A Component-by-Component Construction for the Trigonometric Degree

  • Nico Achtsis
  • Dirk Nuyens
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)


We propose an alternative to the algorithm from Cools, Kuo, Nuyens (Computing 87(1–2):63–89, 2010), for constructing lattice rules with good trigonometric degree. The original algorithm has construction cost \(O(\vert {\mathcal{A}}_{d}(m)\vert + dN\log N)\) for an N-point lattice rule in d dimensions having trigonometric degree m, where the set \({\mathcal{A}}_{d}(m)\) has exponential size in both d and m (in the “unweighted degree” case, which is what we consider here). We reduce the cost to \(O(dN{(\log N)}^{2})\) with an implicit constant governing the needed precision (which is dependent on N and d).


Fourier Coefficient Reproduce Kernel Hilbert Space Dual Lattice Lattice Rule Tensor Product Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors would like to thank the two anonymous referees for useful comments on the manuscript.


  1. 1.
    M. Beckers and R. Cools. A relation between cubature formulae of trigonometric degree and lattice rules. In H. Brass and G. Hämmerlin, editors, Numerical integration IV (Oberwolfach, 1992), pages 13–24. Birkhäuser Verlag, 1993.Google Scholar
  2. 2.
    R. Cools. More about cubature formulas and densest lattice packings. East Journal on Approximations, 12(1):37–42, 2006.Google Scholar
  3. 3.
    R. Cools and H. Govaert. Five- and six-dimensional lattice rules generated by structured matrices. J. Complexity, 19(6):715–729, 2003.Google Scholar
  4. 4.
    R. Cools, F. Y. Kuo, and D. Nuyens. Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput., 28(6):2162–2188, 2006.Google Scholar
  5. 5.
    R. Cools, F. Y. Kuo, and D. Nuyens. Constructing lattice rules based on weighted degree of exactness and worst case error. Computing, 87(1–2):63–89, 2010.Google Scholar
  6. 6.
    R. Cools and J. N. Lyness. Three- and four-dimensional K-optimal lattice rules of moderate trigonometric degree. Math. Comp., 70(236):1549–1567, 2001.Google Scholar
  7. 7.
    R. Cools, E. Novak, and K. Ritter. Smolyak’s construction of cubature formulas of arbitrary trigonometric degree. Computing, 62(2):147–162, 1999.Google Scholar
  8. 8.
    R. Cools and D. Nuyens. A Belgian view on lattice rules. In A. Keller, S. Heinrich, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2006, pages 3–21. Springer-Verlag, 2008.Google Scholar
  9. 9.
    R. Cools and D. Nuyens. Extensions of Fibonacci lattice rules. In P. L’Écuyer and A. B. Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008, pages 1–12. Springer-Verlag, 2009.Google Scholar
  10. 10.
    R. Cools and A. V. Reztsov. Different quality indexes for lattice rules. J. Complexity, 13(2):235–258, 1997.Google Scholar
  11. 11.
    R. Cools and I. H. Sloan. Minimal cubature formulae of trigonometric degree. Math. Comp., 65(216):1583–1600, 1996.Google Scholar
  12. 12.
    J. A. De Loera, J. Rambau, and F. Santos. Triangulations, volume 25 of Algorithms and Computation in Mathematics. Springer-Verlag, 2010.Google Scholar
  13. 13.
    J. Dick, F. Pillichshammer, G. Larcher, and H. Woźniakowski. Exponential convergence and tractability of multivariate integration for Korobov spaces. Math. Comp., 80(274):905–930, 2011.Google Scholar
  14. 14.
    I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press, 7th edition, 2007.Google Scholar
  15. 15.
    F. J. Hickernell. Lattice rules: How well do they measure up? In P. Hellekalek and G. Larcher, editors, Random and Quasi-Random Point Sets, pages 109–166. Springer-Verlag, Berlin, 1998.Google Scholar
  16. 16.
    J. N. Lyness. Notes on lattice rules. J. Complexity, 19(3):321–331, 2003.Google Scholar
  17. 17.
    J. N. Lyness and T. Sørevik. Four-dimensional lattice rules generated by skew-circulant matrices. Math. Comp., 73(245):279–295, 2004.Google Scholar
  18. 18.
    J. N. Lyness and T. Sørevik. Five-dimensional K-optimal lattice rules. Math. Comp., 75(255):1467–1480, 2006.Google Scholar
  19. 19.
    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. Number 63 in Regional Conference Series in Applied Mathematics. SIAM, 1992.Google Scholar
  20. 20.
    D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp., 75(254):903–920, 2006.Google Scholar
  21. 21.
    D. Nuyens and R. Cools. Fast component-by-component construction, a reprise for different kernels. In H. Niederreiter and D. Talay, editors, Monte Carlo and Quasi-Monte Carlo Methods 2004, pages 371–385. Springer-Verlag, 2006.Google Scholar
  22. 22.
    D. Nuyens and R. Cools. Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complexity, 22(1):4–28, 2006.Google Scholar
  23. 23.
    N. N. Osipov, R. Cools, and M. V. Noskov. Extremal lattices and the construction of lattice rules. Appl. Math. Comput., 217(9):4397–4407, 2011.Google Scholar
  24. 24.
    I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration. Oxford Science Publications, 1994.Google Scholar
  25. 25.
    I. H. Sloan and A. V. Reztsov. Component-by-component construction of good lattice rules. Math. Comp., 71(237):263–273, 2002.Google Scholar
  26. 26.
    I. H. Sloan and H. Woźniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complexity, 14(1):1–33, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceK.U.LeuvenHeverleeBelgium

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