Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Quasi-Monte Carlo methods and pseudo-random numbers


Author: Harald Niederreiter
Journal: Bull. Amer. Math. Soc. 84 (1978), 957-1041
MSC (1970): Primary 65-02, 65C05, 65C10, 65D30, 10F40, 10K05; Secondary 10-02, 10A35, 10F10, 10F20, 10G05, 10K30, 12A15, 65D05, 65N05, 65R05, 68A55
DOI: https://doi.org/10.1090/S0002-9904-1978-14532-7
MathSciNet review: 508447
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. J. H. Ahrens, U. Dieter and A. Grube, Pseudo-random numbers: A new proposal for the choice of multiplicators, Computing 6 (1970), 121-138. MR 279958
  • 2. J. L. Altaber, Représentations arithmétiques de grandeurs aléatoires, Ann. Fac. Sci. Univ. Clermont-Ferrand 37 (1967), 1-61. MR 364060
  • 3. I. I. Artobolevskiĭ, M. D. Genkin, V. K. Grinkevič, I. M. Sobol' and R. B. Statnikov, Optimization in the theory of machines by an LP-search, Dokl. Akad. Nauk SSSR 200 (1971), 1287-1290. (Russian)
  • 4. K. I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Dokl. Akad. Nauk SSSR 132 (1960), 982-985 = Soviet Math. Dokl. 1 (1960), 672-675. MR 121607
  • 5. N. S. Bahvalov, Approximate computation of multiple integrals, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him.1959, no. 4, 3-18. (Russian) MR 115275
  • 6. N. S. Bahvalov, Numerical solution of the Dirichlet problem for Laplace's equation, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him.1959, no. 5, 171-195. (Russian) MR 115280
  • 7. N. S. Bahvalov, An estimate of the main remainder term in quadrature formulae, Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961), 64-77 = U.S.S.R. Computational Math. and Math. Phys. 1 (1961), 68-82. MR 136068
  • 8. N. S. Bahvalov, On a rate of convergence of indeterministic integration processes within the functional classes W, Teor. Verojatnost. i Primenen.7 (1962), 238 = Theor. Probability Appl. 7 (1962), 227.
  • 9. N. S. Bahvalov, On the convergence of indeterministic integration processes on slightly smooth functions, Teor. Verojatnost. i Primenen.7 (1962), 473-474 = Theor. Probability Appl. 7 (1962), 463.
  • 10. N. S. Bahvalov, Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions, Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), no. 4, suppl., 5-63. (Russian) MR 172463
  • 11. N. S. Bahvalov, N. M. Korobov and N. N. Čencov, The application of number-theoretic nets to numerical analysis problems, Proc. Fourth All-Union Math. Congr. (Leningrad, 1961), vol. 2, Izdat. "Nauka", Leningrad, 1964, pp. 580-587. (Russian) MR 221724
  • 12. A. Baker, On some diophantine inequalities involving the exponential function, Canad. J. Math. 17 (1965), 616-626. MR 177946
  • 13. C. T. H. Baker, On the nature of certain quadrature formulas and their errors, SIAM J. Numer. Anal. 5 (1968), 783-804. MR 245200
  • 14. J. Bass, Nombres aléatoires, suites arithmétiques, méthode de Monte-Carlo, Publ. Inst. Statist. Univ. Paris 9 (1960), 289-325. MR 134451
  • 15. J. Bass, Stationary functions and their applications to turbulence. I. Stationary functions, II. Turbulent solutions of the Navier-Stokes equations, J. Math. Anal. Appl. 47 (1974), 354-399, 458-503.
  • 16. J. Bass and J. Guilloud, Méthode de Monte-Carlo et suites uniformément denses, Chiffres 1 (1958), 151-155. MR 100338
  • 17. K. Bauknecht, J. Kohlas and C. A. Zehnder, Simulationstechnik, Springer-Verlag, Berlin and New York, 1976. MR 475045
  • 18. J.-P. Bertrandias, Calcul d'une intégrale au moyen de la suite X = An. Evaluation de l'erreur, Publ. Inst. Statist. Univ. Paris 9 (1960), 335-357. MR 136047
  • 19. W. A. Beyer, Lattice structure and reduced bases of random vectors generated by linear recurrences, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 361-370. MR 351032
  • 20. W. A. Beyer, R. B. Roof and D. Williamson, The lattice structure of multiplicative congruential pseudo-random vectors, Math. Comp. 25 (1971), 345-363. MR 309263
  • 21. P. Billingsley and F. Topsøe, Uniformity in weak convergence, Z. Wahrscheinlichkeitstheorie verw. Gebiete 7 (1967), 1-16. MR 209428
  • 22. C. Binder, Über einen Satz von de Bruijn und Post, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 179 (1970), 233-251. MR 296224
  • 23. I. Borosh, Rational continued fractions with small partial quotients (preprint).
  • 24. I. Borosh and H. Niederreiter, Optimal multipliers for pseudo-random number generation by the linear congruential method (to appear).
  • 25. G. W. Brown, Monte Carlo methods, E. F. Beckenbach (ed.), Modern Mathematics for the Engineer, McGraw-Hill, New York, 1956, Chapter 12.
  • 26. O. V. Brušlinskaja, Practical applications of the method of optimal coefficients to the computation of multiple integrals, Questions of Computational Mathematics and Computing Technology (L. A. Ljusternik, ed.), Gos. Naučno-Tehn. Izdat. Mašinostr. Lit., Moscow, 1963, pp. 45-48. (Russian)
  • 27. N. P. Buslenko, Mathematical modeling of production processes on digital computers, Izdat. "Nauka", Moscow, 1964; German transl., Simulation von Produktionsprozessen, Teubner, Leipzig, 1971. MR 353950
  • 28. N.P.Buslenko, D.I. Golenko, I. M. Sobol', V. G. Sragovič and Ju. A. Šreĭder, The method of statistical trials (Monte Carlo method), Fizmatgiz, Moscow, 1962 = The Monte Carlo method, Yu. A. Shreider (ed.), Pergamon Press, Oxford, 1966. MR 211577
  • 29. V. V. Bykov, Digital simulation and statistical radio engineering, Izdat. "Sov. Radio", Moscow, 1971. (Russian)
  • 30. J. W. S. Cassels, An extension of the law of the iterated logarithm, Proc. Cambridge Philos. Soc. 47 (1951), 55-64. MR 40614
  • 31. G. Cenacchi and A. de Matteis, Pseudo-random numbers for comparative Monte Carlo calculations, Numer. Math. 16 (1970), 11-15. MR 275626
  • 32. G. Cenacchi and A. de Matteis, Quasi-random sequences by power residues, Numer. Math. 20 (1972), 54-63. MR 438650
  • 33. N. N. Čencov, On quadrature formulae for functions of an infinitely large number of variables, Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961), 418-424 = U.S.S.R. Computational Math. and Math. Phys. 1 (1961), 455-464. MR 138918
  • 34. N. N. Čencov, Pseudorandom numbers for modeling Markov chains, Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 632-643 = U.S.S.R. Computational Math. and Math. Phys. 7 (1967), no. 3, 218-233. MR 232529
  • 35. G. J. Chaitin, On the length of programs for computing finite binary sequences, J. Assoc. Comput Mach. 13 (1966), 547-569. MR 210520
  • 36. G. J. Chaitin, Randomness and mathematical proof, Sci. Amer. 232 (1975), no. 5, 47-52.
  • 37. C. K. Chui, A convergence theorem for certain Riemann sums, Canad. Math. Bull. 12 (1969), 523-525. MR 248297
  • 38. C. K. Chui, Concerning rates of convergence of Riemann sums, J. Approximation Theory 4 (1971), 279-287. MR 289731
  • 39. C. K. Chui, Convergence of certain quadrature processes, Aequationes Math. 9 (1973), 242-244. MR 329206
  • 40. K. L. Chung, An estimate concerning the Kolmogoroff limit distribution, Trans. Amer. Math. Soc. 67 (1949), 36-50. MR 34552
  • 41. A. Church, On the concept of a random sequence, Bull. Amer. Math. Soc. 46 (1940), 130-135. MR 911
  • 42. H. Conroy, Molecular Schrödinger equation. VIII: A new method for the evaluation of multidimensional integrals, J. Chemical Phys. 47 (1967), 5307-5318.
  • 43. A. H. Copeland, Admissible numbers in the theory of probability, Amer. J. Math. 50 (1928), 535-552. MR 1506687
  • 44. J. Couot, Applications des suites mθ à l'intégration numérique, C. R. Acad. Sci. Paris Sér. A 264 (1967), 183-186. MR 207204
  • 45. J. Couot, Applications des suites mθ à l'intégration multiple sur le tore, C. R. Acad. Sci. Paris Sér. A 266 (1968), 131-134. MR 232530
  • 46. R. R. Coveyou, Serial correlation in the generation of pseudo-random numbers, J. Assoc. Comput. Mach. 7 (1960), 72-74. MR 117869
  • 47. R. R. Coveyou, Random number generation is too important to be left to chance, Studies in Appl. Math., vol. 3, Soc. Industr. Appl. Math., Philadelphia, Pa., 1969, pp. 70-111.
