Prime Numbers

412 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
Let us give an example of the application of such an algorithm. To assess the
volume of the unit D-ball, which is the ball of radius 1, we can take f in terms
of the Heaviside function θ (which is 1 for positive arguments, 0 for negative
arguments, and 1/2 at0),
f(x) =θ(1/4 − (x − y) · (x − y)),
with y =(1/2, 1/2,...,1/2), so that f vanishes everywhere outside a ball of
radius 1/2. (This is the largest ball that fits inside the cube R.) The estimate
of the unit D-ball volume will thus be 2 D I,whereI is the output of Algorithm
8.3.7 for the given, sphere-defining function f.
As we have intimated before, it is a wondrous thing to see firsthand
how much better a qMC algorithm of this type can do, when compared to
a direct Monte Carlo trial. One beautiful aspect of the fundamental qMC
concept is that parallelism is easy: In Algorithm 8.3.7, just start each of, say,
M machines at a different starting seed, ideally in such a way that some
contiguous sequence of NM total vectors is realized. This option is, of course,
the point of having a seed function in the first place. Explicitly, to obtain
a one-billion-point integration, each of 100 machines would use the above
algorithm as is with N =10 7 , except that machine 0 would start with n =0
(and hence start by calling seed(0)), the second machine would start n =1,
through machine 99, which would start with n = 99. The final integral would
be the average of the 100 machine estimates.
Here is a typical numerical comparison: We shall calculate the number π
with qMC methods, and compare with direct Monte Carlo. Noting that the
exact volume of the unit D-ball is
VD =
π D/2
Γ(1 + D/2) ,
let us denote by VD(N) the calculated volume after N vectors are generated,
and denote by πN the “experimental” value for π obtained by solving the
volume formula for π in terms of VD. We shall do two things at once: Display
the typical convergence and convey a notion of the inherent parallelism. For
primes p =2, 3, 5, so that we are assessing the 3-ball volume, the result of
Algorithm 8.3.7 is displayed in Table 8.1.
What is displayed in the left-hand column is the total number of points
“dropped” into the unit D-cube, while the second column is the associated,
cumulative approximation to π. We say cumulative because one may have
runeachintervalof10 6 counts on a separate machine, yet we display the
right-hand column as the answer obtained by combining the machines up to
that N value inclusive. For example, the result π5 can be thought of either as
the result after 5 · 10 6 points are generated, or equivalently, after 5 separate
machines each do 10 6 points. In the latter instance, one would have called
the seed(n) procedure with 5 different initial seeds to start each respective
machine’s interval. How do these data compare with direct Monte Carlo? The
rough answer is that one can expect the error in the last (N =10 7 )rowof