Computing Discrepancies Related to Spaces of Smooth Periodic Functions
نویسندگان
چکیده
A notion of discrepancy is introduced, which represents the integration error on spaces of r-smooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical L2-discrepancy as well as r-smooth versions of it introduced recently by Paskov Pas93]. Based on previous work FH96], we develop an eecient algorithm for computing periodic discrepancies for quadrature formulas possessing certain tensor product structures, in particular, for Smolyak quadrature rules (also called sparse grid methods). Furthermore, fast algorithms of computing periodic discrepancies for lattice rules can easily be derived from well{known properties of lattices. On this basis we carry out numerical comparisons of discrepancies between Smolyak and lattice rules.
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