  • 48. R. R. Coveyou and R. D. MacPherson, Fourier analysis of uniform random number generators, J. Assoc. Comput. Mach. 14 (1967), 100-119. MR 221727
  • 49. R. Cranley and T. N. L. Patterson, Randomization of number theoretic methods for multiple integration, SIAM J. Numer. Anal. 13 (1976), 904-914. MR 494820
  • 50. H. Davenport, Note on irregularities of distribution, Mathematika 3 (1956), 131-135. MR 82531
  • 51. P. J. Davis, On the numerical integration of periodic analytic functions, On Numerical Approximation (Proc. Sympos. Math. Research Center, Madison, Wis., 1958), R. E. Langer, ed., Univ. of Wisconsin Press, Madison, Wis., 1959, pp. 45-59. MR 100354
  • 52. P. J. Davis and P. Rabinowitz, Some Monte Carlo experiments in computing multiple integrals, Math. Tables Aids Comput. 10 (1956), 1-8. MR 76451
  • 53. P. J. Davis and P. Rabinowitz, Methods of numerical integration, Academic Press, New York, 1975. MR 448814
  • 54. N. G. de Bruijn and K. A. Post, A remark on uniformly distributed sequences and Riemann integrability, Nederl. Akad. Wetensch. Proc. Ser. A 71 (1968) = Indag. Math. 30 (1968), 149-150. MR 225946
  • 55. R. Devillers, J. J. Dumont and G. Latouche, Tests de générateurs pseudo-aléatoires, Acad. Roy. Belg. Bull. Cl. Sci. (5) 59 (1973), 703-724. MR 383695
  • 56. U. Dieter, Autokcorrelation multiplikativ erzeugter Pseudo-Zufallszahlen, Operations Research-Verfahren 6 (1969), 69-85.
  • 57. U. Dieter, Pseudo-random numbers: The exact distribution of pairs, Math. Comp. 25 (1971), 855-883. MR 298727
  • 58. U. Dieter, Statistical interdependence of pseudo-random numbers generated by the linear congruential method, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 287-317. MR 353622
  • 59. U. Dieter, Pseudo-random numbers: Permutations of triplets, unpublished manuscript.
  • 60. U. Dieter and J. Ahrens, An exact determination of serial correlation of pseudo-random numbers, Numer. Math. 17 (1971), 101-123. MR 286245
  • 61. D. Y. Downham and F. D. K. Roberts, Multiplicative congruential pseudo-random number generators, Comput. J. 10 (1967), 74-77.
  • 62. H. J. A. Duparc, C. G. Lekkerkerker and W. Peremans, Reduced sequences of integers and pseudo-random numbers, Report ZW 1953-002, Math. Centrum, Amsterdam, 1953. MR 53417
  • 63. P. D. T. A. Elliott, On distribution functions (mod 1): Quantitative Fourier inversion, J. Number Theory 4 (1972), 509-522. MR 311588
  • 64. P. Erdös and P. Turán, On a problem in the theory of uniform distribution. I, Nederl. Akad Wetensch. Proc. 51 (1948), 1146-1154 = Indag. Math. 10 (1948), 370-378. MR 27895
  • 65. S. M. Ermakov, Note on pseudorandom sequences, Ž. Vyčisl. Mat. i Mat. Fiz. 12 (1972), 1077-1082 = U.S.S.R. Computational Math. and Math. Phys. 12 (1972), no. 4, 307-314. MR 329191
  • 66. S. M. Ermakov, The Monte Carlo method and related questions, 2nd ed., Izdat. "Nauka", Moscow, 1975; German transl, of 1st ed., Die Monte-Carlo-Methode und verwandte Fragen, Oldenbourg Verlag, Munich-Vienna, 1975. MR 433799
  • 67. M. Esmenjaud-Bonnardel, Un procédé de génération de nombres "pseudo-aléatoires" pourCAB 500, Rev. Française Traitement Information 7 (1964), 185-197. MR 177498
  • 68. B. M. Fellen, An implementation of the Tausworthe generator, Comm. ACM 12 (1969), 413.
  • 69. G. S. Fishman, Concepts and methods in discrete event digital simulation, Wiley, New York, 1973.
  • 70. L. D. Fosdick, The Monte Carlo method in quantum statistics, SIAM Rev. 10 (1968), 315-328. MR 233488
  • 71. J. N. Franklin, On the equidistribution of pseudo-random numbers, Quart. Appl. Math. 16 (1958), 183-188. MR 93501
  • 72. J. N. Franklin, Deterministic simulation of random processes, Math. Comp. 17 (1963), 28-59. MR 149640
  • 73. J. N. Franklin, Numerical simulation of stationary and nonstationary Gaussian random processes, SIAM Rev. 7 (1965), 68-80. MR 179913
  • 74. K. K. Frolov, Upper error bounds for quadrature formulas on function classes, Dokl. Akad. Nauk SSSR 231 (1976), 818-821 = Soviet Math. Dokl. 17 (1976), 1665-1669. MR 427922
  • 75. H. Gabai, On the discrepancy of certain sequences mod 1, Nederl. Akad. Wetensch. Proc. Ser. A 66 (1963) = Indag. Math. 25 (1963), 603-605. MR 155812
  • 76. H. Gabai, On the discrepancy of certain sequences mod 1, Illinois J. Math. 11 (1967), 1-12. MR 209252
  • 77. I. M. Gel'fand, S. M. Feĭnberg, A. S. Frolov and N. N. Čencov, On application of the method of random trials (Monte Carlo method) for the solution of a kinetic equation, Proc. 2nd Internat. Conf. on the Peaceful Uses of Atomic Energy (Geneva, 1958), vol. 2, Atomizdat, Moscow, 1959, pp. 628-633. (Russian)
  • 78. I. M. Gel'fand, A. S. Frolov and N. N. Čencov, The computation of continuous integrals by the Monte Carlo method, Izv. Vysš. Učebn. Zaved. Matematika 1958, no. 5, 32-45. (Russian) MR 135694
  • 79. V. S. Gladkiĭ, Probabilistic computational models, Izdat. "Nauka", Moscow, 1973. (Russian) MR 362824
  • 80. D. I. Golenko, Simulation and statistical analysis of pseudo-random numbers on electronic computers, Izdat. "Nauka", Moscow, 1965. (Russian) MR 187370
  • 81. S. W. Golomb, Sequences with randomness properties, Glenn L. Martin Co. Report, Baltimore, Md., 1955.
  • 82. I. J. Good and R. A. Gaskins, Some relationships satisfied by additive and multiplicative recurrent congruential sequences, with implications for pseudorandom number generation, Computers in Number Theory (A. O. L. Atkin and B. J. Birch, eds.), Academic Press, London, 1971, pp. 125-136. MR 319876
  • 83. L. K. Gorskiĭ, Statistical algorithms for investigating reliability, Izdat. "Nauka", Moscow, 1970. (Russian)
  • 84. B. L. Granovskiĭ and S. M. Ermakov, The Monte Carlo method, Itogi Nauki i Tekhniki 13 (1976), 59-108 = J. Soviet Math. 7 (1977), 161-192. MR 451620
  • 85. B. F. Green, Jr., J. E. K. Smith and L. Klem, Empirical tests of an additive random number generator, J. Assoc. Comput. Mach. 6 (1959), 527-537. MR 107957
  • 86. M. Greenberger, An a priori determination of serial correlation in computer generated random numbers, Math. Comp. 15 (1961), 383-389; Corrigenda, ibid. 16 (1962), 126, 406. MR 144489
  • 87. M. Greenberger, Method in randomness, Comm. ACM 8 (1965), 177-179.
  • 88. J. A. Greenwood, A fast machine-independent long-period generator for 31-bit pseudorandom integers, Compstat 1976: Proceedings in Computational Statistics (J. Gordesch and P. Naeve, eds.), Physica-Verlag, Vienna, 1976, pp. 30-37.
