Some properties of rank- lattice rules

Authors:
J. N. Lyness and I. H. Sloan

Journal:
Math. Comp. **53** (1989), 627-637

MSC:
Primary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1989-0982369-6

MathSciNet review:
982369

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Abstract: A rank-2 lattice rule is a quadrature rule for the (unit) *s*-dimensional hypercube, of the form

*f*, and , are integer vectors. In this paper we discuss these rules in detail; in particular, we categorize a special subclass, whose leading one- and two-dimensional projections contain the maximum feasible number of abscissas. We show that rules of this subclass can be expressed uniquely in a simple tricycle form.

**[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]**S. Haber, "Numerical evaluation of multiple integrals,"*SIAM Rev.*, v. 12, 1970, pp. 481-526. MR**0285119 (44:2342)****[3]**S. Haber, "Parameters for integrating periodic functions of several variables,"*Math. Comp.*, v. 41, 1983, pp. 115-129. MR**701628 (85g:65033)****[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, 1981. MR**617192 (83g:10034)****[6]**P. Keast, "Optimal parameters for multidimensional integration,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 831-838. MR**0353636 (50:6119)****[7]**N. M. Korobov, "Properties and calculation of optimal coefficients,"*Dokl. Akad. Nauk SSSR*, v. 132, 1960, pp. 1009-1012; English transl, in Soviet Math. Dokl., v. 1, 1960, pp. 696-700. MR**0120768 (22:11517)****[8]**D. Maisonneuve, "Recherche et utilisation des 'bons treillis'. Programmation et résultats numériques," in*Applications of Number Theory to Numerical Analysis*(S. K. Zaremba, ed.), Academic Press, London, 1972, pp. 121-201. MR**0343529 (49:8270)****[9]**H. Niederreiter, "Quasi-Monte Carlo methods and pseudo-random numbers,"*Bull. Amer. Math. Soc.*, v. 84, 1978, pp. 957-1041. MR**508447 (80d:65016)****[10]**I. H. Sloan, "Lattice methods for multiple integration,"*J. Comput Appl. Math.*, v. 12 and 13, 1985, pp. 131-143. MR**793949 (86f:65045)****[11]**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)****[12]**I. H. Sloan & J. N. Lyness, "The representation of lattice quadrature rules as multiple sums,"*Math. Comp.*, v. 52, 1989, pp. 81-94. MR**947468 (90a:65053)****[13]**S. K. Zaremba, "Good lattice points, discrepancy and numerical integration,"*Ann. Mat. Pura Appl.*, v. 73, 1966, pp. 293-317. MR**0218018 (36:1107)****[14]**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. 39-119. MR**0343530 (49:8271)**

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0982369-6

Article copyright:
© Copyright 1989
American Mathematical Society