VAN_DER_CORPUT
The van der Corput Quasirandom Sequence

VAN_DER_CORPUT is a set of FORTRAN90 routines, using double precision arithmetic, that compute the van der Corput quasirandom sequence.

VAN_DER_CORPUT includes routines 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

Related Data and Programs:

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

FAURE is a FORTRAN90 library of routines which computes elements of a Faure quasirandom sequence.

GRID is a FORTRAN90 library of routines which computes elements of a grid dataset.

HALTON is a FORTRAN90 library of routines which computes elements of a Halton quasirandom sequence.

HAMMERSLEY is a FORTRAN90 library of routines which computes elements of a Hammersley quasirandom sequence.

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

HEX_GRID_ANGLE is a FORTRAN90 library of routines which computes elements of an angled hexagonal grid dataset.

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

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

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

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

LCVT is a FORTRAN90 library of routines which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2 is a FORTRAN90 library of routines which computes elements of a Niederreiter quasirandom sequence with base 2.

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

SEQUENCE_STREAK is a MATLAB program that can make a "streak file" of a van der Corput sequence.

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

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

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

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

VAN_DER_CORPUT_DATASET is an interactive program which creates a van der Corput dataset and writes it to a file.

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.f90, the source code.
• van_der_corput.csh, commands to compile the source code.

Examples and Tests:

• van_der_corput_prb.f90, a sample problem.
• van_der_corput_prb.csh, commands to compile, link and run the sample problem.
• van_der_corput_prb.out, sample problem output.

List of Routines:

• CIRCLE_UNIT_VAN_DER_CORPUT picks a van der Corput point on the unit circle.
• GET_SEED returns a seed for the random number generator.
• GET_UNIT returns a free FORTRAN unit number.
• 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.
• I4VEC_INDICATOR sets an I4VEC to the indicator vector A(I)=I.
• PRIME returns any of the first PRIME_MAX prime numbers.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING writes the current YMDHMS date into 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_MEMORY stores data for the 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 sequence to a file.

You can go up one level to the FORTRAN90 source codes.