An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules
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Abstract:
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 ${\mathbb {R}}^{s}$ and are generalizations of ‘number-theoretic’ rules introduced by Korobov (1959) and Hlawka (1962)—themselves 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 $\rho$, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order $N$ (that is, having $N$ nodes) then becomes that of finding rules with large values of $\rho$. 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.References
- N. S. Bahvalov, Approximate computation of multiple integrals, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 (1959), no. 4, 3–18 (Russian). MR 0115275
- M. Beckers and A. Haegemans, Transformation of integrands for lattice rules, Numerical Integration: Recent Developments, Software and Applications (T. O. Espelid and A. Genz, eds.), Kluwer Academic Publishers, Dordrecht, 1992, pp. 329–340.
- Marc Bourdeau and Alain Pitre, Tables of good lattices in four and five dimensions, Numer. Math. 47 (1985), no. 1, 39–43. MR 797876, DOI 10.1007/BF01389874
- A. J. Brentjes, Multidimensional continued fraction algorithms, Mathematical Centre Tracts, vol. 145, Mathematisch Centrum, Amsterdam, 1981. MR 638474
- Arne J. Brentjes, A two-dimensional continued fraction algorithm for best approximations with an application in cubic number fields, J. Reine Angew. Math. 326 (1981), 18–44. MR 622343, DOI 10.1515/crll.1981.326.18
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- T. W. Cusick, The Szekeres multidimensional continued fraction, Math. Comp. 31 (1977), no. 137, 280–317. MR 429765, DOI 10.1090/S0025-5718-1977-0429765-5
- L. E. Dickson, Studies in the Theory of Numbers, Chelsea, New York, 1961.
- Shaun Disney, Error bounds for rank $1$ lattice quadrature rules modulo composites, Monatsh. Math. 110 (1990), no. 2, 89–100. MR 1076324, DOI 10.1007/BF01302778
- Shaun Disney and Ian H. Sloan, Error bounds for the method of good lattice points, Math. Comp. 56 (1991), no. 193, 257–266. MR 1052090, DOI 10.1090/S0025-5718-1991-1052090-6
- Shaun Disney and Ian H. Sloan, Lattice integration rules of maximal rank formed by copying rank $1$ rules, SIAM J. Numer. Anal. 29 (1992), no. 2, 566–577. MR 1154284, DOI 10.1137/0729036
- K. K. Frolov, The connection of quadrature formulas and sublattices of the lattice of integer vectors, Dokl. Akad. Nauk SSSR 232 (1977), no. 1, 40–43 (Russian). MR 0427237
- Seymour Haber, Parameters for integrating periodic functions of several variables, Math. Comp. 41 (1983), no. 163, 115–129. MR 701628, DOI 10.1090/S0025-5718-1983-0701628-X
- Edmund Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140–151 (German). MR 143329, DOI 10.1007/BF01387711
- Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192
- S. Joe and S. A. R. Disney, Intermediate rank lattice rules for multidimensional integration, SIAM J. Numer. Anal. 30 (1993), no. 2, 569–582. MR 1211405, DOI 10.1137/0730027
- Gershon Kedem and S. K. Zaremba, A table of good lattice points in three dimensions, Numer. Math. 23 (1974), 175–180. MR 373239, DOI 10.1007/BF01459950
- A. Ya. Khinchin, Continued fractions, University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
- N. M. Korobov, Approximate evaluation of repeated integrals, Dokl. Akad. Nauk SSSR 124 (1959), 1207–1210 (Russian). MR 0104086
- N. M. Korobov, Properties and calculation of optimal coefficients, Soviet Math. Dokl. 1 (1960), 696–700. MR 0120768
- J. C. Lagarias, Some new results in simultaneous Diophantine approximation, Queen’s Papers in Pure and Applied Math. 54 (1980), 453–474.
