The representation of lattice quadrature rules as multiple sums

Authors:
Ian H. Sloan and James N. Lyness

Journal:
Math. Comp. **52** (1989), 81-94

MSC:
Primary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1989-0947468-3

MathSciNet review:
947468

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Abstract: We provide a classification of lattice rules. Applying elementary group theory, we assign to each *s*-dimensional lattice rule a *rank m* and a set of positive integer invariants . The number of abscissas required by the rule is the product , and the rule may be expressed in a canonical form with *m* independent summations. Under this classification an *N*-point number-theoretic rule in the sense of Korobov and Conroy is a rank rule having invariants *N*, 1, 1,..., 1, and the product trapezoidal rule using points is a rank rule having invariants *n, n,..., n*. Besides providing a canonical form, we give some of the properties of copy rules and of projections into lower dimensions.

**[1]**H. Conroy, "Molecular Schrödinger equation, VIII: A new method for the evaluation of multidimensional integrals,"*J. Chem Phys.*, v. 47, 1967, pp. 5307-5318.**[2]**R. Cranley & T. N. L. Patterson, "Randomization of number-theoretic methods for multiple-integration,"*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 904-914. MR**0494820 (58:13605)****[3]**M. Hall, Jr.,*The Theory of Groups*, Macmillan, New York, 1959. MR**0103215 (21:1996)****[4]**E. Hlawka, "Zur angenäherten Berechnung mehrfacher Integrale,"*Monatsh. Math.*, v. 66, 1962, pp. 140-151. MR**0143329 (26:888)****[5]**Hua Loo Keng & Wang Yuan,*Applications of Number Theory to Numerical Analysis*, Springer-Verlag, Berlin, Science Press, Beijing, 1981. MR**617192 (83g:10034)****[6]**T. W. Hungerford,*Algebra*, Springer-Verlag, New York, 1974. MR**600654 (82a:00006)****[7]**P. Keast, "Optimal parameters for multidimensional integration,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 831-838. MR**0353636 (50:6119)****[8]**N. M. Korobov, "The approximate computation of multiple integrals,"*Dokl. Akad. Nauk SSSR*, v. 124, 1959, pp. 1207-1210. (Russian) MR**0104086 (21:2848)****[9]**W. Ledermann,*Introduction to the Theory of Finite Groups*, Oliver and Boyd, Edinburgh, 1964. MR**0054593 (14:945c)****[10]**H. Niederreiter, "Quasi-Monte Carlo methods and pseudo-random numbers,"*Bull. Amer. Math. Soc.*, v. 84, 1978, pp. 957-1041. MR**508447 (80d:65016)****[11]**I. H. Sloan, "Lattice methods for multiple integration,"*J. Comput. Appl. Math.*, v. 12 and 13, 1985, pp. 131-143. MR**793949 (86f:65045)****[12]**I. H. Sloan & P. J. Kachoyan, "Lattice methods for multiple integration: theory, error analysis and examples,"*SIAM J. Numer. Anal.*, v. 24, 1987, pp. 116-128. MR**874739 (88e:65023)****[13]**S. K. Zaremba, "La méthode des "bons treillis" pour le calcul des intégrales multiples," in*Applications of Number Theory to Numerical Analysis*(S. K. Zaremba, ed.), Academic Press, London, 1972, pp. 39-116. MR**0343530 (49:8271)**

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0947468-3

Article copyright:
© Copyright 1989
American Mathematical Society