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Bad Lattice Points

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An Erratum to this article was published on 01 February 2006

Abstract

We introduce and discuss the term “bad lattice points” which can be seen as a counterpart to the method of good lattice points for Monte Carlo and quasi-Monte Carlo integration. We show several examples of the occurrence of bad lattice points in the latter fields and perform a computer search for such point sets.

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References

  • Afflerbach, L., Gruber, G.: Assessment of random number generators in high accuracy. In: New directions in simulation for manufacturing and communications (Morito, S., Sakasegawa, H., Fushimi, M., Nakano, K., eds.), pp. 128–133. OR Society of Japan, 1994.

  • S. L. Anderson (1990) ArticleTitleRandom number generators on vector supercomputers and other advanced architectures SIAM Rev 32 221–251 Occurrence Handle10.1137/1032044 Occurrence Handle0708.65004 Occurrence Handle91g:65015

    Article  MATH  MathSciNet  Google Scholar 

  • R. F. Boisvert M. McClain B. Miller (1998) GAMS: The guide to available mathematical software National Institute of Standards and Technology Gaithersburg, MD, USA

    Google Scholar 

  • Cohen, H.: A course in computational algebraic number theory. Graduate Texts in Mathematics, vol. 38. Springer 1993.

  • R. Couture P. L’Ecuyer (1994) ArticleTitleOn the lattice structure of certain linear congruential sequences related to AWC/SWB generators Math. Comp. 62 799–808 Occurrence Handle94g:65007

    MathSciNet  Google Scholar 

  • Couture, R., L’Ecuyer, P.: Linear recurrences with carry as uniform random number generators. In: Proc. 1995 Winter Simulation Conf. (Alexopoulos, C., Goldsman, D., Kang, K., Lilegdon, W. R., eds.), pp. 263–267 (1995).

  • R. R. Coveyou R. D. MacPherson (1967) ArticleTitleFourier analysis of uniform random number generators J. Assoc. Comput. Mach. 14 100–119 Occurrence Handle36 #4779

    MathSciNet  Google Scholar 

  • A. DeMatteis J. Eichenauer-Herrmann H. Grothe (1992) ArticleTitleComputation of critical distances within multiplicative congruential pseudorandom number sequences J. Comp. Appl. Math. 39 49–55 Occurrence Handle93b:65011

    MathSciNet  Google Scholar 

  • A. DeMatteis S. Pagnutti (1988) ArticleTitleParallelization of random number generators and long-range correlations Numer. Math. 53 595–608 Occurrence Handle89h:65016

    MathSciNet  Google Scholar 

  • A. DeMatteis S. Pagnutti (1993) ArticleTitleLong-range correlation analysis of the Wichmann-Hill random number generator Stat. Comput. 3 67–70

    Google Scholar 

  • A. DeMatteis S. Pagnutti (1995) ArticleTitleControlling correlations in parallel Monte Carlo Parallel Comput. 21 73–84 Occurrence Handle1314379

    MathSciNet  Google Scholar 

  • U. Dieter (1975) ArticleTitleHow to calculate shortest vectors in a lattice Math. Comp. 29 IssueID131 827–833 Occurrence Handle0306.10012 Occurrence Handle52 #291

    MATH  MathSciNet  Google Scholar 

  • J. Eichenauer-Herrmann H. Grothe (1989) ArticleTitleA remark on long-range correlations in multiplicative congruential pseudo random number generators Numer. Math. 56 609–611 Occurrence Handle10.1007/BF01396346 Occurrence Handle90m:65017

    Article  MathSciNet  Google Scholar 

  • J. Eichenauer-Herrmann H. Grothe (1990) ArticleTitleUpper bounds for the beyer ratios of linear congruential generators J. Comput. Appl. Math. 31 IssueID1 73–80 Occurrence Handle91i:65011

    MathSciNet  Google Scholar 

  • K. Entacher (1998) ArticleTitleBad subsequences of well-known linear congruential pseudorandom number generators ACM Trans. Modeling Comput. Simul. 7 IssueID1 61–70 Occurrence Handle1622246

    MathSciNet  Google Scholar 

  • K. Entacher (1999) ArticleTitleParallel streams of linear random numbers in the spectral test ACM Trans. Modeling Comput. Simul. 9 IssueID1 31–44

    Google Scholar 

  • Entacher, K., Hellekalek, P., L’Ecuyer, P.: Quasi–Monte Carlo node sets from linear congruential generators. In: Monte Carlo and quasi–Monte Carlo methods 1998. Berlin: Springer 2000, pp 188–198.