- J. C. Lagarias, Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators, Trans. Amer. Math. Soc. 272 (1982), no. 2, 545–554. MR 662052, DOI 10.1090/S0002-9947-1982-0662052-7
- T. N. Langtry, The determination of canonical forms for lattice quadrature rules, J. Comput. Appl. Math. 59 (1995), no. 2, 129–143. MR 1346008, DOI 10.1016/0377-0427(94)00025-V
- J. N. Lyness and P. Keast, Application of the Smith normal form to the structure of lattice rules, SIAM J. Matrix Anal. Appl. 16 (1995), no. 1, 218–231. MR 1311428, DOI 10.1137/S089547989121793X
- J. N. Lyness and T. Sørevik, A search program for finding optimal integration lattices, Computing 47 (1991), no. 2, 103–120 (English, with German summary). MR 1139431, DOI 10.1007/BF02253429
- J. N. Lyness and T. Sørevik, An algorithm for finding optimal integration lattices of composite order, BIT 32 (1992), no. 4, 665–675. MR 1191020, DOI 10.1007/BF01994849
- J. N. Lyness and T. Sørevik, Lattice rules by component scaling, Math. Comp. 61 (1993), no. 204, 799–820. MR 1185247, DOI 10.1090/S0025-5718-1993-1185247-6
- Dominique Maisonneuve, Recherche et utilisation des “bons treillis”. Programmation et résultats numériques, Applications of number theory to numerical analysis (Proc. Sympos., Univ. Montréal, Montreal, Que., 1971) Academic Press, New York, 1972, pp. 121–201 (French, with English summary). MR 0343529
- Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI 10.1090/S0002-9904-1978-14532-7
- Harald Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
- Harald Niederreiter, The existence of efficient lattice rules for multidimensional numerical integration, Math. Comp. 58 (1992), no. 197, 305–314, S7–S16. MR 1106976, DOI 10.1090/S0025-5718-1992-1106976-5
- Harald Niederreiter, Improved error bounds for lattice rules, J. Complexity 9 (1993), no. 1, 60–75. Festschrift for Joseph F. Traub, Part I. MR 1213487, DOI 10.1006/jcom.1993.1005
- Harald Niederreiter and Ian H. Sloan, Integration of nonperiodic functions of two variables by Fibonacci lattice rules, J. Comput. Appl. Math. 51 (1994), no. 1, 57–70. MR 1286416, DOI 10.1016/0377-0427(92)00004-S
- Harald Niederreiter and Ian H. Sloan, Quasi-Monte Carlo methods with modified vertex weights, Numerical integration, IV (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 112, Birkhäuser, Basel, 1993, pp. 253–265. MR 1248409
- A. I. Saltykov, Tables for evaluating multiple integrals by the method of optimal coefficients, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 181–186 (Russian). MR 150976
- Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
- Ian H. Sloan, Lattice methods for multiple integration, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 131–143. MR 793949, DOI 10.1016/0377-0427(85)90012-3
- Ian H. Sloan, Numerical integration in high dimensions—the lattice rule approach, Numerical integration (Bergen, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 357, Kluwer Acad. Publ., Dordrecht, 1992, pp. 55–69. MR 1198898
- I. H. Sloan and S. Joe, Lattice Methods for Multiple Integration, Oxford University Press, Oxford, 1994.
- Ian H. Sloan and Philip J. Kachoyan, Lattice methods for multiple integration: theory, error analysis and examples, SIAM J. Numer. Anal. 24 (1987), no. 1, 116–128. MR 874739, DOI 10.1137/0724010
- Ian H. Sloan and James N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp. 52 (1989), no. 185, 81–94. MR 947468, DOI 10.1090/S0025-5718-1989-0947468-3
- Ian H. Sloan and Linda Walsh, A computer search of rank-$2$ lattice rules for multidimensional quadrature, Math. Comp. 54 (1990), no. 189, 281–302. MR 1001485, DOI 10.1090/S0025-5718-1990-1001485-4
- Jerome Spanier and Earl H. Maize, Quasi-random methods for estimating integrals using relatively small samples, SIAM Rev. 36 (1994), no. 1, 18–44. MR 1267048, DOI 10.1137/1036002
- G. Szekeres, Multidimensional continued fractions, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13 (1970), 113–140 (1971). MR 313198
- G. Szekeres, Search for the three-dimensional approximation constant, Diophantine analysis (Kensington, 1985) London Math. Soc. Lecture Note Ser., vol. 109, Cambridge Univ. Press, Cambridge, 1986, pp. 139–146. MR 874124
- R. T. Worley, On integration lattices, BIT 31 (1991), no. 3, 529–539. MR 1127489, DOI 10.1007/BF01933265
- S. C. Zaremba, Good lattice points, discrepancy, and numerical integration, Ann. Mat. Pura Appl. (4) 73 (1966), 293–317. MR 218018, DOI 10.1007/BF02415091
- S. K. Zaremba, La méthode des “bons treillis” pour le calcul des intégrales multiples, Applications of number theory to numerical analysis (Proc. Sympos., Univ. Montréal, Montreal, Que., 1971) Academic Press, New York, 1972, pp. 39–119 (French, with English summary). MR 0343530
- P. Zinterhof, Gratis lattice points for multidimensional integration, Computing 38 (1987), no. 4, 347–353 (English, with German summary). MR 902029, DOI 10.1007/BF02278712
Additional Information
- T. N. Langtry
- Affiliation: School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW, 2007, Australia
- Email: tim@maths.uts.edu.au
- Received by editor(s): February 22, 1995
- Received by editor(s) in revised form: July 26, 1995
- Additional Notes: This work was carried out as part of a doctoral program under the supervision of Prof. I. H. Sloan and Dr. S. A. R. Disney of the University of New South Wales. The author expresses his appreciation of their guidance and support. The comments of an anonymous referee also helped to improve the paper.
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 1635-1662
- MSC (1991): Primary 65D30; Secondary 65D32, 11J25, 11J70
- DOI: https://doi.org/10.1090/S0025-5718-96-00758-2
- MathSciNet review: 1351203