mathematics of computation volume 54,number 189 january 1990, pages 303-312 LATTICE RULES FOR MULTIPLE INTEGRATION AND DISCREPANCY HARALD NIEDERREITER AND IAN H. SLOAN Abstract. Upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established. In this way the applicability of lattice rules is extended to nonperiodic integrands. 1. Introduction Lattice rules for numerical integration over the s-dimensional unit cube [0, if were introduced by Sloan [11] and Sloan and Kachoyan [13], and the theory of lattice rules was developed further by Sloan and Walsh [12], Sloan and Kachoyan [14], and Sloan and Lyness [15]. An N-point lattice rule approx- imates the integral of a function / over [0, if by 1 A'"1 (1) A/£/W> N=0 with distinct nodes x0, ... , xjV_, e Us = [0, 1 )s for which the corresponding residue classes \0 + Zs, ... , *N_¡ + Z5 form a subgroup L of the torus group R!/Zs. Geometrically, this means that L = U„Jo (x« + ^) > considered as a subset of Rs, is a lattice in Rs, whence the name "lattice rule". The special case where L is a finite cyclic subgroup of Is /Zs yields the number-theoretic method of good lattice points due to Korobov [5] and Hlawka [3] (see also [4, 8] for expository accounts of this method and the recent survey in [10]). Lattice rules were originally conceived for the numerical integration of peri- odic functions having [0, if as their period interval, but the approximation ( 1) can of course also be used for nonperiodic integrands /. An upper bound for the integratiop error is obtained from the classical Koksma-Hlawka inequality [2] whenever the total variation V(f) of / over [0, if in the sense of Hardy and Krause is finite (compare also with [6, Chapter 2]). The resulting error bound is V(f)DN , where DN is the discrepancy of the nodes xQ, ... , xAr_, . We recall that the discrepancy of any points t0, ... , tN_, 6 Us is defined by dn = dn%> ••• >t/v-|) = sup Received December 20, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 65D32; Secondary 11H06, 11K38. ©1990 American Mathematical Society 0025-5718/90 $1.00+ $.25 per page A(J;N) N voi(y) 303 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use