Algorithm 823: Implementing scrambled digital sequences

ACM Transactions on Mathematical Software

Volumn 29, Issue 2, 2003, Pages 95-109

  Hong, Hee Sun ^a   Hickernell, Fred J ^a  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

Digital net; Scrambling

Indexed keywords

CONVERGENCE RATES; DIGITAL NETS; DIGITAL SEQUENCES; SCRAMBLING;

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; ERROR DETECTION; MATHEMATICAL MODELS; MONTE CARLO METHODS;

RANDOM PROCESSES;

EID: 1842651631     PISSN: 00983500     EISSN: None     Source Type: Journal    
DOI: 10.1145/779359.779360     Document Type: Article

References (45)

1. ABRAMOWITZ, M. AND STEGUN, I. A., Eds. 1964. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. U. S. Government Printing Office, Washington, DC.
2. BRATLEY, P. AND FOX, B. L. 1988. Algorithm 659: Implementing Sobol's quasirandom sequence generator. ACM Trans. Math. Softw. 14, 88-100.
3. BRATLEY, P., FOX, B. L., AND NIEDERREITER, H. 1992. Implementation and tests of low-discrepancy sequences. ACM Trans. Model. Comput. Simul. 2, 195-213.
4. FANG, K. T., HICKERNELL, F. J., AND NIEDERREITER, H., Eds. 2002. Monte Carlo and Quasi-Monte Carlo Methods 2000. Springer-Verlag, Berlin, Germany.
5. FAURE, H. 1982. Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41, 337-351.
6. FAURE, H. AND TEZUKA, S. 2002. Another random scrambling of digital (t, s)-sequences. See Fang et al. [2002], 242-256.
7. FOX, B. L. 1999. Strategies for Quasi-Monte Carlo. Kluwer Academic Publishers, Boston, MA.
8. GENZ, A. 1982. A Lagrange extrpolation algorithm for sequences of approximations to multiple integrals. SIAM J. Sci. Stat. Comput. 3, 160-172.
9. GENZ, A. 1992. Numerical computation of multivariate normal probabilities. J. Comput. Graph. Statist. 1, 141-150.
10. GENZ, A. 1993. Comparison of methods for the computation of multivariate normal probabilities. Comput. Sci. and Statist. 25, 400-405.
11. HELLEKALEK, P. AND LARCHER, G., Eds. 1998. Random and Quasi-Random Point Sets. Lecture Notes in Statistics, vol. 138. Springer-Verlag, New York, NY.
12. HICKERNELL, F. J. 1996. The mean square discrepancy of randomized nets. ACM Trans. Model. Comput. Simul. 6, 274-296.
13. HICKERNELL, F. J. 1998. A generalized discrepancy and quadrature error bound. Math. Comp. 67, 299-322.
14. HICKERNELL, F. J. 1999. Goodness-of-fit statistics, discrepancies and robust designs. Statist. Probab. Lett. 44, 73-78.
15. HICKERNELL, F. J. 2000. What affects the accuracy of quasi-Monte Carlo quadrature? See Niederreiter and Spanier [2000], 16-55.
16. HICKERNELL, F. J. AND HONG, H. S. 1997. Computing multivariate normal probabilities using rank-1 lattice sequences. In Proceedings of the Workshop on Scientific Computing, G. H. Golub, S. H. Lui, F. T. Luk, and R. J. Plemmons, Eds. Springer-Verlag, Singapore, Hong Kong, 209-215.
17. HICKERNELL, F. J. AND HONG, H. S. 1999. The asymptotic efficiency of randomized nets for quadrature. Math. Comp. 68, 767-791.
18. HICKERNELL, F. J., HONG, H. S., L'ÉCUYER, P., AND LEMIEUX, C. 2000. Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J. Sci. Comput. 22, 1117-1138.
19. HICKERNELL, F. J. AND YUE, R. X. 2000. The mean square discrepancy of scrambled (t, s)-sequences. SIAM J. Numer. Anal. 38, 1089-1112.
20. KEISTER, B. D. 1996. Multidimensional quadrature algorithms. Comput. Phys. 10, 119-122.
