{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:51:17Z","timestamp":1778496677401,"version":"3.51.4"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n 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\n \n \n \n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:msup>\n {\\mathcal L}^p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -star discrepancy and\n \n \n \n \n P<\/mml:mi>\n \n \u03b1\n \n <\/mml:mi>\n <\/mml:msub>\n P_\\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n that arises in the study of lattice rules.\n <\/p>","DOI":"10.1090\/s0025-5718-98-00894-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"299-322","source":"Crossref","is-referenced-by-count":553,"title":["A generalized discrepancy and quadrature error bound"],"prefix":"10.1090","volume":"67","author":[{"given":"Fred","family":"Hickernell","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"key":"2","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Methods of numerical integration","author":"Davis, Philip J.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0122063600","edition":"2"},{"key":"3","unstructured":"[FH95] K. T. Fang and F. J. Hickernell, The uniform design and its applications, Bulletin of the International Statistical Institute, 50th Session, Book 1 (Beijing), 1995, pp. 333\u2013349."},{"key":"4","series-title":"Monographs on Statistics and Applied Probability","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3095-8","volume-title":"Number-theoretic methods in statistics","volume":"51","author":"Fang, K.-T.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0412465205"},{"issue":"216","key":"5","doi-asserted-by":"publisher","first-page":"1621","DOI":"10.1090\/S0025-5718-96-00756-9","article-title":"Efficient algorithms for computing the \ud835\udc3f\u2082-discrepancy","volume":"65","author":"Heinrich, S.","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"[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\u2013227.","DOI":"10.1007\/978-1-4612-2552-2_13"},{"key":"7","doi-asserted-by":"crossref","unstructured":"[Hic96] F. J. Hickernell, Quadrature error bounds with applications to lattice rules, SIAM J. Numer. Anal. 33 (1996), 1995\u20132016.","DOI":"10.1137\/S0036142994261439"},{"issue":"6","key":"8","doi-asserted-by":"publisher","first-page":"1251","DOI":"10.1137\/0915077","article-title":"Quasi-random sequences and their discrepancies","volume":"15","author":"Morokoff, William J.","year":"1994","journal-title":"SIAM J. Sci. 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Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995, pp. 299\u2013317.","DOI":"10.1007\/978-1-4612-2552-2_19"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[SJ94] I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford University Press, Oxford, 1994.","DOI":"10.1093\/oso\/9780198534723.001.0001"},{"key":"13","volume-title":"{\\cyr Mnogomernye kvadraturnye formuly i funktsii Khaara}","author":"Tsobol\u2032, I. M.","year":"1969"},{"key":"14","first-page":"319","article-title":"Computational investigations of low-discrepancy point sets","author":"Warnock, Tony T.","year":"1972"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1090\/S0273-0979-1991-15985-9","article-title":"Average case complexity of multivariate integration","volume":"24","author":"Wo\u017aniakowski, H.","year":"1991","journal-title":"Bull. Amer. Math. Soc. 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