Prime Numbers

8.3 Quasi-Monte Carlo (qMC) methods 411
Algorithm 8.3.6 is usually used in floating-point mode, i.e., with stored
floating-point inverse powers qi,j but integer digits ni,j. However, there is
nothing wrong in principle with an exact generator in which actual integer
powers are kept for the qi,j. In fact, the integer mode can be used for testing of
the algorithm, in the following interesting way. Take, for example, N = 1000,
so vectors x0,...,x999 are allowed, and choose D = 2 dimensions so that the
primes 2,3 are involved. Then call seed(701), which sets the variable x to be
the vector
x701 = (757/1024, 719/729).
Now, calling random() exactly 9 times produces
x710 = (397/1024, 674/729),
and sure enough, we can test the integrity of the algorithm by going back and
calling seed(710) to verify that starting over thus with seed value 701+9 gives
precisely the x710 shown.
It is of interest that Algorithm 8.3.6 really is fast, at least in this
sense: In practice, it tends to be faster even than calling a system’s built-in
random-number function. And this advantage has meaning even outside the
numerical-integration paradigm. When one really wants an equidistributed,
random number in [0, 1), say, a system’s random function should certainly be
considered, especially if the natural tendency for random samples to clump
and separate is supposed to remain intact. But for many statistical studies,
one simply wants some kind if irregular “coverage” of [0, 1), one might say a
“fair” coverage that does not bias any particular subinterval, in which case
such a fast qMC algorithm should be considered.
Now we may get a multidimensional integral by calling, in a very simple
way, the procedures of Algorithm 8.3.6:
Algorithm 8.3.7 (qMC multidimensional integration). Given a dimension
D, and integrable function f : R → R, where R = [0, 1] D , this algorithm
estimates the multidimensional integral
I = f(x) d D x,
x∈R
via the generation of N0 qMC vectors, starting with the n-th of a sequence
(x0,x1,...,xn,...,xn+N0−1,...). It is assumed that Algorithm 8.3.6 is initialized
with an index bound N ≥ n + N0.
1. [Initialize via Algorithm 8.3.6]
seed(n); // Start the qMC process, to set a global x = xn.
I =0;
2. [Perform qMC integration]
// Function random() updates a global qMC vector (Algorithm 8.3.6).
for(0 ≤ j