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

VAN_DER_CORPUT is a Python 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:

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

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

LATIN_RANDOM, a Python program which computes elements of a Latin Hypercube dataset, choosing points at random.

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

RANDLC, a Python library which generates a sequence of pseudorandom numbers, used by the NAS Benchmark programs.

RNGLIB, a Python library which implements a random number generator (RNG) with splitting facilities, allowing multiple independent streams to be computed, by L'Ecuyer and Cote.

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

### Reference:

1. J G van der Corput,
Verteilungsfunktionen I & II,