On the complexity of real root isolation using continued fractions

Theoretical Computer Science

Volumn 392, Issue 1-3, 2008, Pages 158-173

  Tsigaridas, Elias P ^a   Emiris, Ioannis Z ^b  

^a LORIA   (France)

^b UNIVERSITY OF ATHENS   (Greece)

Author keywords

Complexity; Continued fraction; Real root isolation; Synaps

Indexed keywords

C (PROGRAMMING LANGUAGE); COMPUTATIONAL COMPLEXITY; COMPUTER PROGRAMMING LANGUAGES; COMPUTER SOFTWARE; INTEGER PROGRAMMING;

CONTINUED FRACTION; FRACTION EXPANSION; REAL ROOT ISOLATION; SYNAPS;

POLYNOMIALS;

EID: 39149138443     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2007.10.010     Document Type: Article

References (60)

1. A. Akritas, There is no "Uspensky's method". in: Proc. Symposium on Symbolic and Algebraic Computation, Waterloo, Ontario, Canada, 1986, pp. 88-90 (extended abstract)
2. Akritas A. An implementation of Vincent's theorem. Numerische Mathematik 36 (1980) 53-62
3. Akritas A., and Strzeboński A. A comparative study of two real root isolation methods. Nonlinear Analysis: Modelling and Control 10 4 (2005) 297-304
4. A. Akritas, A. Bocharov, A. Strzeboński, Implementation of real root isolation algorithms in Mathematica, in: Abstracts of the International Conference on Interval and Computer-Algebraic Methods in Science and Engineering, Interval '94, St. Petersburg, Russia, March 1994, pp. 23-27
5. Akritas A., Strzeboński W., and Vigklas P.S. Implementations of a new theorem for computing bounds for positive roots of polynomials. Computing 78 (2006) 355-367
6. Akritas A. Elements of Computer Algebra with Applications (1989), J. Wiley & Sons, New York
7. Alesina A., and Galuzzi M. A new proof of Vincent's theorem. L'Enseignement Mathématique 44 (1998) 219-256
8. Bailey D., Borwein J., and Crandall R. On the Khintchine constant. Mathematics of Computation 66 (1997) 417-431
9. Basu S., Pollack R., and Roy M.-F. Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics vol. 10 (2003), Springer-Verlag. 3-540-00973-6
10. Bini D. Numerical computation of polynomial zeros by means of Aberth's method. Numerical Algorithms 13 3-4 (1996) 179-200
11. Bini D., and Fiorentino G. Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms (2000) 127-173
12. Bombieri E., and van der Poorten A. Continued fractions of algebraic numbers. Computational Algebra and Number Theory. Sydney, 1992 (1995), Kluwer Acad. Publ., Dordrecht 137-152
13. Brent R., van der Poorten A., and Riele H. A comparative study of algorithms for computing continued fractions of algebraic numbers. In: Cohen H. (Ed). ANTS. LNCS (1996), Springer 35-47
14. Brent R.P. Analysis of the binary Euclidean algorithm. In: Traub J.F. (Ed). New Directions and Recent Results in Algorithms and Complexity (1976), Academic Press, New York 321-355
15. R.P. Brent, Twenty years' analysis of the binary Euclidean algorithm, in: J. Davies, A. W. Roscoe, and J. Woodcock (Ed.), Millenial Perspectives in Computer Science: Proc. of the 1999 Oxford-Microsoft Symposium in honour of Professor Sir Antony Hoare, Palgrave, New York, 2000, pp. 41-53
16. Cantor D., Galyean P., and Zimmer H. A continued fraction algorithm for real algebraic numbers. Mathematics of Computation, 0025-5718 26 119 (1972) 785-791
17. Chen J. A new algorithm for the isolation of real roots of polynomial equations. Proc. 2nd International Conference on Computers and Applications (1987), IEEE Computer Soc. press 714-719
18. Collins G., and Akritas A. Polynomial real root isolation using Descartes' rule of signs. SYMSAC '76 (1976), ACM Press, New York, USA 272-275
19. Collins G., and Loos R. Real zeros of polynomials. In: Buchberger B., Collins G.E., and Loos R. (Eds). Computer Algebra: Symbolic and Algebraic Computation. 2nd edition (1982), Springer-Verlag, Wien 83-94
20. J.H. Davenport, Cylindrical algebraic decomposition, Technical Report 88-10, School of Mathematical Sciences, University of Bath, England, 1988. Available at: http://www.bath.ac.uk/masjhd/
21. Du Z., Sharma V., and Yap C.K. Amortized bound for root isolation via Sturm sequences. In: Wang D., and Zhi L. (Eds). Int. Workshop on Symbolic Numeric Computing, School of Science, Beihang University (2005), Birkhauser, Beijing, China 113-129
22. Eigenwillig A., Sharma V., and Yap C.K. Almost tight recursion tree bounds for the Descartes method. ISSAC '06: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (2006), ACM Press, New York, NY, USA. 1-59593-276-3 71-78
23. Emiris I.Z., and Tsigaridas E.P. Computing with real algebraic numbers of small degree. In: Albers S., and Radzik T. (Eds). Proc. 12th European Symposium of Algorithms. ESA. LNCS vol. 3221 (2004), Springer Verlag 652-663
24. I.Z. Emiris, E.P. Tsigaridas, A note on the complexity of univariate root isolation, Research Report RR-6043, INRIA, October 2006. URL: http://hal.inria.fr/inria-00116985/en/
25. I.Z. Emiris, E.P. Tsigaridas, G.M. Tzoumas, The predicates for the Voronoi diagram of ellipses, in: Proc. 22th Annual ACM Symp. on Computational Geometry, Sedona, USA, 2006, pp. 227-236
