VAN_DER_CORPUT, a C++ code 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) |
---|---|---|---|
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 |
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
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.
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