  • 89. A. Grube, Mehrfach rekursiv-erzeugte Pseudo-Zufallszahlen, Z. Angew. Math. Mech. 53 (1973), T223-T225. MR 351033
  • 90. V. S. Gubenko, N. E. Kirillov, K. A. Meškovskiĭ and A. I. Čerkunov, Formation of pseudo-random uniformly distributed numbers from noise-like signals, Izv. Akad. Nauk SSSR Tehn. Kibernet. 1969, no. 1, 57-63. (Russian)
  • 91. F. G. Gustavson and W. Liniger, A fast random number generator with good statistical properties, Computing 6 (1970), 221-226. MR 290520
  • 92. S. Haber, On a sequence of points of interest for numerical quadrature, J. Res. Nat. Bur. Standards Sect. B 70 (1966), 127-136. MR 203938
  • 93. S. Haber, A modified Monte Carlo quadrature, Math. Comp. 20 (1966), 361-368. MR 210285
  • 94. S. Haber, Sequences of numbers that are approximately completely equidistributed, J. Assoc. Comput. Mach. 17 (1970), 269-272. MR 323746
  • 95. S. Haber, Numerical evaluation of multiple integrals, SIAM Rev. 12 (1970), 481-526. MR 285119
  • 96. S. Haber, Experiments on optimal coefficients, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 11-37. MR 391479
  • 97. S. Haber and C. F. Osgood, On a theorem of Piatetsky-Shapiro and approximation of multiple integrals, Math. Comp. 23 (1969), 165-168. MR 239758
  • 98. S. Haber and C. F. Osgood, On the sum Σ<>, Pacific J. Math. 31 (1969), 383-394. MR 260173
  • 99. J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 84-90; Berichtigung, ibid., 196. MR 121961
  • 100. J. H. Halton, A retrospective and prospective survey of the Monte Carlo method, SIAM Rev. 12 (1970), 1-63. MR 258231
  • 101. J. H. Halton, Estimating the accuracy of quasi-Monte Carlo integration, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 345-360. MR 351029
  • 102. J. H. Halton and G. B. Smith, Algorithm 247: Radical-inverse quasi-random point sequence [G5], Comm. ACM 7 (1964), 701-702.
  • 103. J. H. Halton and S. K. Zaremba, The extreme and L2 discrepancies of some plane sets, Monatsh. Math. 73 (1969), 316-328. MR 252329
  • 104. H. C. Hamaker, A simple technique for producing random sampling numbers, Nederl. Akad. Wetensch. Proc. 52 (1949), 145-150. MR 32154
  • 105. J. M. Hammersley, Monte Carlo methods for solving multivariable problems, Ann. New York Acad. Sci. 86 (1960), 844-874. MR 117870
  • 106. J. M. Hammersley and D. C. Handscomb, Monte Carlo methods, Methuen, London, 1964. MR 223065
  • 107. N. Harada, Optimal multipliers for the spectral test of uniform random number generators, Information Processing in Japan 14 (1974), 120-126.
  • 108. G. H. Hardy and J. E. Littlewood, Notes on the theory of series. XXIV: A curious power series, Proc. Cambridge Philos. Soc. 42 (1946), 85-90. MR 15529
  • 109. C. B. Haselgrove, A method for numerical integration, Math. Comp. 15 (1961), 323-337. MR 146960
  • 110. G. Helmberg, Gleichverteilte Folgen in lokal kompakten Räumen, Math. Z. 86 (1964), 157-189. MR 170879
  • 111. E. Hlawka, Funktionen von beschränkter Variation in der Theorie der Gleichverteilung, Ann, Mat Pura Appl. 54 (1961), 325-333. MR 139597
  • 112. E. Hlawka, Über die Diskrepanz mehrdimensionaler Folgen mod 1, Math. Z. 77 (1961), 273-284. MR 150106
  • 113. E. Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140-151. MR 143329
  • 114. E. Hlawka, Lösung von Integralgleichungen mittels zahlentheoretischer Methoden. I, Österreich. Akad. Wiss. Math.-Nat KI. S.-B. II 171 (1962), 103-123. MR 150552
  • 115. E. Hlawka, Discrepancy and uniform distribution of sequences, Compositio Math. 16 (1964), 83-91. MR 174544
  • 116. E. Hlawka, Uniform distribution modulo 1 and numerical analysis, Compositio Math. 16 (1964), 92-105. MR 175278
  • 117. E. Hlawka, Trigonometrische Interpolation bei Funktionen von mehreren Variablen, Acta Arith. 9 (1964), 305-320. MR 173901
  • 118. E. Hlawka, Interpolation analytischer Funktionen auf dem Einheitskreis, Number Theory and Analysis (P. Turán, ed.), Plenum Press, New York, 1969, pp. 97-118. MR 271362
  • 119. E. Hlawka, Zur Definition der Diskrepanz, Acta Arith. 18 (1971), 233-241. MR 286757
  • 120. E. Hlawka, Discrepancy and Riemann integration, Studies in Pure Mathematics (L. Mirsky, ed.), Academic Press, New York, 1971, pp. 121-129. MR 277674
  • 121. E. Hlawka, Über eine Methode von E. Hecke in der Theorie der Gleichverteilung, Acta Arith. 24 (1973), 11-31. MR 417092
  • 122. E. Hlawka, Anwendung zahlentheoretischer Methoden auf Probleme der numerischen Mathematik I, Osterreich. Akad. Wiss. Math.-Nat. Kl. S.-B. II 184 (1975), 217-225. MR 424737
  • 123. E. Hlawka, Numerische analytische Fortsetzung in Polyzylindern, Österreich. Akad. Wiss. Math.-Nat Kl. S.-B. II 184 (1975), 307-331. MR 478568
  • 124. E. Hlawka and K. Kreiter, Lösung von Integralgleichungen mittels zahlentheoretischer Methoden II, Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B. II 172 (1963), 229-250. MR 179558
  • 125. E. Hlawka and R. Mück, A transformation of equidistributed sequences, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 371-388. MR 447161
  • 126. E. Hlawka and R. Mück, Über eine Transformation von gleichverteilten Folgen II, Computing 9 (1972), 127-138. MR 453682
  • 127. E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, vol. 1, 3rd ed., Cambridge Univ. Press, London, 1927.
  • 128. L. C. Hsu, Concerning the numerical integration of periodic functions of several variables, Acta Sci. Math. (Szeged) 20 (1959), 230-233. MR 111141
  • 129. L. C. Hsu, Note on the numerical integration of periodic functions and of partially periodic functions, Numer. Math. 3 (1961), 169-173. MR 128077
  • 130. L.-K. Hua and Y. Wang, Remarks concerning numerical integration, Sci. Record (N.S.) 4 (1960), 8-11. MR 128074
  • 131. L.-K. Hua and Y. Wang, Numerical integration and its applications, Science Press, Peking, 1963. (Chinese) MR 203940
  • 132. L.-K. Hua and Y. Wang, On diophantine approximations and numerical integrations. I, II, Sci. Sinica 13 (1964), 1007-1010. MR 172845
  • 133. L.-K. Hua and Y. Wang, On numerical integration of periodic functions of several variables, Sci. Sinica 14 (1965), 964-978. MR 187371
  • 134. L.-K. Hua and Y. Wang, On uniform distribution and numerical analysis (Number-theoretic method). I, II, III, Sci. Sinica 16 (1973), 483-505; 17 (1974), 331-348; 18 (1975), 184-198.
  • 135. T. E. Hull and A. R. Dobell, Random number generators, SIAM Rev. 4 (1962), 230-254. MR 148202
  • 136. D. W. Hutchinson, A new uniform pseudorandom number generator, Comm. ACM 9 (1966), 432-433. MR 193739
  • 137. R. Iglisch, Zum Aufbau der Wahrscheinlichkeitsrechnung, Math. Ann. 107 (1932), 471-484.
  • 138. M. Isida and H. Ikeda, Random number generator, Ann. Inst. Statist. Math. Tokyo 8 (1956), 119-126. MR 86369
  • 139. M. I. Israilov and T. S. Maksudov, Cubature formulae for singular integrals with Hilbert kernel on the class of functions $E\sbn\sp{\alpha }$, Dokl. Akad. Nauk UzSSR 1974, no. 8, 10-12. (Russian) MR 381256
  • 140. D. L. Jagerman, Some theorems concerning pseudo-random numbers, Math. Comp. 19 (1965), 418-426. MR 184405
  • 141. B. Jansson, Autocorrelations between pseudo-random numbers, Nordisk Tidskr. Informations-Behandling 4 (1964), 6-27. MR 165654
  • 142. B. Jansson, Random number generators, Almqvist & Wiksell, Stockholm, 1966. MR 224253
  • 143. D. L. Johnson, Generating and testing pseudo random numbers on the IBM type 701, Math. Tables Aids Comp. 10 (1956), 8-13. MR 76467
  • 144. M. Kadyrov, Tables of random numbers, Izdat. Sredne-Aziatkogo Gos. Univ., Taškent, 1936. (Russian)
  • 145. T. Kamae, Subsequences of normal sequences, Israel J. Math. 16 (1973), 121-149. MR 338321
  • 146. T. Kamae and B. Weiss, Normal numbers and selection rules, Israel J. Math. 21 (1975), 101-110. MR 401695
  • 147. G. Kedem, The search for good lattice points in N dimensions, Technical Report no. 1570, Math. Research Center, Madison, Wis., 1975.
  • 148. G. Kedem and S. K. Zaremba, A table of good lattice points in three dimensions, Numer. Math. 23 (1974), 175-180. MR 373239
  • 149. M. G. Kendall and B. Babington Smith, Random sampling numbers, Tracts for Computers, no. 24, Cambridge Univ. Press, London, 1939.
