# van_der_corput

van_der_corput, a FORTRAN90 code which computes the van der Corput Quasi Monte Carlo (QMC) sequence, using a simple interface.

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)
00.00.0
11.10.5
210.010.25
311.110.75
4100.0010.125
5101.1010.625
6110.0110.375
7111.1110.875
81000.00010.0625

### Licensing:

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

### Languages:

van_der_corput is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

cvt, a FORTRAN90 code which computes elements of a Centroidal Voronoi Tessellation.

faure, a FORTRAN90 code which computes elements of a Faure sequence.

HALTON, a FORTRAN90 code which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HAMMERSLEY, a FORTRAN90 code which computes elements of a Hammersley Quasi Monte Carlo (QMC) sequence, using a simple interface.

IEEE_UNIFORM_SAMPLE, a FORTRAN90 code which tries to uniformly sample the discrete set of values that represent the legal IEEE real numbers;

IHS, a FORTRAN90 code which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a FORTRAN90 code which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a FORTRAN90 code which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a FORTRAN90 code which computes elements of a Latin Hypercube dataset, choosing points at random.

LCVT, a FORTRAN90 code which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2, a FORTRAN90 code which computes elements of a Niederreiter sequence using base 2.

NORMAL, a FORTRAN90 code which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a FORTRAN90 code which computes elements of a Sobol quasirandom sequence.

UNIFORM, a FORTRAN90 code which computes elements of a uniform pseudorandom sequence.

### 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:

Last revised on 10 September 2020.