Numerical Algorithms

, Volume 59, Issue 3, pp 403–431 | Cite as

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

  • Jan Baldeaux
  • Josef Dick
  • Gunther Leobacher
  • Dirk Nuyens
  • Friedrich Pillichshammer
Original Paper

Abstract

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base 2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.

Keywords

Numerical integration Quasi-Monte Carlo Polynomial lattice rules Digital nets 

Mathematics Subject Classifications (2000)

65D30 65C05 

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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Jan Baldeaux
    • 1
  • Josef Dick
    • 2
  • Gunther Leobacher
    • 3
  • Dirk Nuyens
    • 4
  • Friedrich Pillichshammer
    • 3
  1. 1.School of Finance and EconomicsThe University of TechnologySydneyAustralia
  2. 2.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia
  3. 3.Institut für FinanzmathematikUniversität LinzLinzAustria
  4. 4.Department of Computer ScienceK.U.LeuvenHeverleeBelgium

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