On the discrepancy of quadratic congruential pseudorandom numbers
Journal of Computational and Applied Mathematics
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
On the distribution of inversive congruential pseudorandom numbers in parts of the period
Mathematics of Computation
Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation
SIAM Journal on Scientific Computing
Exact error estimates and optimal randomized algorithms for integration
NMA'06 Proceedings of the 6th international conference on Numerical methods and applications
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We consider the b-adic diaphony as a tool to measure the uniform distribution of sequences, as well as to investigate pseudo-random properties of sequences. The study of pseudo-random properties of uniformly distributed nets is extremely important for quasi-Monte Carlo integration. It is known that the error of the quasi-Monte Carlo integration depends on the distribution of the points of the net. On the other hand, the b-adic diaphony gives information about the points distribution of the net. Several particular constructions of sequences (xi) are considered. The b-adic diaphony of the two dimensional nets {yi = (xi, xi+1)} is calculated numerically. The numerical results show that if the two dimensional net {yi} is uniformly distributed and the sequence (xi) has good pseudorandom properties, then the value of the b-adic diaphony decreases with the increase of the number of the points. The analysis of the results shows a direct relation between pseudo-randomness of the points of the constructed sequences and nets and the b-adic diaphony as well as the discrepancy.