Prime Numbers

408 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
These sequences are easy to envision and likewise easy to generate in practice;
in fact, their generation is easier than one might suspect. Say we desire the
van der Corput sequence for base B = 2. Then we simply count from n =0,
in binary
n =0, 1, 10, 11, 100,...,
and form the reversals of the bits to obtain (also in binary)
S =(0.0, 0.10, 0.01, 0.11, 0.001,...).
To put it symbolically, if we are counting and happen to be at integer index
n = nknk−1 ...n1n0,
then the term ρB(n) ∈ S is given by reversing the digits thus:
ρB(n) =0.n0n1 ...nk.
It is known that every van der Corput sequence has
D ∗ N(SB) =O
ln N
N
where the implied big-O constant depends only on B. It turns out that B =3
has the smallest such constant, but the main point affecting implementations
is that the constant generally increases for larger bases B [Faure 1981].
For D>1 dimensions, it is possible to generate qMC sequences based on
the van der Corput forms, in the following manner:
Definition 8.3.4. Let ¯ B = {B1,B2,...,BD} be a set of pairwise-coprime
bases, each Bi > 1. We define the Halton sequence for bases ¯ B by
where
,
S ¯ B =(xn) , n =0, 1, 2,...,
xn =(ρB1(n),...,ρBD (n)).
In other words, a Halton sequence involves a specific base for each vector
coordinate, and the respective bases are to be pairwise coprime. Thus for
example, a qMC sequence of points in the (D = 3)-dimensional unit cube can
be generated by choosing prime bases {B1,B2,B3} = {2, 3, 5} and counting
n =0, 1, 2,... in those bases simultaneously, to obtain
x0 =(0, 0, 0),
x1 =(1/2, 1/3, 1/5),
x2 =(1/4, 2/3, 2/5),
x3 =(3/4, 1/9, 3/5),
and so on. The manner in which these points deposit themselves in the unit
3-cube is interesting. We can see once again the basic, qualitative aspect