Continued fraction algorithms, functional operators, and structure constants

Theoretical Computer Science

Volumn 194, Issue 1-2, 1998, Pages 1-34

  Flajolet, Philippe ^a   Vallée, Brigitte ^b  

^a INRIA ROCQUENCOURT   (France)

^b UNIVERSITÉ DE CAEN   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001905407     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(97)00123-0     Document Type: Review

References (41)

1. F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, M. Yvinec, Evaluation of a new method to compute signs of determinants, in: 11th Annual ACM Symp. on Computational Geometry, 1995, pp. C16-C17. Full paper to appear in Algorithmica, 1997.
2. F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, M. Yvinec, Evaluation of a new method to compute signs of determinants, in: 11th Annual ACM Symp. on Computational Geometry, 1995, pp. C16-C17. Full paper to appear in Algorithmica, 1997.
3. K.I. Babenko, On a problem of Gauss, Sov. Math. Dokl. 19 (1) (1978) 136-140.
4. T. Bedford, M. Keane, C. Series (Eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, Oxford, 1991.
5. M. Beeler, R.W. Gosper, R. Schroeppel, HAKMEM, Memorandum 239, M.I.T., Artificial Intelligence Laboratory, 1972.
6. P. Billingsley, Probability and Measure, 2nd ed., Wiley, 1986.
7. E.B. Bogomolny, M. Carioli, Quantum maps from transfer operators, Physica D 67 (1993) 88-112.
8. H. Daudé, P. Flajolet, B. Vallée, An average-case analysis of the Gaussian algorithm for lattice reduction, Combin. Probab. Comput., to appear, 1997.
9. J.D. Dixon, The number of steps in the Euclidean algorithm, J. Number Theory 2 (1970) 414-422.
10. I. Efrat, Dynamics of the continued fraction map and the spectral theory of SL(2, Z), Invent. Math. 114 (1993) 207-218.
11. C. Faivre, Distribution of Lévy's constants for quadratic numbers, Acta Arithmetica 61 (1) (1992) 13-34.
12. S. Finch, Favorite mathematical constants. Available under World Wide Web at http://www. mathsoft.com/asolve/constant/constant.html, 1995.
13. P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoret. Comput. Sci. 144 (1-2) (1995) 3-58.
14. H. Heilbronn, On the average length of a class of continued fractions, in: P. Turan (Ed.), Number Theory and Analysis, Plenum Press, New York, 1969, pp. 87-96.
15. D. Hensley, The Hausdorff dimensions of some continued fraction Cantor sets, J. Number Theory 33 (1989) 182-198.
16. D. Hensley, The number of steps in the Euclidean algorithm, J. Number Theory 49 (2) (1994) 142-182.
17. H.-K. Hwang, Théorèmes limites pour les structures combinatoires et les fonctions arithmetiques, PhD thesis, École Polytechnique, 1994.
18. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980.
19. A.I. Khinchin, Continued Fractions, University of Chicago Press, Chicago, 1964. A translation of the Russian original published in 1935.
20. D.E. Knuth, The Art of Computer Programming, vol. 3: Sorting and Searching, Addison-Wesley, Reading, MA, 1973.
21. D.E. Knuth, The Art of Computer Programming, 2nd ed., vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading, MA, 1981.
22. M. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964.
23. E.R. Lorch, Spectral Theory, Oxford University Press, New York, 1962.
24. H. Mahmoud, Evolution of Random Search Trees, Wiley, New York, 1992.
25. D. Mayer, G. Roepstorff, On the relaxation time of Gauss's continued fraction map. I. The Hilbert space approach, J. Statist. Phys. 47 (1/2) (1987) 149-171.
26. D. Mayer, G. Roepstorff, On the relaxation time of Gauss's continued fraction map. II. The Banach space approach (transfer operator approach), J. Statist. Phys. 50 (1/2) (1988) 331-344.
27. D.H. Mayer, On a ζ function related to the continued fraction transformation, Bull. Soc. Math. France 104 (1976) 195-203.
28. D.H. Mayer, Continued fractions and related transformations, in: T. Bedford, M. Keane, C. Series (Eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, Oxford, 1991, pp. 175-222.
29. L.M. Milne-Thomson, The Calculus of Finite Differences, Chelsea, New York, 1981. Reprinted from the original edition, London, 1933.
30. G.A. Mischyavichyus, Estimate of the remainder in the limit theorem for the denominators of continued fractions, Litovskiǐ Matematičeskiǐ Sbornik 21 (3) (1987) 63-74.
31. T. Morita, Local limit theorem and distribution of periodic orbits of Lasota-Yorke transformations with infinite Markov partitions, J. Math. Soc. Japan 46 (2) (1994) 309-343. Corrections in Vol. 47 (1) (1997) 191-192.
32. O. Perron, Die Lehre von der Kettenbrüchen, vol. 1. Teubner, Stuttgart, 1954.
33. W. Philipp, Ein zentraler Grenzwertsatz mit Anwendungen auf die Zahlentheorie, Zeitschrift für Wahrscheinlichkeitstheorie 8 (1967) 195-203.
34. A. Rockett, P. Szüsz, Continued Fractions, World Scientific, Singapore, 1992.
35. D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval, CRM Monograph Series, vol. 4, American Mathematical Society, Providence, RI, 1994.
36. R. Sitaramachandrarao, A formula of S. Ramanujan, J. Number Theory 25 (1987) 1-19.
37. G. Tenenbaum, Introduction à la théorie analytique des nombres, vol. 13, Institut Élie Cartan, Nancy, France, 1990.
38. B. Vallée, Opérateurs de Ruelle-Mayer généralisés et analyse des algorithmes d'Euclide et de Gauss. Acta Arithmetica, LXXXI.2 (1997), 101-144.
39. B. Vallée, Algorithms for computing signs of 2 × 2 determinants: dynamics and average-case analysis, To appear in European Symposium of Algorithms (ESA), 1997.
40. B. Vallée, Dynamique des fractions continues à contraintes périodiques, Les cahiers du GREYC, Université de Caen, 1997, submitted.
41. E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces, Acta Arithmetica 24 (1974) 507-528.