Prime Numbers

410 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
Algorithm 8.3.6 (Fast qMC sequence generation). This algorithm generates
D-dimensional Halton-sequence vectors. Let p1,...,pD denote the first D
primes. For starting index n, aseed() procedure creates xn whose components are
for clarity denoted by xn[1],...,xn[D]. Thenarandom() function may be used
to generate subsequent vectors xn+1,xn+2,..., where we assume an upper bound
of N for all indices. For high efficiency, global digits (di,j) are initially seeded to
represent the starting index n, then upon subsequent calls to a random() function,
are incremented in “odometer” fashion for subsequent indices exceeding n.
1. [Procedure seed]
seed(n) { // n is the desired starting index.
}
for(1 ≤ i ≤ D) {
ln(N+1)
Ki = ln ; // A precision parameter.
pi
qi,0 =1;
k = n;
x[i] =0; // x is the vector xn.
for(1 ≤ j ≤ Ki) {
qi,j = qi,j−1/pi; // qi,j = p −j
i .
di,j = k mod pi; // The di,j start as base-pi digits of n.
k =(k− di,j)/pi;
x[i] =x[i]+di,jqi,j;
}
}
return; // xn now available as (x[1],...,x[D]).
2. [Function random]
random() {
for(1 ≤ i ≤ D) {
for(1 ≤ j ≤ Ki) {
}
}
di,j = di,j +1; // Increment the “odometer.”
x[i] =x[i]+qi,j;
if(di,j