,
   Verteilungsfunktionen I & II,
   Nederl. Akad. Wetensch. Proc.,
   Volume 38, 1935, pages 813-820, pages 1058-1066.

Source Code:

• van_der_corput.c, the source code.
• van_der_corput.h, the include file.

Examples and Tests:

• van_der_corput_test.c, a sample calling program.
• van_der_corput_test.txt, the output file.

List of Routines:

• R8VEC_TRANSPOSE_PRINT prints an R8VEC "transposed".
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VDC_BASE computes an element of the van der Corput sequence in any base.
• VDC_INVERSE inverts an element of the van der Corput sequence.
• VDC computes an element of the van der Corput sequence.
• VDC_SEQUENCE computes a sequence of elements of the van der Corput sequence.

You can go up one level to the C source codes.