Extensions of Atanassov’s Methods for Halton Sequences
We extend Atanassov’s methods for Halton sequences in two different directions: (1) in the direction of Niederreiter (t, s) − sequences, (2) in the direction of generating matrices for Halton sequences. It is quite remarkable that Atanassov’s method for classical Halton sequences applies almost “word for word” to (t, s) − sequences and gives an upper bound quite comparable to those of Sobol’, Faure, and Niederreiter. But Atanassov also found a way to improve further his bound for classical Halton sequences by means of a clever scrambling producing sequences which he named modified Halton sequences. We generalize his method to nonsingular lower triangular matrices in the last part of this article.
KeywordsTruncation Operator Lower Triangular Matrice Diophantine Geometry Pairwise Coprime Elementary Interval
We wish to thank the referee for his/her detailed comments, which were very helpful to improve the presentation of this manuscript. The second author acknowledges the support of NSERC for this work.
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