Eichenauer, Jürgen; Lehn, Jürgen On the structure of quadratic congruential sequences. (English) Zbl 0598.65004 Manuscr. Math. 58, 129-140 (1987). Sequences of integers defined by a quadratic congruential formula are divided into non-overlapping subsequences of length d. The structure of the set of the resulting points in the d-dimensional Euclidean space \({\mathbb{R}}^ d\) is studied. The analysis is restricted to the case of sequences with maximal period length since such sequences are of special interest in connection with pseudo random number generation. Cited in 18 Documents MSC: 65C10 Random number generation in numerical analysis 11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms 11B99 Sequences and sets Keywords:quadratic congruential sequences; pseudo random number generators; superimposition of lattice structures PDFBibTeX XMLCite \textit{J. Eichenauer} and \textit{J. Lehn}, Manuscr. Math. 58, 129--140 (1987; Zbl 0598.65004) Full Text: DOI EuDML References: [1] Afflerbach, L.: The sub-lattice structure of linear congruential random number generators, manuscripta math. 55, 455-465 (1986) · Zbl 0601.65005 [2] Afflerbach, L. and Grothe, H.: Calculation of Minkowski-reduced lattice bases, Computing 35, 269-276 (1985) · Zbl 0557.10025 [3] Beyer, W.A.: Lattice structure and reduced bases of random vectors generated by linear recurrences. In: S.K. Zaremba (ed.): Applications of number theory to numerical analysis, 361-370 (1972) [4] Beyer, W.A., Roof, R.B. and Williamson, D.: The lattice structure of multiplicative pseudo-random vectors, Math. Comp. 25, 345-363 (1971) · Zbl 0269.65003 [5] Dieter, U. and Ahrens, J.H.: Uniform random numbers, Institut f. Math. Stat., Technische Hochschule Graz (1974) [6] Eichenauer, J. and Lehn, J.: A non-linear congruential pseudo random number generator. Fachbereich Mathematik, Technische Hochschule Darmstadt, Preprint Nr. 988 (1986); Statistical Papers (to appear) · Zbl 0607.65001 [7] Knuth, D.E.: The art of computer programming, vol. 2, 2nd ed., Addison-Wesley 1981 · Zbl 0477.65002 [8] Lehmer, D.E.: Mathematical methods in large-scale computing units, Ann. Comp. Lab. Harvard Univ. 26, 141-146 (1951) · Zbl 0045.40001 [9] Marsaglia, G.: Random numbers fall mainly in the planes, Proc. Nat. Acad. Sci. 61, 25-28 (1968) · Zbl 0172.21002 [10] Marsaglia, G.: Regularities in congruential random number generators, Numer. Math. 16, 8-10 (1970) · Zbl 0212.18204 [11] Marsaglia, G.: The structure of linear congruential sequences. In: S.K. Zaremba (ed.): Applications of number theory to numerical analysis, 249-285 (1972) · Zbl 0266.65007 [12] Rotenberg, A.: A new pseudo-random number generator, Journ. ACM 7, 75-77 (1960) · Zbl 0096.33902 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.