VAN_DER_CORPUT
The van der Corput
Quasirandom Sequence

VAN_DER_CORPUT is a C++ library which computes the van der Corput quasirandom sequence.

VAN_DER_CORPUT includes several subroutines to make it easy to manipulate this computation, to compute the next N entries, to compute a particular entry, or to restart the sequence at a particular point.

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)
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.

Related Data and Programs:

CVT is a C++ library which computes points in a Centroidal Voronoi Tessellation.

FAURE is a C++ library which computes Faure sequences.

GRID is a C++ library which computes points on a grid.

HALTON is a C++ library which computes Halton sequences.

HAMMERSLEY is a C++ library which computes Hammersley sequences.

HEX_GRID is a C++ library which computes sets of points in a 2D hexagonal grid.

IHS is a C++ library which computes improved Latin Hypercube datasets.

LATIN_CENTER is a C++ library which computes Latin square data choosing the center value.

LATIN_EDGE is a C++ library which computes Latin square data choosing the edge value.

LATIN_RANDOM is a C++ library which computes Latin square data choosing a random value in the square.

NIEDERREITER2 is a C++ library which computes Niederreiter sequences with base 2.

NORMAL is a C++ library which computes a sequence of pseudorandom normally distributed values.

SEQUENCE_STREAK_DISPLAY is a MATLAB program which makes a "streak file" of a van der Corput sequence.

SOBOL is a C++ library which computes Sobol sequences.

UNIFORM is a C++ library which computes uniform random values.

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

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

Reference:

1. John Halton,
   On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,
   Numerische Mathematik,
   Volume 2, pages 84-90, 1960.
2. John Hammersley,
   Monte Carlo methods for solving multivariable problems,
   Proceedings of the New York Academy of Science,
   Volume 86, pages 844-874, 1960.
3. Johannes 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.
• van_der_corput.csh, commands to compile the source code.

Examples and Tests:

• van_der_corput_prb.C, a sample problem.
• van_der_corput_prb.csh, commands to compile, link and run the sample problem.
• van_der_corput_prb_output.txt, the output file.

List of Routines:

• DIGIT_TO_CH returns the base 10 digit character corresponding to a digit.
• GET_SEED returns a random seed for the random number generator.
• I4_LOG_10 returns the whole part of the logarithm base 10 of an integer.
• I4_TO_S converts an integer to a string.
• I4_TO_VAN_DER_CORPUT computes an element of a van der Corput sequence.
• I4_TO_VAN_DER_CORPUT_SEQUENCE: next N elements of a van der Corput sequence.
• R8_EPSILON returns the R8 round off unit.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• VAN_DER_CORPUT computes the next element in the van der Corput sequence.
• VAN_DER_CORPUT_BASE_GET gets the base for a van der Corput sequence.
• VAN_DER_CORPUT_BASE_SET sets the base for a van der Corput sequence.
• VAN_DER_CORPUT_SEED_GET gets the "seed" for the van der Corput sequence.
• VAN_DER_CORPUT_SEED_SET sets the "seed" for the van der Corput sequence.
• VAN_DER_CORPUT_SEQUENCE: next N elements in the van der Corput sequence.
• VAN_DER_CORPUT_WRITE writes a van der Corput dataset to a file.

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