21. LARCHER, G. 1998a. Digital point sets: Analysis and applications. See Hellekalek and Larcher [1998], 167-222.
22. LARCHER, G. 1998b. On the distribution of digital sequences. In Monte Carlo and quasi-Monte Carlo methods 1996, H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, Eds. Lecture Notes in Statistics, vol. 127. Springer-Verlag, New York, NY, 109-123.
23. LICHTNER, J. 1998. Iterating an α-ary gray code. SIAM J. Discrete Math. 11, 3, 381-386.
24. 2-discrepancy for anchored boxes. J. Complexity 14, 527-556.
25. MATOUŠEK, J. 1999. Geometric Discrepancy: An Illustrated Guide. Number 18 in Algorithms and Combinatorics. Springer-Verlag, Berlin, Germany.
26. MCNAMEE, J. AND STENGER, F. 1967. Construction of fully symmetric numerical integration formulas. Numer. Math. 10, 327-344.
27. NIEDERREITER, H. 1988. low-discrepancy and low dispersion sequences. J. Numb. Theor. 30, 51-70.
28. NIEDERREITER, H. 1992. Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA.
29. NIEDERREITER, H. AND SPANIER, J., Eds. 2000. Monte Carlo and Quasi-Monte Carlo Methods 1998. Springer-Verlag, Berlin, Germany.
30. NIEDERREITER, H. AND XING, C. 1996. Quasirandom points and global function fields. In Finite Fields and Applications, S. Cohen and H. Niederreiter, Eds. Number 233 in London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, England, 269-296.
31. NIEDERREITER, H. AND XING, C. 1998. Nets, (t, s)-sequences and algebraic geometry. See Hellekalek and Larcher [1998], 267-302.
32. OWEN, A. B. 1995. Randomly permuted (t, m, s)-nets and (t, s)-sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, H. Niederreiter and P. J.-S. Shiue, Eds. Lecture Notes in Statistics, vol. 106. Springer-Verlag, New York, NY, 299-317.
33. OWEN, A. B. 1997a. Monte Carlo variance of scrambled equidistribution quadrature. SIAM J. Numer. Anal. 34, 1884-1910.
34. OWEN, A. B. 1997b. Scrambled net variance for integrals of smooth functions. Ann. Stat. 25, 1541-1562.
35. OWEN, A. B. 1998. Scrambling Sobol' and Niederreiter-Xing points. J. Complexity 14, 466-489.
36. OWEN, A. B. 2000. Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo. See Niederreiter and Spanier [2000], 86-97.
37. PAPAGEORGIOU, A. AND TRAUB, J. F. 1997. Faster evaluation of multidimensional integrals. Comput. Phys. 11, 574-578.
38. PATTERSON, T. N. L. 1968. The optimum addition of points to quadrature formulae. Math. Comp. 22, 847-856.
39. PIRSIC, G. 2002. A software implementation of niederreiter-xing sequences. See Fang et al. [2002], 434-445.
40. SCHMID, W. C. 1999. The exact quality parameter of nets derived from Sobol' and Niederreiter sequences. In Recent Advances in Numerical Methods and Applications, O. P. Iliev et al., Eds. World Scientific, Sigapore, 287-295.
41. SOBOL', I. M. 1967. The distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Comput. Math. Math. Phys. 7, 86-112.
42. TEZUKA, S. 1995. Uniform Random Numbers: Theory and Practice. Kluwer Academic Publishers, Boston, MA.
43. YUE, R. X. 1999. Variance of quadrature over scrambled unions of nets. Statist. Sinica 9, 451-473.
44. YUE, R. X. AND HICKERNELL, F. J. 2002. The discrepancy and gain coefficients of scrambled digital nets. J. Complex. 18, 135-151.
45. YUE, R. X. AND MAO, S. S. 1999. On the variance of quadrature over scrambled nets and sequences. Statist. Probab. Lett. 44, 267-280.