  • K. Entacher Th. Schell A. Uhl (2002) ArticleTitleEfficient lattice assessment for LCG and GLP parameter searches Math. Comp. 71 1231–1242 Occurrence Handle10.1090/S0025-5718-01-01415-6 Occurrence Handle2003c:11088

    Article  MathSciNet  Google Scholar 

  • Entacher, K., Uhl, A., Wegenkittl, S.: Parallel random number generation: long-range correlations among multiple processors. In: Parallel computation (Zinterhof, P., Vajteršic, M., Uhl, A., eds.). Proc. 4th Int. Conf. of the ACPC (ACPC99), Lecture Notes in Computer Science, pp. 107–116. Springer 1999.

  • A. M. Ferrenberg D. P. Landau Y. J. Wong (1992) ArticleTitleMonte Carlo simulations: hidden errors from ‘‘good’’ random number generators Phys. Rev. Lett. 69 3382–3384 Occurrence Handle10.1103/PhysRevLett.69.3382

    Article  Google Scholar 

  • U. Fincke M. Pohst (1985) ArticleTitleImproved methods for calculating vectors of short length in a lattice, including a complexity analysis Math. Comp. 44 463–471 Occurrence Handle86e:11050

    MathSciNet  Google Scholar 

  • Fishman, G.S.: Monte Carlo: Concepts, algorithms, and applications. Springer Series in Operations Research, vol. 1. New York: Springer 1996

  • S. Haber (1983) ArticleTitleParameters for integrating periodic functions of several variables Math. Comp. 41 IssueID163 115–129 Occurrence Handle0532.65015 Occurrence Handle85g:65033

    MATH  MathSciNet  Google Scholar 

  • P. Hellekalek (1998) ArticleTitleGood random number generators are (not so) easy to find Math. Comp. Simul. 46 485–505 Occurrence Handle0931.65001 Occurrence Handle1638550

    MATH  MathSciNet  Google Scholar 

  • Hellekalek, P., Larcher, G.: Random and quasi-random point sets. Lecture Notes in Statistics, vol. 138. Berlin: Springer 1998

  • P. Hellekalek H. Niederreiter (1998) ArticleTitleThe weighted spectral test: Diaphony ACM Trans. Modeling Comput. Simul. 8 43–60

    Google Scholar 

  • Hickernell, F. J.: Lattice rules: How well do they measure up? In [25], pp. 109–166.

  • J. Honerkamp (1990) Stochastische dynamische Systeme VCH Verlagsgesellschaft Weinheim

    Google Scholar 

  • D. W. Hutchinson (1966) ArticleTitleA new uniform pseudo-random number generator Commun. ACM 9 IssueID6 432–433 Occurrence Handle10.1145/365696.365712 Occurrence Handle0141.14603 Occurrence Handle33 #1954

    Article  MATH  MathSciNet  Google Scholar 

  • R. Jain (1991) The art of computer systems performance analysis Wiley New York

    Google Scholar 

  • Jansson, B.: Random number generators. PhD thesis, University of Stockholm, 1966. Victor Pettersons Bokindustri AB (also published by Almqvist and Wiksell).

  • C. Kao J. Y. Wong (1998) ArticleTitleRandom number generators with long period and sound statistical properties Comp. Math. Appl. 36 IssueID3 113–121 Occurrence Handle10.1016/S0898-1221(98)00133-3 Occurrence Handle99c:65009

    Article  MathSciNet  Google Scholar 

  • W. J. Kennedy J. E. Gentle (1980) Statistical computing Dekker New York

    Google Scholar 

  • Knuth, D. E.: The art of computer programming, vol. 2: Seminumerical algorithms, 2nd ed. Reading, MA: Addison-Wesley 1981.

  • N. M. Korobov (1963) Number-theoretic methods in approximate analysis (in Russian) Fizmatgiz Moscow

    Google Scholar 

  • Law, A. M., Kelton, W. D.: Simulation modeling and analysis, 2nd ed.. New York: McGraw-Hill 1991.

  • L’Ecuyer, P.: Testing random number generators. In: Proc. 1992 Winter Simulation Conf., pp. 305–313. IEEE Press 1992.

  • P. L’Ecuyer (1994) ArticleTitleUniform random number generation Ann. Oper. Res. 53 77–120 Occurrence Handle95k:65007

    MathSciNet  Google Scholar 

  • L’Ecuyer, P.: Software for uniform random number generation: Distinguishing the good and the bad. In: Proc. 2001 Winter Simulation Conf. 2001.

