A Generalized Discrepancy and

Quadrature Error Bound

Author:
Fred J. Hickernell

Journal:
Math. Comp. **67** (1998), 299-322

MSC (1991):
Primary 65D30, 65D32

DOI:
https://doi.org/10.1090/S0025-5718-98-00894-1

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 -star discrepancy and that arises in the study of lattice rules.

**[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, 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 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**

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Additional Information

**Fred J. Hickernell**

Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Email:
fred@hkbu.edu.hk

DOI:
https://doi.org/10.1090/S0025-5718-98-00894-1

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