Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
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In contrast to random points that may cluster, quasi-random points keep their distance. These distances are investigated.1.If N independent random points in the n-dimensional unit hypercube are selected, two of these points may be arbitrarily close. However, if Q"0, Q"1, ..., Q"N"-"1, are quasi-random points, the minimum distance between pairs of these points, d"N, has a positive lower bound. For the Sobol sequence d"N=1/2nN^-^1. Numerical experiments suggest that for large N(1)d"N@?N^-^1^/^n. 2.For certain search algorithms, it is important to know points Q"i and Q"i"+"1 that are not close. For the Sobol sequence, the distances@r(Q"2"k,Q"2"k"+"1)=12n,and14n@?@r(Q"4"k"+"1,Q"4"k"+"2)@?145n+c,where c=0 for even n and c=4 for odd n. 3.Numerical estimations of d"N for the Halton and Faure sequences were carried out. It is likely that for these sequences, (1) is true also.