  • 150. J. Kiefer, On large deviations of the empiric d. f. of vector chance variables and a law of the iterated logarithm, Pacific J. Math. 11 (1961), 649-660. MR 131885
  • 151. G. W. King, The Monte Carlo method as a natural mode of expression in operations research, J. Operations Res. Soc. Amer. 1 (1953), 46-51.
  • 152. P. Kirschenmann, Concepts of randomness, J. Philos. Logic 1 (1972), 395-414. MR 426081
  • 153. D. E. Knuth, Construction of a random sequence, Nordisk Tidskr. Informations-Behandling 5 (1965), 246-250. MR 197434
  • 154. D. E. Knuth, The art of computer programming, vol. 2: Seminumerical algorithms, Addison-Wesley, Reading, Mass., 1969. MR 378456
  • 155. D. E. Knuth, Notes on generalized Dedekind sums, Computer Sci. Dept., Stanford Univ., Stanford, Ca., 1975; Acta Arith. 33 (1977), 297-325. MR 485660
  • 156. J. F. Koksma, Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1, Mathematica B (Zutphen) 11 (1942/43), 7-11. MR 15094
  • 157. J. F. Koksma, Some theorems on diophantine inequalities, Scriptum no. 5, Math. Centrum, Amsterdam, 1950. MR 38379
  • 158. A. N. Kolmogorov, On tables of random numbers, Sankhyā Ser. A 25 (1963), 369-376. MR 178484
  • 159. J. F. Koksma, Three approaches to the definition of the concept "quantity of information", Problemy Peredači Informacii 1 (1965), no. 1, 3-11. (Russian) MR 184801
  • 160. N. M. Korobov, Approximate calculation of multiple integrals with the aid of methods in the theory of numbers, Dokl. Akad. Nauk SSSR 115 (1957), 1062-1065. (Russian) MR 98714
  • 161. N. M. Korobov, The approximate computation of multiple integrals, Dokl. Akad. Nauk SSSR 124 (1959), 1207-1210. (Russian) MR 104086
  • 162. N. M. Korobov, On some number-theoretic methods for the approximate computation of multiple integrals, Uspehi Mat. Nauk 14 (1959), no. 2, 227-230. (Russian)
  • 163. N. M. Korobov, Computation of multiple integrals by the method of optimal coefficients, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959, no. 4, 19-25. (Russian) MR 114086
  • 164. N. M. Korobov, On the approximate solution of integral equations, Dokl. Akad. Nauk SSSR 128 (1959), 235-238. (Russian) MR 112260
  • 165. N. M. Korobov, Properties and calculation of optimal coefficients, Dokl. Akad. Nauk SSSR 132 (1960), 1009-1012 = Soviet Math. Dokl. 1 (1960), 696-700. MR 120768
  • 166. N. M. Korobov, Application of number-theoretic nets to integral equations and interpolation formulas, Trudy Mat. Inst. Steklov. 60 (1961), 195-210. (Russian) MR 132979
  • 167. N. M. Korobov, On applications of number-theoretic nets, Computational Methods and Programming, Izdat. Moskov. Gos. Univ., Moscow, 1962, pp. 80-102. (Russian)
  • 168. N. M. Korobov, On number-theoretic methods in approximate analysis, Questions of Computational Mathematics and Computing Technology (L. A. Ljusternik, ed.), Gos. Naučno-Tehn. Izdat. Mašinostr. Lit, Moscow, 1963, pp. 36-44. (Russian) MR 189241
  • 169. N. M. Korobov, Number-theoretic methods in approximate analysis, Fizmatgiz, Moscow, 1963. (Russian) MR 157483
  • 170. N. M. Korobov, Some problems in the theory of diophantine approximation, Uspehi Mat. Nauk 22 (1967), no. 3, 83-118 = Russian Math. Surveys 22 (1967), no. 3, 80-118.
  • 171. N. M. Korobov, Trigonometric sums with exponential functions and the distribution of signs in repeating decimals, Mat. Zametki 8 (1970), 641-652 = Math. Notes 8 (1970), 831-837. MR 280445
  • 172. N. M. Korobov, On the distribution of digits in periodic fractions, Mat. Sb. (N.S.) 89 (1972), 654-670 = Math. USSR-Sb. 18 (1972), 659-676. MR 424660
  • 173. V. I. Krylov, Approximate calculation of integrals, Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1959; Macmillan, New York, 1962. MR 144464
  • 174. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974. MR 419394
  • 175. R.-D. Kulle and A. Reich, Flächenmessung mit gleichverteilten Folgen, Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl., 1973, no. 12, 217-225. MR 361010
  • 176. D. H. Lehmer, Mathematical methods in large-scale computing units, Proc. 2nd Sympos. on Large-Scale Digital Calculating Machinery (Cambridge, Mass., 1949), Harvard Univ. Press, Cambridge, Mass., 1951, pp. 141-146. MR 44899
  • 177. W. J. LeVeque, An inequality connected with Weyl's criterion for uniform distribution, Proc. Sympos. Pure Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1965, pp. 22-30. MR 179150
  • 178. L. A. Levin, On the notion of a random sequence, Dokl. Akad. Nauk SSSR 212 (1973), 548-550 = Soviet Math. Dokl. 14 (1973), 1413-1416. MR 366096
  • 179. L. A. Levin, Uniform tests of randomness, Dokl. Akad. Nauk SSSR 227 (1976), 33-35 = Soviet Math. Dokl. 17 (1976), 337-340; Erratum, ibid. 231 (1976), 264. MR 414222
  • 180. M. B. Levin, On the uniform distribution of the sequence {αλ, Mat. Sb. (N.S.) 98 (1975), 207-222 = Math. USSR-Sb. 27 (1975), 183-197. MR 406947
  • 181. P. A. W. Lewis, A. S. Goodman and J. M. Miller, A pseudo-random number generator for the System/360, IBM Systems J. 8 (1969), 136-146.
  • 182. T. G. Lewis, Distribution sampling for computer simulation, Lexington Books, Farnborough, 1975.
  • 183. W. Liniger, On a method by D. H. Lehmer for the generation of pseudo-random numbers, Numer. Math. 3 (1961), 265-270. MR 129529
  • 184. M. D. MacLaren and G. Marsaglia, Uniform random number generators, J. Assoc. Comput. Mach. 12 (1965), 83-89. MR 170449
  • 185. D. Maisonneuve, Recherche et utilisation des "bons treillis". Programmation et résultats numériques, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 121-201. MR 343529
  • 186. G. Marsaglia, Random numbers fall mainly in the planes, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 25-28. MR 235695
  • 187. G. Marsaglia, Regularities in congruential random number generators, Numer. Math. 16 (1970), 8-10. MR 273775
  • 188. G. Marsaglia, The structure of linear congruential sequences, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 249-285. MR 411115
  • 189. G. Marsaglia and T. A. Bray, One-line random number generators and their use in combinations, Comm. ACM 11 (1968), 757-759. MR 243719
  • 190. F. F. Martin, Computer modeling and simulation, Wiley, New York, 1968.
  • 191. P. Martin-Löf, The definition of random sequences, Information and Control 9 (1966), 602-619. MR 223179
  • 192. P. Martin-Löf, The literature on von Mises' Kollektivs revisited, Theoria 35 (1969), 12-7. MR 240841
  • 193. J. Maurin, Simulation déterministe du hasard, Masson, Paris, 1975. MR 423753
  • 194. P. McShane, Randomness, statistics and emergence, Univ. of Notre Dame Press, Notre Dame, Ind., 1970.
  • 195. H. G. Meijer, The discrepancy of a g-adic sequence, Nederl. Akad. Wetensch, Proc. Ser. A 71 (1968) = Indag. Math. 30 (1968), 54-66. MR 228450
  • 196. H. G. Meijer and H. Niederreiter, Equirépartition et théorie des nombres premiers, Répartition Modulo 1 (Colloque de Marseille-Luminy, 1974), Lecture Notes in Math., vol. 475, Springer-Verlag, Berlin and New York, 1975, pp. 104-112. MR 389819
  • 197. M. Mendès France, Calcul des moyennes des fonctions aléatoires ou pseudo-aléatoires par échantillonnage, Publ. Inst. Statist. Univ. Paris 11 (1962), 225-256. MR 150793
  • 198. M. Mendès France, Suites de nombres au hasard (d'après Knuth), Sém. Théorie des Nombres 1974-1975, Univ. Bordeaux, Exp. 6. MR 392884
  • 199. N. Metropolis and S. M. Ulam, The Monte Carlo method, J. Amer. Statist. Assoc. 44 (1949), 335-341. MR 31341
  • 200. H. A. Meyer (ed.), Symposium on Monte Carlo methods, Wiley, New York, 1956.