26. I.Z. Emiris, B. Mourrain, E.P. Tsigaridas, Real algebraic numbers: Complexity analysis and experimentation, in: P. Hertling, C. Hoffmann, W. Luther, N. Revol (Eds.), Reliable Implementations of Real Number Algorithms: Theory and Practice, in: LNCS, Springer Verlag, 2008 (in press). Also available in www.inria.fr/rrrt/rr-5897.html
27. Hensley D. The largest digit in the continued fraction expansion of a rational number. Pacific Journal of Mathematics 151 2 (1991) 237-255
28. Hong H. Bounds for absolute positiveness of multivariate polynomials. Journal of Symbolic Computation 25 5 (1998) 571-585
29. C.P. Hughes, A. Nikeghbali, The zeros of random polynomials cluster uniformly near the unit circle, March 15 2004. URL: http://arxiv.org/abs/math/0406376v3
30. I. Ibragimov, D. Zaporozhets, On the distribution of roots of random polynomials, Technical Report CRC 06-060, Collaborative Research Centre 701, Universität Bielefeld, 2006
31. Ibragimov I., and Zeitouni O. On the roots of random polynomials. Transactions of the American Mathematical Society 349 (1997) 2427-2441
32. J.R. Johnson, Algorithms for Polynomial real root isolation, Ph.D. Thesis, The Ohio State University, 1991
33. Khintchine A. Continued Fractions (1964), University of Chicago Press, Chicago
34. Kioustelidis J. Bounds for the positive roots of polynomials. Journal of Computational and Applied Mathematics 16 (1986) 241-244
35. Knuth D.E. Seminumerical Algorithms. 3rd edition. The Art of Computer Programming vol. 2 (1998), Addison-Wesley, Reading, MA
36. Krandick W. Isolierung reeller nullstellen von polynomen. In: Herzberger J. (Ed). Wissenschaftliches Rechnen (1995), Akademie-Verlag, Berlin 105-154
37. Krandick W., and Mehlhorn K. New bounds for the Descartes method. Journal of Symbolic Computation 41 1 (2006) 49-66
38. Kwek S., and Mehlhorn K. Optimal search for rationals. Information Processing Letters 86 1 (2003) 23-26
39. Lévy P. Sur les lois de probabilitié dont dependent les quotients complets et incomplets d' une fraction continue. Bulletin of the Society of Mathematicians 57 (1929) 178-194
40. M. Michelini, Kuzmin's Extraordinary Zero Measure Set, Senior Thesis, Mathematics Department, Princeton University, 2004
41. Mignotte M. Mathematics for Computer Algebra (1991), Springer-Verlag, New York
42. Mignotte M. On the distance between the roots of a polynomial. Applicable Algebra in Engineering Communication and Computing 6 6 (1995) 327-332
43. Mignotte M., and Ştefǎnescu D. Polynomials: An Algorithmic Approach (1999), Springer
44. B. Mourrain, J-P. Pavone, P. Trébuchet, E.P. Tsigaridas, synaps: A library for symbolic-numeric computing, in: Proc. 8th Int. Symp. on Effective Methods in Algebraic Geometry, MEGA, Italy, May 2005 (software presentation)
45. Pan V.Y. Univariate polynomials: Nearly optimal algorithms for numerical factorization and rootfinding. Journal of Symbolic Computation 33 5 (2002) 701-733
46. Pan V.Y. Solving a polynomial equation: Some history and recent progress. SIAM Review 39 2 (1997) 187-220
47. Richtmyer R., Devaney M., and Metropolis N. Continued fraction expansions of algebraic numbers. Numerische Mathematik 4 (1962) 68-84
48. Rosen D., and Shallit J. A continued fraction algorithm for approximating all real polynomial roots. Mathematics Magazine 51 (1978) 112-116
49. Rouillier F., and Zimmermann Z. Efficient isolation of polynomial's real roots. Journal of Computational and Applied Mathematics 162 1 (2004) 33-50
50. A. Schönhage, The fundamental theorem of algebra in terms of computational complexity, Manuscript. Univ. of Tübingen, Germany, 1982
51. V. Sharma, Complexity of real root isolation using Continued Fractions, in: C.W. Brown (Ed.), Proc. Annual ACM ISSAC, Waterloo, Canada, 2007, pp. 339-346
52. Ştefǎnescu D. New bounds for the positive roots of polynomials. Journal of Universal Computer Science 11 12 (2005) 2132-2141
53. Tsigaridas E.P., and Emiris I.Z. Univariate polynomial real root isolation: Continued fractions revisited. In: Azar Y., and Erlebach T. (Eds). Proc. 14th European Symposium of Algorithms. ESA. LNCS vol. 4168 (2006), Springer Verlag, Zurich, Switzerland 817-828
54. Uspensky J.V. Theory of Equations (1948), McGraw-Hill
55. van der Poorten A. An introduction to continued fractions. Diophantine Analysis (1986), Cambridge University Press 99-138
56. van der Sluis A. Upper bounds for the roots of polynomials. Numerische Mathematik 15 (1970) 250-262
57. Vincent A.J.H. Sur la résolution des équations numériques. Joumalde Mathématiques Pures et Appliquées 1 (1836) 341-372
58. J. von zur Gathen, J. Gerhard, Fast algorithms for Taylor shifts and certain difference equations, in: ISSAC, 1997, pp. 40-47
59. Wang X. A Method for Isolating Roots of Algebraic Equations, Number N. 1 (1960), Univ. Academic Press
60. Yap C.K. Fundamental Problems of Algorithmic Algebra (2000), Oxford University Press, New York