VAN_DER_CORPUT
The van der Corput
Quasirandom Sequence

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

It also 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 disttinct 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

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 MATLAB program which computes elements of a Centroidal Voronoi Tessellation.

FAURE is a MATLAB program which computes elements of a Faure sequence.

GRID is a MATLAB program which computes elements of a grid sequence.

HALTON is a MATLAB program which computes elements of a Halton sequence.

HAMMERSLEY is a MATLAB program which computes elements of a Hammersley sequence.

HEX_GRID is a MATLAB program which computes elements of a hexagonal grid dataset.

IHS is a MATLAB program which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER is a MATLAB program which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE is a MATLAB program which computes elements of a Latin Hypercube dataset, choosing edge points.

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

LCVT is a MATLAB program which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2 is a MATLAB program which computes elements of a Niederreiter sequence using base 2.

NORMAL is a MATLAB library which computes elements of a sequence of pseudorandom normally distributed values.

SEQUENCE_STREAK_DISPLAY is a MATLAB program which can display a "streak plot" of a van der Corput sequence.

SOBOL is a MATLAB library which computes elements of a Sobol quasirandom sequence.

UNIFORM is a MATLAB library which computes elements of a uniform pseudorandom sequence.

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:

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.
John Hammersley,
Monte Carlo methods for solving multivariable problems,
Proceedings of the New York Academy of Science,
Volume 86, pages 844-874, 1960.
J G van der Corput,
Verteilungsfunktionen I & II,
Nederl. Akad. Wetensch. Proc.,
Volume 38, 1935, pages 813-820, pages 1058-1066.

Source Code:

i4_to_van_der_corput.m, computes the SEED-th element of the van der Corput sequence in base BASE.
i4_to_van_der_corput_sequence.m, computes the next N elements, starting at SEED, of the van der Corput sequence in base BASE.
r8_epsilon.m, returns the machine precision for R8's.
timestamp.m, prints the current YMDHMS date as a timestamp.
timestring.m, returns the current YMDHMS date as a string.
van_der_corput.m, computes the next element of the van der Corput sequence in the current base.
van_der_corput_base_get.m, returns the current base.
van_der_corput_base_set.m, sets the current base.
van_der_corput_seed_get.m, returns the current seed.
van_der_corput_seed_set.m, sets the current seed.
van_der_corput_sequence.m, computes the next N elements of the van der Corput sequence, starting at the current seed, in the current base.
van_der_corput_write.m, writes a van der Corput sequence to a file.

Examples and Tests:

van_der_corput_test.m, runs all the tests.
van_der_corput_test01.m, tests van_der_corput.
van_der_corput_test02.m, tests van_der_corput_sequence.
van_der_corput_test03.m, tests van_der_corput_seed_get and van_der_corput_seed_set.
van_der_corput_test04.m, tests van_der_corput_base_get and van_der_corput_base_set.
van_der_corput_test045.m, tests van_der_corput_seed_get and van_der_corput_seed_set.
van_der_corput_test05.m, tests i_to_van_der_corput.
van_der_corput_test06.m, tests i_to_van_der_corput_sequence.
van_der_corput_test_output.txt, the output file.

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