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

VAN_DER_CORPUT is a FORTRAN90 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)
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

### 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:

CVT, a FORTRAN90 library which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a FORTRAN90 library which computes elements of a Faure sequence.

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

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

HEX_GRID, a FORTRAN90 library which computes elements of a hexagonal grid dataset.

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

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

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

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

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

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

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

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

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

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

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

VAN_DER_CORPUT_ADVANCED, a FORTRAN90 library which computes elements of a 1D van der Corput Quasi Monte Carlo (QMC) sequence, allowing the user more advanced and sophisticated input.

### Reference:

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

### 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 FORTRAN90 source codes.

Last revised on 10 August 2016.