# VAN_DER_CORPUT_DATASET Generate van der Corput Datasets

VAN_DER_CORPUT_DATASET is a C++ program which creates a van der Corput sequence dataset and writes it to a file.

The program is interactive, and allows the user to choose the parameters that define the sequence.

The NDIM-dimensional Halton sequence is derived from the 1-dimensional van der Corput sequence by using a set of different (usually distinct prime) bases for each dimension, and the Hammersley sequence is derived in almost the same way.

The van der Corput sequence is often used to generate a "subrandom" sequence of points which have a better covering property than pseudorandom points.

The van der Corput sequence generates a sequence of points in [0,1] which (theoretically) never repeats. Except for SEED = 0, the elements of the van der Corput sequence are strictly between 0 and 1.

The van der Corput sequence writes an integer in a given base B, and then its digits are "reflected" about the decimal point. This maps the numbers from 1 to N into a set of numbers in [0,1], which are especially nicely distributed if N is one less than a power of the base.

Hammersley suggested generating a set of N nicely distributed points in two dimensions by setting the first component of the Ith point to I/N, and the second to the van der Corput value of I in base 2.

Halton suggested that in many cases, you might not know the number of points you were generating, so Hammersley's formulation was not ideal. Instead, he suggested that to generate a nicely distributed sequence of points in M dimensions, you simply choose the first M primes, P(1:M), and then for the J-th component of the I-th point in the sequence, you compute the van der Corput value of I in base P(J).

Thus, to generate a Halton sequence in a 2 dimensional space, it is typical practice to generate a pair of van der Corput sequences, the first with prime base 2, the second with prime base 3. Similarly, by using the first K primes, a suitable sequence in K-dimensional space can be generated.

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

Here is an example in base 2:
SEED (decimal) SEED (binary) VDC (binary) VDC (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

### Usage:

van_der_corput_dataset base seed n
where
• base is the base of the sequence;
• seed is the index of the first element to be computed;
• n is the number of points to generate.
The program generates the data and writes it to the file van_der_corput_base_seed_n.txt.

### Languages:

VAN_DER_CORPUT_DATASET is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FAURE_DATASET, a C++ program which creates a Faure quasirandom dataset;

GRID_DATASET, a C++ program which creates a grid sequence and writes it to a file.

LATIN_CENTER_DATASET, a C++ program which creates a Latin Center Hypercube dataset;

LATIN_EDGE_DATASET, a C++ program which creates a Latin Edge Hypercube dataset;

LATIN_RANDOM_DATASET, a C++ program which creates a Latin Random Hypercube dataset;

NIEDERREITER2_DATASET, a C++ program which creates a Niederreiter quasirandom dataset with base 2;

NORMAL_DATASET, a C++ program which generates a dataset of multivariate normal pseudorandom values and writes them to a file.

SOBOL_DATASET, a C++ program which computes a Sobol quasirandom sequence and writes it to a file.

UNIFORM_DATASET, a C++ program which generates a dataset of uniform pseudorandom values and writes them to a file.

VAN_DER_CORPUT, a C++ library which computes elements of a van der Corput sequence.

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

### Reference:

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

### List of Routines:

• MAIN is the main program for VAN_DER_CORPUT_DATASET.
• I4_TO_VAN_DER_CORPUT_SEQUENCE: next N elements of a van der Corput sequence.
• R8MAT_WRITE writes an R8MAT file.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 31 August 2005.