{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:38Z","timestamp":1776721958360,"version":"3.51.2"},"reference-count":49,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in\n \n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n {\\mathbb {R}}^{s}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and are generalizations of \u2018number-theoretic\u2019 rules introduced by Korobov (1959) and Hlawka (1962)\u2014themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit\n \n \n \n \n \u03c1\n \n <\/mml:mi>\n \\rho<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (that is, having\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n nodes) then becomes that of finding rules with large values of\u00a0\n \n \n \n \n \u03c1\n \n <\/mml:mi>\n \\rho<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.\n <\/p>","DOI":"10.1090\/s0025-5718-96-00758-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1635-1662","source":"Crossref","is-referenced-by-count":6,"title":["An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules"],"prefix":"10.1090","volume":"65","author":[{"given":"T.","family":"Langtry","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"4","key":"1","first-page":"3","article-title":"Approximate computation of multiple integrals","volume":"1959","author":"Bahvalov, N. S.","year":"1959","journal-title":"Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. 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