  • P. L’Ecuyer (1997) ArticleTitleBad lattice structures for vectors of non-successive values produced by some linear recurrences INFORMS J. Comp. 9 57–60 Occurrence Handle98f:65014

    MathSciNet  Google Scholar 

  • P. L’Ecuyer (1999) ArticleTitleGood parameter sets for combined multiple recursive random number generators Operations Res. 47 159–164

    Google Scholar 

  • P. L’Ecuyer (1999) ArticleTitleTables of linear congruential generators of different sizes and good lattice structure Math. Comp. 68 IssueID225 249–260 Occurrence Handle99c:11101

    MathSciNet  Google Scholar 

  • P. L’Ecuyer F. Blouin R. Couture (1993) ArticleTitleA search for good multiple recursive generators ACM Trans. Modeling Comput. Simul. 3 87–98

    Google Scholar 

  • P. L’Ecuyer R. Couture (1997) ArticleTitleAn implementation of the lattice and spectral tests for multiple recursive linear random number generators INFORMS J. Comp. 9 IssueID2 209–217 Occurrence Handle1477315

    MathSciNet  Google Scholar 

  • C. Lemieux P. L’Ecuyer (2001) ArticleTitleOn selection criteria for lattice rules and other quasi-Monte Carlo point sets Math. Comp. Simul. 55 139–148 Occurrence Handle2001m:65010

    MathSciNet  Google Scholar 

  • P. A. Lewis A. S. Goodman J. M. Miller (1969) ArticleTitleA pseudo-random number generator for the system/360 IBM Syst. J. 8 136–146

    Google Scholar 

  • G. Marsaglia (1972) The structure of linear congruential sequences S. K. Zaremba (Eds) Applications of number theory to numerical analysis Academic Press New York 248–285

    Google Scholar 

  • G. Marsaglia B. Narasimhan A. Zaman (1990) ArticleTitleA random number generator for PC’s Comp. Phys. Comm. 60 345–349 Occurrence Handle1076268

    MathSciNet  Google Scholar 

  • G. Marsaglia A. Zaman (1991) ArticleTitleA new class of random number generators Ann. Appl. Probability 1 462–480 Occurrence Handle92h:65009

    MathSciNet  Google Scholar 

  • B. J. T. Morgan (1986) Elements of simulation Chapman and Hall London New York

    Google Scholar 

  • H. R. Neave (1973) ArticleTitleOn using the Box-Müller transformation with multiplicative congruential pseudo-random number generators Appl. Stat. 22 92–97

    Google Scholar 

  • H. Niederreiter (1978) ArticleTitleQuasi–Monte Carlo methods and pseudo-random numbers Bull. Amer. Math. Soc. 84 957–1041 Occurrence Handle0404.65003 Occurrence Handle80d:65016

    MATH  MathSciNet  Google Scholar 

  • H. Niederreiter (1992) Random number generation and quasi–Monte Carlo methods SIAM Philadelphia

    Google Scholar 

  • Niederreiter, H. et al. (eds.): Monte Carlo and quasi–Monte Carlo methods 1996, 1998, 2000, 2002. Proc. for the Conferences MCQMC 1996–2002. Springer.

  • S. K. Park K. W. Miller (1988) ArticleTitleRandom number generators: good ones are hard to find Comm. ACM 31 1192–1201 Occurrence Handle10.1145/63039.63042 Occurrence Handle91h:65012

    Article  MathSciNet  Google Scholar 

  • B. D. Ripley (1983) ArticleTitleThe lattice structure of pseudo-random number generators Proc. Roy. Soc. London Ser. A 389 197–204 Occurrence Handle0516.65003 Occurrence Handle85i:65010

    MATH  MathSciNet  Google Scholar 

  • B. D. Ripley (1990) ArticleTitleThoughts on pseudorandom number generators J. Comput. Appl. Math. 31 153–163 Occurrence Handle0701.65006

    MATH  Google Scholar 

  • S. M. Ross (1990) A course in simulation Macmillan New York

    Google Scholar 

  • A. Rotenberg (1960) ArticleTitleA new pseudo-random number generator J. Assoc. Comp. Mach. 7 75–77 Occurrence Handle0096.33902 Occurrence Handle22 #8642

    MATH  MathSciNet  Google Scholar 

  • I. H. Sloan S. Joe (1994) Lattice methods for multiple integration Oxford University Press New York

    Google Scholar 

  • I. H. Sloan P. J. Kachoyan (1987) ArticleTitleLattice methods for multiple integration: Theory, error analysis and examples SIAM J. Numer. Anal. 24 116–128 Occurrence Handle10.1137/0724010 Occurrence Handle88e:65023

    Article  MathSciNet  Google Scholar 

  • Tezuka, S.: Uniform random numbers: Theory and practice. Kluwer Academic Publishers 1995.

  • S. Tezuka P. L’Ecuyer R. Couture (1993) ArticleTitleOn add-with-carry and subtract-with-borrow random number generators ACM Trans. Modeling Comput. Simul. 3 315–331

    Google Scholar 

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Correspondence to K. Entacher.

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An erratum to this article is available at http://dx.doi.org/10.1007/s00607-005-0156-9.

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Entacher, K., Schell, T. & Uhl, A. Bad Lattice Points. Computing 75, 281–295 (2005). https://doi.org/10.1007/s00607-004-0105-z

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