Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A Generalized Discrepancy and
Quadrature Error Bound

Author: Fred J. Hickernell
Journal: Math. Comp. 67 (1998), 299-322
MSC (1991): Primary 65D30, 65D32
MathSciNet review: 1433265
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the ${\mathcal L}^p$-star discrepancy and $P_\alpha$ that arises in the study of lattice rules.

References [Enhancements On Off] (What's this?)

  • [AS64] M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables, U. S. Government Printing Office, Washington, DC, 1964. MR 29:4914
  • [DR84] P. J. Davis and P. Rabinowitz, Methods of numerical integration, Academic Press, Orlando, Florida, 1984. MR 86d:65004
  • [FH95] K. T. Fang and F. J. Hickernell, The uniform design and its applications, Bulletin of the International Statistical Institute, $50^{\textup{th}}$ Session, Book 1 (Beijing), 1995, pp. 333-349.
  • [FW94] K. T. Fang and Y. Wang, Number theoretic methods in statistics, Chapman and Hall, New York, 1994. MR 95g:65189
  • [Hei96] S. Heinrich, Efficient algorithms for computing the $L_2$ discrepancy, Math. Comp. 65 (1996), 1621-1633. MR 97a:65024
  • [Hic95] F. J. Hickernell, A comparison of random and quasirandom points for multidimensional quadrature, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 213-227.
  • [Hic96] F. J. Hickernell, Quadrature error bounds with applications to lattice rules, SIAM J. Numer. Anal. 33 (1996), 1995-2016. CMP 97:02
  • [MC94] W. J. Morokoff and R. E. Caflisch, Quasi-random sequences and their discrepancies, SIAM J. Sci. Comput. 15 (1994), 1251-1279. MR 95e:65009
  • [Nie92] H. Niederreiter, Random number generation and quasi-Monte Carlo methods, SIAM, Philadelphia, 1992. MR 93h:65008
  • [Owe92] A. B. Owen, Orthogonal arrays for computer experiments, integration and visualization, Statist. Sinica 2 (1992), 439-452. MR 93h:62135
  • [Owe95] A. B. Owen, Equidistributed Latin hypercube samples, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 299-317.
  • [SJ94] I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford University Press, Oxford, 1994.
  • [Sob69] I. M. Sobol', Multidimensional quadrature formulas and Haar functions (in Russian), Izdat. ``Nauka'', Moscow, 1969. MR 54:10952
  • [War72] T. T. Warnock, Computational investigations of low discrepancy point sets, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 319-343. MR 50:3526
  • [Wo\'{z}91] H. Wo\'{z}niakowski, Average case complexity of multivariate integration, Bull. Amer. Math. Soc. 24 (1991), 185-194. MR 91i:65224
  • [Zar68] S. K. Zaremba, Some applications of multidimensional integration by parts, Ann. Polon. Math. 21 (1968), 85-96. MR 38:4034

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65D30, 65D32

Retrieve articles in all journals with MSC (1991): 65D30, 65D32

Additional Information

Fred J. Hickernell
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Keywords: Figure of merit, multidimensional integration, number-theoretic nets and sequences, quasi-random sets, variation
Received by editor(s): April 5, 1996
Received by editor(s) in revised form: September 4, 1996
Additional Notes: This research was supported by a Hong Kong RGC grant 94-95/38 and HKBU FRG grant 95-96/II-01
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society