Random Number Generation and Quasi-Monte Carlo Methods by Harald Niederreiter

By Harald Niederreiter

Great development has taken position within the comparable components of uniform pseudorandom quantity new release and quasi-Monte Carlo tools within the final 5 years. This quantity includes contemporary very important paintings in those parts, and stresses the interaction among them. a few advancements contained the following have by no means ahead of seemed in booklet shape. comprises the dialogue of the built-in therapy of pseudorandom numbers and quasi-Monte Carlo equipment; the systematic improvement of the speculation of lattice ideas and the speculation of nets and (t,s)-sequences; the development of latest and higher low-discrepancy element units and sequences; Nonlinear congruential tools; the initiation of a scientific learn of tools for pseudorandom vector iteration; and shift-register pseudorandom numbers.

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This leads to the family of Halton sequences. For a given dimension 3 > 1, let 61,... ,6S be integers > 2. 1, we define the Halton sequence in the bases 61,... , ba as the sequence XQ, xi,... with For 3 = 1 this definition reduces to that of a van der Corput sequence. 6. If S is the Halton sequence in the pairwise relatively prime bases 61,... , ba, then Proof. Fix N > 1 and write D(J) = A(J-,SN) - N\S(J) for an interval J C Is, where SN is the point set consisting of the first N terms of the Halton sequence.

Hm)  Cm(b). For 1 < d < m, there are bd~l elements (hi,... , hm) of Cm(b) with d(hi,... , hm) = d and a fixed nonzero value of hd. 17) we obtain, for 6 > 2, LOW-DISCREPANCY POINT SETS AND SEQUENCES 41 By [222, p. 574], we have and, by combining these results, we obtain the first part of the lemma. 16) to obtain and the second part of the lemma follows. D We now consider some general principles of obtaining lower bounds for the discrepancy. The following is an elementary lower bound for the discrepancy of point sets comprising only points with rational coordinates.

1. Let 0 < t < m be integers. A (t,m,s)-net in base b is a point set P of 6m points in I8 such that A(E\ P) = b* for every elementary interval E in base b with \8(E) = 6*~m. 2. Let t > 0 be an integer. A sequence XQ, xi,... of points in I3 is a (t, s)-sequence in base b if, for all integers k > 0 and m > t, the point set consisting of the xn with kbm < n < (k + l)6m is a (£, m, s)-net in base b. In this language, the van der Corput sequence in base b is a (0, l)-sequence in base 6. 2 were introduced by Sobol' [323] in the case where 6 = 2; the general definitions were first given by Niederreiter [244].

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