  • 201. G. A. Mihram, Simulation: Statistical foundations and methodology, Academic Press, New York, 1972. MR 336951
  • 202. O. Miyatake, Generation of uniform random numbers of good quality, Math. Japon. 17 (1972), 79-84. MR 341796
  • 203. O. Miyatake, H. Inoue and Y. Yoshizawa, Generation of physical random numbers, Math. Japon. 20 (1975), 207-217. MR 405788
  • 204. L. J. Mordell, On the exponential sum $\sum \sb{x=1}\spX$ ${\rm exp} (2\pi i(ax+bg\spx)/p)$, Mathematika 19 (1972), 84-87. MR 318073
  • 205. L. J. Mordell, A new type of exponential series, Quart. J. Math. 23 (1972), 373-374. MR 319912
  • 206. R. Mück and W. Philipp, Distances of probability measures and uniform distribution mod 1, Math. Z. 142 (1975), 195-202. MR 369304
  • 207. R. E. Nance and C. Overstreet, Jr., Bibliography 29: A bibliography on random number generation, Comput. Rev. 13 (1972), 495-508.
  • 208. T. H. Naylor, Bibliography 19: Simulation and gaming, Comput. Rev. 10 (1969), 61-69.
  • 209. T. H. Naylor, J. L. Balintey and D. S. Burdick, Computer simulation techniques, Wiley, New York, 1966.
  • 210. H. Neunzert and J. Wick, Die Theorie der asymptotischen Verteilung und die numerische Lösung von Integrodifferentialgleichungen, Numer. Math. 21 (1973), 234-243. MR 349054
  • 211. H. Neunzert and J. Wick, Die Darstellung von Funktionen mehrerer Variabler durch Punktmengen, Report no. 996-MA, Kernforschungsanlage Jülich (West Germany), 1973.
  • 212. H. Neunzert and J. Wick, Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen, Numerische Behandlung nichtlinearer Integrodifferential- und Differentialgleichungen (R. Ansorge and W. Törnig, eds.), Lecture Notes in Math., vol. 395, Springer-Verlag, Berlin and New York, 1974, pp. 275-290. MR 371338
  • 213. T. G. Newman and P. L. Odell, The generation of random variates, Hafner, New York, 1971.
  • 214. H. Niederreiter, Diskrepanz in kompakten abelschen Gruppen II, Manuscripta Math. 1 (1969), 293-306. MR 249370
  • 215. H. Niederreiter, Discrepancy and convex programming, Ann. Mat. Pura Appl. 93 (1972), 89-97. MR 389828
  • 216. H. Niederreiter, On a number-theoretical integration method, Aequationes Math. 8 (1972), 304-311. MR 319910
  • 217. H. Niederreiter, Methods for estimating discrepancy, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 203-236. MR 354593
  • 218. H. Niederreiter, On the distribution of pseudo-random numbers generated by the linear congruential method, Math. Comp. 26 (1972), 793-795. MR 326979
  • 219. H. Niederreiter, Metric theorems on the distribution of sequences, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 195-212. MR 337872
  • 220. H. Niederreiter, Application of diophantine approximations to numerical integration, Diophantine Approximation and Its Applications (C. F. Osgood, ed.), Academic Press, New York, 1973, pp. 129-199. MR 357357
  • 221. H. Niederreiter, On the distribution of pseudo-random numbers generated by the linear congruential method. II, Math. Comp. 28 (1974), 1117-1132. MR 457391
  • 222. H. Niederreiter, Quantitative versions of a result of Hecke in the theory of uniform distribution mod 1, Acta Arith. 28 (1975), 321-339. MR 389778
  • 223. H. Niederreiter, Résultats nouveaux dans la théorie quantitative de l'équirépartition, Répartition Modulo 1 (Colloque de Marseille-Luminy, 1974), Lecture Notes in Math., vol. 475, Springer-Verlag, Berlin and New York, 1975, pp. 132-154. MR 389822
  • 224. H. Niederreiter, Some new exponential sums with applications to pseudo-random numbers, Topics in Number Theory (Debrecen, 1974), Colloq. Math. Soc. János Bolyai, vol. 13, North-Holland, Amsterdam, 1976, pp. 209-232. MR 447141
  • 225. H. Niederreiter, On the cycle structure of linear recurring sequences, Math. Scand. 38 (1976), 53-77. MR 406918
  • 226. H. Niederreiter, On the distribution of pseudo-random numbers generated by the linear congruential method. III, Math. Comp. 30 (1976), 571-597. MR 457392
  • 227. H. Niederreiter, Statistical independence of linear congruential pseudo-random numbers, Bull. Amer. Math. Soc. 82 (1976), 927-929. MR 419395
  • 228. H. Niederreiter, Weights of cyclic codes, Information and Control 34 (1977), 130-140. MR 449872
  • 229. H. Niederreiter, Pseudo-random numbers and optimal coefficients, Advances in Math. 26 (1977), 99-181. MR 476679
  • 230. H. Niederreiter, The serial test for linear congruential pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), 273-274. MR 458791
  • 231. H. Niederreiter, Existence of good lattice points in the sense of Hlawka, Monatsh. Math. (to appear). MR 517026
  • 232. H. Niederreiter, A quasi-Monte Carlo method for the approximate computation of the extreme values of a function, Paul Turán Memorial Volume (to appear). MR 820248
  • 233. H. Niederreiter, The serial test for pseudo-random numbers generated by the linear congruential method (in preparation).
  • 234. H. Niederreiter and W. Philipp, Berry-Esseen bounds and a theorem of Erdös and Turán on uniform distribution mod 1, Duke Math. J. 40 (1973), 633-649. MR 337873
  • 235. H. Niederreiter and J. M. Wills, Diskrepanz und Distanz von Massen bezüglich konvexer und Jordanscher Mengen, Math. Z. 144 (1975), 125-134; Berichtigung, ibid. 148 (1976), 99. MR 376588
  • 236. S. M. Nikol'skiĭ, Quadrature formulae, Fizmatgiz, Moscow, 1958 = Hindustan Publ. Corp., Delhi, 1964.
  • 237. O. Onicescu, Nombres et systèmes aléatoires, Editions Eyrolles, Paris, 1964. MR 172314
  • 238. W. H. Payne, Fortran Tausworthe pseudorandom number generator, Comm. ACM 13 (1970), 57.
  • 239. W. H. Payne, J. R. Rabung and T. P. Bogyo, Coding the Lehmer pseudo-random number generator, Comm. ACM 12 (1969), 85-86.
  • 240. L. G. Peck, On uniform distribution of algebraic numbers, Proc. Amer. Math. Soc. 4 (1953), 440-443. MR 55385
  • 241. O. Perron, Die Lehre von den Kettenbrüchen, vol. 1, 3rd ed., Teubner, Stuttgart, 1954. MR 64172
  • 242. W. Philipp, Das Gesetz vom iterierten Logarithmus mit Anwendungen auf die Zahlentheorie, Math. Ann. 180 (1969), 75-94; Corrigendum, ibid. 190 (1971), 338. MR 240069
  • 243. W. Philipp, Mixing sequences of random variables and probabilistic number theory, Mem. Amer. Math. Soc. no. 114, Amer. Math. Soc., Providence, R.I., 1971. MR 437481
  • 244. W. Philipp, Empirical distribution functions and uniform distribution mod 1, Diophantine Approximation and Its Applications (C. F. Osgood, ed.), Academic Press, New York, 1973, pp. 211-234. MR 354594
  • 245. Ju. G. Polljak, On the analysis of pseudorandom numbers, Avt. i Vyčisl. Tehn. 1968, no. 5, 31-35. (Russian) MR 286248
  • 246. K. Popper, Logik der Forschung: Zur Erkenntnistheorie der modernen Naturwissenschaft, Springer, Vienna, 1935.
  • 247. A. G. Postnikov, Arithmetic modeling of random processes, Trudy Mat. Inst. Steklov. 57 (1960) = Selected Transl. Math. Statist. Probability, vol. 13, Amer. Math. Soc., Providence, R.I., 1973, pp. 41-122. MR 148639
  • 248. A. G. Postnikov, Ergodic problems in the theory of congruences and of diophantine approximations, Trudy Mat. Inst. Steklov. 82 (1966) = Proc. Steklov Inst. Math., vol. 82, Amer. Math. Soc., Providence, R.I., 1967. MR 214560
  • 249. P. D. Proĭnov, The square discrepancy of symmetric lattices, Vestnik Moskov. Univ. Ser. I Mat. Meh. 30 (1975), no. 2, 41-47 = Moscow Univ. Math. Bull. 30 (1975), no. 1/2, 105-109. MR 417093
  • 250. C. M. Rader, L. R. Rabiner and R. W. Schafer, A fast method of generating digital random numbers, Bell System Tech. J. 49 (1970), 2303-2310.
  • 251. RAND Corporation, One million random digits and 100, 000 normal deviates, Free Press, Glencoe, III., 1955.
  • 252. G. Rauzy, Fonctions entières et répartition modulo un. II, Bull. Soc. Math. France 101 (1973), 185-192. MR 342483
  • 253. H. Reichenbach, Axiomatik der Wahrscheinlichkeitsrechnung, Math. Z. 34 (1932), 568-619. MR 1545274
  • 254. R. D. Richtmyer, On the evaluation of definite integrals and a quasi-Monte Carlo method based on properties of algebraic numbers, Report LA-1342, Los Alamos Sci. Lab., Los Alamos, N.M., 1951.
  • 255. R. D. Richtmyer, A non-random sampling method based on congruences for Monte-Carlo problems, AEC Research and Development Rep. NYO-8674, AEC Comp. Appl. Math. Center, New York Univ., New York, 1958.
  • 256. R. D. Richtmyer, M. Devaney and N. Metropolis, Continued fraction expansions of algebraic numbers, Numer. Math. 4 (1962), 68-84. MR 136574
  • 257. V. S. Rjaben'kiĭ, Tables and interpolation of a certain class of functions, Dokl. Akad. Nauk SSSR 131 (1960), 1025-1027 = Soviet Math. Dokl. 1 (1960), 382-384. MR 123131
  • 258. V. S. Rjaben'kiĭ, A way of obtaining difference schemes and the use of number-theoretic nets for the solution of the Cauchy problem by the method of finite differences, Trudy Mat. Inst. Steklov. 60 (1961), 232-237. (Russian) MR 144478
  • 259. P. Roos and L. Arnold, Numerische Experimente zur mehrdimensionalen Quadratur, Österreich. Akad. Wiss. Math.-Nat. KI. S.-B. II 172 (1963), 271-286. MR 170475
  • 260. M. Rosenblatt, Multiply schemes and shuffling, Math. Comp. 29 (1975), 929-934. MR 381231
  • 261. K. F. Roth, On irregularities of distribution, Mathematika 1 (1954), 73-79. MR 66435
  • 262. K. F. Roth, On irregularities of distribution. II, Comm. Pure Appl. Math. 29 (1976), 749-754. MR 460266
  • 263. K. F. Roth, On irregularities of distribution. III, Acta Arith. (to appear). MR 598865
  • 264. K. F. Roth, On irregularities of distribution. IV (to appear). MR 460266
  • 265. Ju. N. Šahov, Approximate solution of second kind Volterra equations by means of iterations, Dokl. Akad. Nauk SSSR 128 (1959), 1136-1139. (Russian) MR 112261
  • 266. Ju. N. Šahov, The approximate solution of Volterra equations of the second hind by the method of iterations, Dokl. Akad. Nauk SSSR 136 (1961), 1302-1305 = Soviet Math. Dokl. 2 (1961), 206-209. MR 131735
  • 267. Ju. N. Šahov, On calculating the eigenvalues of a multidimensional symmetric kernel using number-theoretic nets, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 988-997 = U.S.S.R. Computational Math. and Math. Phys. 3 (1963), 1350-1362. MR 158569
  • 268. Ju. N. Šahov, On the approximate solution of higher-dimensional linear Volterra equations of second kind by an iteration method, Ž. Vyčisl. Mat i Mat. Fiz. 4 (1964), no. 4, suppl., 75-100. (Russian) MR 178318
  • 269. Ju. N. Šahov, The calculation of integrals of increasing multiplicity, Ž. Vyčisl. Mat. i Mat. Fiz. 5 (1965), 911-916 = U.S.S.R. Computational Math. and Math. Phys. 5 (1965), no. 5, 184-192. MR 199979
  • 270. Ju. N. Šahov, On the error made in recovering functions of a certain class on parallelepiped-type grids, Mat. Zametki 15 (1974), 749-756 = Math. Notes 15 (1974), 448-452. MR 407508
  • 271. M. Saint-André, Calcul de la moyenne d'une fonction presque-périodique: application au calcul d'intégrales, Rev. Française Informat. Recherche Opérationnelle 4 (1970), Sér. R-3, 141-146. MR 300007
  • 272. M. Saint-André, Détermination d'un vecteur optimal pour le calcul d'intégrales (simples ou multiples), Rev. Française Informat. Recherche Opérationnelle 5 (1971), Sér. R-2, 141-149. MR 315337
  • 273. R. Salfi, A long-period random number generator with application to permutations, Compstat 1974: Proceedings in Computational Statistics (G. Bruckmann, F. Ferschl and L. Schmetterer, eds.), Physica-Verlag, Vienna, 1974, pp. 28-35. MR 345251
  • 274. A. I. Saltykov, Tables for computing multiple integrals by the method of optimal coefficients, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 181-186 = U.S.S.R Computational Math. and Math. Phys. 3 (1963), 235-242. MR 150976
  • 275. I. F. Šarygin, The use of number-theoretic methods of integration in the case of nonperiodic functions, Dokl. Akad. Nauk SSSR 132 (1960), 71-74 = Soviet Math. Dokl. 1 (1960), 506-509. MR 128075
  • 276. I. F. Šarygin, A lower estimate for the error of quadrature formulas for certain classes of functions, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 370-376 = U.S.S.R. Computational Math. and Math. Phys. 3 (1963), 489-497. MR 150952
  • 277. M. Sato, On the periods of certain pseudorandom sequences, Publ. Res. Inst. Math. Sci. 10 (1974/75), 77-89. MR 381232
  • 278. K. Schmidt, Über die C-Gleichverteilung von Massen, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 17 (1971), 327-332. MR 287594
  • 279. K. Schmidt and P. Zinterhof, Über Quadraturformeln auf T, Computing 6 (1970), 94-96. MR 280004
  • 280. W. M. Schmidt, Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc. 110 (1964), 493-518. MR 159802
  • 281. W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 125 (1970), 189-201. MR 268129
  • 282. W. M. Schmidt, Irregularities of distribution. VII, Acta Arith. 21 (1972), 45-50. MR 319933
  • 283. W. M. Schmidt, Lectures on irregularities of distribution, Lecture notes, Boulder, Co., 1973.
  • 284. W. M. Schmidt, Irregularities of distribution. IX, Acta Arith. 27 (1975), 385-396. MR 376593
    284a. W. M. Schmidt, Irregularities of distribution. X, Number Theory and Algebra (H. Zassenhaus, ed.), Academic Press, New York, 1977, pp. 311-329. MR 491574

  • 285. C. P. Schnorr, Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Math., vol. 218, Springer-Verlag, Berlin and New York, 1971. MR 414225
  • 286. C. S. Smith, Multiplicative pseudo-random number generators with prime modulus, J. Assoc. Comput. Mach. 18 (1971), 586-593. MR 295522
  • 287. S. A. Smoljak, Interpolation and quadrature formulas for the classes $W\sb{s\sp{\alpha }}$ and $E\sb{s\sp{\alpha }}$, Dokl. Akad. Nauk SSSR 131 (1960), 1028-1031 = Soviet Math. Dokl. 1 (1960), 384-387. MR 123132
  • 288. S. A. Smoljak, Quadrature and interpolation formulae on tensor products of certain function classes, Dokl. Akad. Nauk SSSR 148 (1963), 1042-1045 = Soviet Math. Dokl. 4 (1963), 240-243. MR 147825
  • 289. I. M. Sobol', Multidimensional integrals and the Monte Carlo method, Dokl. Akad. Nauk SSSR 114 (1957), 706-709. (Russian) MR 92205
  • 290. I. M. Sobol', Pseudo-random numbers for the machine "Strela", Teor. Verojatnost. i Primenen. 3 (1958), 205-211 = Theor. Probability Appl. 3 (1958), 192-197. MR 94923
  • 291. I. M. Sobol', An accurate error estimate for multidimensional quadrature formulae for the functions of the class S, Dokl. Akad. Nauk SSSR 132 (1960), 1041-1044 = Soviet Math. Dokl. 1 (1960), 726-729. MR 138198
  • 292. I. M. Sobol', An exact estimate of the error in multidimensional quadrature formulae for functions of the classes W1 and H1, Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961), 208-216 = U.S.S.R. Computational Math. and Math. Phys. 1 (1961), 228-240. MR 136513
  • 293. I. M. Sobol', On the evaluation of infinite-dimensional integrals, Ž. Vyčisl. Mat i Mat. Fiz. 1 (1961), 917-922 = U.S.S.R. Computational Math. and Math. Phys. 1 (1961), 1086-1091. MR 138919
  • 294. I. M. Sobol', On the evaluation of multidimensional integrals, Dokl. Akad. Nauk SSSR 139 (1961), 821-823 = Soviet Math. Dokl. 2 (1961), 1022-1025. MR 140186
  • 295. I. M. Sobol', The use of the ω, Ž. Vyčisl. Mat. i Mat. Fiz. 2 (1962), 717-723 = U.S.S.R. Computational Math. and Math. Phys. 2 (1962), 808-816. MR 155418
  • 296. I. M. Sobol', The application of Haar series in the theory of quadrature formulae, Questions of Computational Mathematics and Computing Technology (L. A. Ljusternik, ed.), Gos Naučno-Tehn. Izdat. Mašinostr. Lit., Moscow, 1963, pp. 31-35. (Russian) MR 184433
  • 297. I. M. Sobol', On the periods of pseudo-random sequences, Teor. Verojatnost. i Primenen. 9 (1964), 367-373 = Theor. Probability Appl. 9 (1964), 333-338. MR 163334
  • 298. I. M. Sobol', Distribution of points in a cube and integration nets, Uspehi Mat. Nauk 21 (1966), no. 5, 271-272. (Russian) MR 198678
  • 299. I. M. Sobol', An integral encountered in quadrature formulae theory, Ž. Vyčisl. Mat. i Mat. Fiz. 6 (1966), 1084-1089 = U.S.S.R. Computational Math. and Math. Phys. 6 (1966), no. 6, 189-196. MR 203942
  • 300. I. M. Sobol', The distribution of points in a cube and the approximate evaluation of integrals, Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 784-802 = U.S.S.R. Computational Math. and Math. Phys. 7 (1967), no. 4, 86-112. MR 219238
  • 301. I. M. Sobol', The use of Haar series in estimating the error in the computation of infinite-dimensional integrals, Dokl. Akad. Nauk SSSR 175 (1967), 34-37 = Soviet Math. Dokl. 8 (1967), 810-813. MR 215527
  • 302. I. M. Sobol', A Monte Carlo method for critical calculation in multigroup approximation, The Monte Carlo Method in Problems of Radiation Transfer, Atomizdat, Moscow, 1967, pp. 232-254. (Russian)
  • 303. I. M. Sobol', Multidimensional quadrature formulas and Haar functions, Izdat. "Nauka", Moscow, 1969. (Russian) MR 422968
  • 304. I. M. Sobol', On an approach to the computation of multiple integrals, Voprosy Vyčisl. i Prikl. Mat. (Taškent) 1970, no. 38, 100-111. (Russian)
  • 305. I. M. Sobol', The problem of the minimum of $\phi\infty$ in the three-dimensional cube, Voprosy Vyčisl. i Prikl. Mat. (Taškent) 1970, no. 38, 112-115. (Russian)
  • 306. I. M. Sobol', The Monte Carlo method, Izdat. "Nauka", Moscow, 1972 = Mir Publishers, Moscow, 1975. MR 353620
  • 307. I. M. Sobol', A deterministic interpretation of goodness-of-fit tests and a test of pseudo-random numbers, Operations Research and Statistical Modeling, vol. 1, Izdat Leningrad. Univ., Leningrad, 1972, pp. 162-169. (Russian) MR 373221
  • 308. I. M. Sobol', A probabilistic estimate of the error for nonrandom integration nets, Voprosy Vyčisl. i Prikl. Mat. (Taskent) 1972, no. 14, 5-11. (Russian)
  • 309. I. M. Sobol', Numerical Monte Carlo methods, Izdat. "Nauka", Moscow, 1973. (Russian) MR 405786
  • 310. I. M. Sobol', A probabilistic estimate of the integration error for P, Ž. Vyčisl. Mat. i Mat. Fiz. 13 (1973), 1035-1037 = U.S.S.R. Computational Math. and Math. Phys. 13 (1973), no. 4, 259-262. MR 368394
  • 311. I. M. Sobol', Calculation of improper integrals using equidistributed sequences, Dokl. Akad. Nauk SSSR 210 (1973), 278-281 = Soviet Math. Dokl. 14 (1973), 734-738. MR 375726
  • 312. I. M. Sobol', Pseudo-random numbers for constructing discrete Markov chains by the Monte Carlo method, Ž. Vyčisl. Mat. i Mat. Fiz. 14 (1974), 36-44 = U.S.S.R. Computational Math. and Math. Phys. 14 (1974), no. 1, 36-45. MR 339444
  • 313. I. M. Sobol', Infinite-dimensional uniformly distributed sequences in numerical mathematics, Preprint no. 22, Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow, 1974. (Russian)
  • 314. I. M. Sobol', On convergence of infinite-dimensional cubature and simulation of Markov chains, Voprosy Vyčisl. i Prikl. Mat. (Taškent) 1975, no. 32, 162-167. (Russian)
  • 315. I. M. Sobol', Uniformly distributed sequences with an additional uniformity property, Ž. Vyčisl. Mat. i Mat. Fiz. 16 (1976), 1332-1337 = U.S.S.R. Computational Math. and Math. Phys. 16 (1976), no. 5, 236-242. MR 483283
  • 316. I. M. Sobol' and Ju. L. Levitan, Generation of points uniformly distributed in a multidimensional cube, Preprint no. 40, Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow, 1976. (Russian)
  • 317. I. M. Sobol' and R. B. Statnikov, LP-search and problems of optimal design, Problems of Random Search, vol. 1, Izdat. "Zinatne", Riga, 1972, pp. 117-135. (Russian)
  • 318. I. M. Sobol' R. B. Statnikov and N. F. Ovčinnikova, Localization of the characteristic roots of a matrix, Z. Vyčisl. Mat. i Mat. Fiz. 13 (1973), 1581-1583 = U.S.S.R. Computational Math. and Math. Phys. 13 (1973), no. 6, 255-258. MR 356472
  • 319. V. M. Solodov, On the calculation of multiple integrals, Dokl. Akad. Nauk SSSR 127 (1959), 753-756. (Russian) MR 111146
  • 320. V. M. Solodov, On the error involved in a numerical integration, Dokl. Akad. Nauk SSSR 148 (1963), 284-287 = Soviet Math. Dokl. 4 (1963), 85-88. MR 144467
  • 321. V. M. Solodov, Integration over regions different from the unit cube, Ž. Vyčisl. Mat. i Mat. Fiz. 8 (1968), 1334-1341 = U.S.S.R. Computational Math. and Math. Phys. 8 (1968), no. 6, 198-207. MR 242373
  • 322. V. M. Solodov, An application of the method of optimal coefficients to numerical integration, Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 14-29 = U.S.S.R. Computational Math. and Math. Phys. 9 (1969), no. 1, 14-34. MR 250487
    322a. E. R. Sowey, A chronological and classified bibliography on random number generation and testing, Internat. Statist. Rev. 40 (1972), 355-371. MR 423754

  • 323. J. Spanier and E. M. Gelbard, Monte Carlo principles and neutron transport problems, Addison-Wesley, Reading, Mass., 1969. MR 272339
  • 324. V. T. Stojancev, Indeterminate methods of integration with a finite number of feasible methods, Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1235-1246 = U.S.S.R. Computational Math. and Math. Phys. 9 (1969), no. 6, 1-16. MR 266433
  • 325. V. T. Stojancev, Solution of the Cauchy problem for a parabolic equation by a quasi-Monte Carlo method, Ž. Vyčisl. Mat. i Mat. Fiz. 13 (1973), 1153-1160 = U.S.S.R. Computational Math. and Math. Phys. 13 (1973), no. 5, 67-75. MR 329278
  • 326. V. T. Stojancev, Solution of the Dirichlet problem by a quasi-Monte Carlo method, Uspehi Mat. Nauk 30 (1975), no. 1, 263-264. (Russian) MR 405787
  • 327. R. G. Stoneham, On a new class of multiplicative pseudo-random number generators, Nordisk Tidskr. Informations-Behandling 10 (1970), 481-500. MR 278488
  • 328. R. G. Stoneham, On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers, Acta Arith. 22 (1973), 371-389.
  • 329. A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, Englewood Cliffs, NJ., 1971. MR 327006
  • 330. W. Stute, Convergence rates for the isotrope discrepancy, Ann. Probability (to appear). MR 455096
  • 331. P. Szüsz, On a problem in the theory of uniform distribution, Compt. Rend. Premier Congrès Hongrois, Budapest, 1952, pp. 461-472. (Hungarian) MR 56036
  • 332. O. Taussky and J. Todd, Generation and testing of pseudo-random numbers, Symposium on Monte Carlo Methods (H. A. Meyer, ed.), Wiley, New York, 1956, pp. 15-28. MR 80382
  • 333. R. C. Tausworthe, Random numbers generated by linear recurrence modulo two, Math. Comp. 19 (1965), 201-209. MR 184406
  • 334. D. Teichroew, A history of distribution sampling prior to the era of the computer and its relevance to simulation, J. Amer. Statist. Assoc. 60 (1965), 27-49. MR 175214
  • 335. L. H. C. Tippett, Random sampling numbers, Tracts for Computers, no. 15, Cambridge Univ. Press, London, 1927.
  • 336. K. D. Tocher, The application of automatic computers to sampling experiments, J. Roy. Statist. Soc. Ser. B 16 (1954), 39-61. MR 64476
  • 337. J. P. R. Tootill, W. D. Robinson and A. G. Adams, The runs up-and-down performance of Tausworthe pseudo-random number generators, J. Assoc. Comput. Mach. 18 (1971), 381-399.
  • 338. J. P. R. Tootill, W. D. Robinson and D. J. Eagle, An asymptotically random Tausworthe sequence, J. Assoc. Comput. Mach. 20 (1973), 469-481.
  • 339. T. Tsuda, Numerical integration of functions of very many variables, Numer. Math. 20 (1973), 377-391. MR 433828
  • 340. S. M. Ulam, Monte Carlo calculations in problems of mathematical physics, E. F. Beckenbach (ed.), Modern mathematics for the engineer, 2nd series, McGraw-Hill, New York, 1961, Chapter 11. MR 129165
  • 341. J. G. van der Corput, Verteilungsfunktionen. I, II, Nederl. Akad. Wetensch. Proc. 38 (1935), 813-821, 1058-1066.
  • 342. A. van Gelder, Some new results in pseudo-random number generation, J. Assoc. Comput Mach. 14 (1967), 785-792.
  • 343. A. van Wijngaarden, Mathematics and computing, Proc. Sympos. Automatic Digital Computation (London, 1954), H. M. Stationery Office, London, 1954, pp. 125-129. MR 1665378
  • 344. J. Venn, The logic of chance, Macmillan, London, 1876.
  • 345. P. H. Verdier, Relations within sequences of congruential pseudo-random numbers, J. Res. Nat. Bur. Standards Sect. B 73 (1969), 41-44. MR 239724
  • 346. I. V. Vilenkin, Plane nets of integration, Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 189-196 = U.S.S.R. Computational Math. and Math. Phys. 7 (1967), no. 1, 258-267. MR 205464
  • 347. I. V. Vilenkin, More on plane nets of integration, Ž. Vyčisl. Mat. i Mat Fiz. 13 (1973), 854-864 = U.S.S.R. Computational Math. and Math. Phys. 13 (1973), no. 4, 43-56. MR 371032
  • 348. J. Ville, Étude critique de la notion de collectif, Gauthier-Villars, Paris, 1939.
  • 349. S. von Hoerner, Herstellung von Zufallszahlen auf Rechenautomaten, Z. Angew. Math. Physik 8 (1957), 26-52. MR 83186
  • 350. R. von Mises, Grundlagen der Wahrscheinlichkeitsrechnung, Math. Z. 5 (1919), 52-99. MR 1544374
  • 351. R. von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, Springer, Vienna, 1928. MR 350796
  • 352. J. von Neumann, Various techniques used in connection with random digits, NBS Appl. Math. Series, no. 12, U. S. Government Printing Office, Washington, D. C., 1951, pp. 36-38 = Collected Works, vol. 5, Pergamon Press, Oxford, 1963, pp. 768-770.
  • 353. Ju. V. Voroncov and Ju. G. Polljak, On the use of quasirandom sequences in the direct probabilistic simulation of systems, Avt. i Vyčisl. Tehn. 1971, no. 6, 23-27. (Russian)
  • 354. J. E. Walsh, An experimental method for obtaining random digits and permutations, Sankhyā 17 (1957), 355-360.
  • 355. Y. Wang, A note on interpolation of a certain class of functions, Sci. Sinica 10 (1961), 632-636. MR 151764
  • 356. Y. Wang, On numerical integration and its applications (Number-theoretic method), Shuxue Jinzhan 5 (1962), no. 1, 1-44. (Chinese)
  • 357. T. T. Warnock, Computational investigations of low-discrepancy point sets, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 319-343. MR 351035
  • 358. Y. Watanabe, An improvement for the Richtmyer-Haselgrove method, Sci. Rep. Osaka 22 (1973), no. 1, 33-44. MR 822714
  • 359. W. J. Westlake, A uniform random number generator based on the combination of two congruential generators, J. Assoc. Comput. Mach. 14 (1967), 337-340.
  • 360. B. E. White, Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix, Monatsh. Math. 80 (1975), 219-229. MR 417102
  • 361. B. E. White, On optimal extreme-discrepancy point sets in the square, Numer. Math. 27 (1977), 157-164. MR 468180
  • 362. J. R. B. Whittlesey, A comparison of the correlational behavior of random number generators for the IBM 360, Comm. ACM 11 (1968), 641-644. MR 255011
  • 363. J. R. B. Whittlesey, On the multidimensional uniformity of pseudorandom generators, Comm. ACM 12 (1969), 247.
  • 364. A. C. Yao and D. E. Knuth, Analysis of the subtractive algorithm for greatest common divisors, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 4720-4722. MR 417041
  • 365. S. K. Zaremba, Good lattice points, discrepancy, and numerical integration, Ann. Mat. Pura Appl. 73 (1966), 293-317. MR 218018
  • 366. S. K. Zaremba, Some applications of multidimensional integration by parts, Ann. Polon. Math. 21 (1968), 85-96. MR 235731
  • 367. S. K. Zaremba, Good lattice points in the sense of Hlawka and Monte Carlo integration, Monatsh. Math. 72 (1968), 264-269. MR 228155
  • 368. S. K. Zaremba, The mathematical basis of Monte Carlo and quasi-Monte Carlo methods, SIAM Rev. 10 (1968), 303-314. MR 233489
  • 369. S. K. Zaremba, A quasi-Monte Carlo method for computing double and other multiple integrals, Aequationes Math. 4 (1970), 11-22. MR 263213
  • 370. S. K. Zaremba, La discrépance isotrope et l'intégration numérique, Ann. Mat. Pura Appl. 87 (1970), 125-136.
  • 371. S. K. Zaremba, A remarkable lattice generated by Fibonacci numbers, Fibonacci Quart 8 (1970), 185-198. MR 260696
  • 372. S. K. Zaremba, Sur la discrépance des suites aléatoires, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 20 (1971), 236-248. MR 312553
  • 373. S. K. Zaremba, La méthode des "bons treillis" pour le calcul des intégrales multiples, Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), Academic Press, New York, 1972, pp. 39-119. MR 343530
  • 374. S. K. Zaremba, Good lattice points modulo primes and composite numbers, Diophantine Approximation and Its Applications (C. F. Osgood, ed.), Academic Press, New York, 1973, pp. 327-356. MR 354595
  • 375. S. K. Zaremba, Good lattice points modulo composite numbers, Monatsh. Math. 78 (1974), 446-460. MR 371845
  • 376. S. K. Zaremba, Computing the isotropic discrepancy of point sets in two dimensions, Discrete Math. 11 (1975), 79-92. MR 354596
  • 377. S. K. Zaremba, L'erreur dans le calcul des intégrales doubles par la méthode des bons treillis, Demonstratio Math. 8 (1975), 347-364. MR 381264
  • 378. S. K. Zaremba, On Cartesian products of good lattices, Math. Comp. 30 (1976), 546-552. MR 423770
  • 379. N. Zierler, Linear recurring sequences, J. Soc. Industr. Appl. Math. 7 (1959), 31-48. MR 101979
  • 380. Ja. M. Zileĭkin, Approximate solution of the Dirichlet problem for the Laplace equation, Dokl. Akad. Nauk SSSR 155 (1964), 999-1002 = Soviet Math. Dokl. 5 (1964), 528-531. MR 170029
  • 381. Ja. M. Zileĭkin, On the approximate solution of integral equations, Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), 749-753 = U.S.S.R. Computational Math. and Math. Phys. 4 (1964), no. 4, 176-181. MR 166573
  • 382. Ja. M. Zileĭkin, An approximate method of solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped, Ž. Vyčisl. Mat. i Mat. Fiz. 5 (1965), 345-347 = U.S.S.R. Computational Math. and Math. Phys. 5 (1965), no. 2, 246-249. MR 196967
  • 383. Ja. M. Zileĭkin, Quadrature formulae on classes of functions, Ž. Vyčisl. Mat. i Mat. Fiz. 8 (1968), 507-516 = U.S.S.R. Computational Math. and Math. Phys. 8 (1968), no. 3, 1-14. MR 237098
  • 384. P. Zinterhof, Einige zahlentheoretische Methoden zur numerischen Quadratur und Interpolation, Österreich. Akad. Wiss. Math.-Nat. KI. S.-B. II 177 (1969), 51-77. MR 248982

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1970): 65-02, 65C05, 65C10, 65D30, 10F40, 10K05, 10-02, 10A35, 10F10, 10F20, 10G05, 10K30, 12A15, 65D05, 65N05, 65R05, 68A55

Retrieve articles in all journals with MSC (1970): 65-02, 65C05, 65C10, 65D30, 10F40, 10K05, 10-02, 10A35, 10F10, 10F20, 10G05, 10K30, 12A15, 65D05, 65N05, 65R05, 68A55


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1978-14532-7

American Mathematical Society