Complexity of real root isolation using continued fractions

Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Volumn , Issue , 2007, Pages 339-346

  Sharma, Vikram ^a,b  

^a INRIA   (France)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Continued fractions; Davenport Mahler bound; Descartes rule of signs; Polynomial real root isolation

Indexed keywords

COMPUTATIONAL COMPLEXITY; NUMERICAL METHODS; POLYNOMIALS;

CONTINUED FRACTIONS; DAVENPORT-MAHLER BOUND; DESCARTES RULE OF SIGNS; POLYNOMIAL REAL ROOT ISOLATION;

ALGORITHMS;

EID: 36349000609     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1277548.1277594     Document Type: Conference Paper

References (30)

1. A. G. Akritas. Vincent's theorem in algebraic manipulation. PhD thesis, Operations Research Program, North Carolina State University, Raleigh, North Carolina, 1978.
2. A. G. Akritas. There is no "Uspensky's method". In Proc. of the 1986 Symp. on Symbolic and Algebraic Computation, pages 88-90, Waterloo, Ontario, Canada, 1986.
3. A. G. Akritas. Elements of Computer Algebra with Applications. John Wiley Interscience, New York, 1989.
4. A. G. Akritas and A. Strzébonski. A comparative study of two real root isolation methods. Nonlinear Analysis:Modelling and Control, 10(4):297-304, 2005.
5. A. G. Akritas, A. Strzeboński, and P. Vigklas. Implementations of a New Theorem for Computing Bounds for Positive Roots of Polynomials. Computing, 78:355-367, 2006. Published online.
6. A. Alesina and M. Galuzzi. A new proof of Vincent's theorem. L'Enseignement Mathémathique, 44:219-256, 1998.
7. S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry. Springer, 2nd edition, 2006.
8. G. E. Collins and A. G. Akritas. Polynomial real root isolation using Descartes' rule of signs. In R. D. Jenks, editor, Proc. of the 1976 ACM Symp. on Symbolic and Algebraic Computation, pages 272-275. ACM Press, 1976.
9. J. H. Davenport. Computer algebra for cylindrical algebraic decomposition. Technical report, The Royal Institute of Technology, Department of Numerical Analysis and Computing Science, S-100 44, Stockholm, Sweden, 1985. Reprinted as: Tech. Rep. 88-10, School of Math. Sciences, U. of Bath, Bath, England, http://www.bath.ac.uk/masjhd/TRITA.pdf.
10. Z. Du, V. Sharma, and C. Yap. Amortized bounds for root isolation via Sturm sequences. In D. Wang and L. Zhi, editors, Proc. Int'l Workshop on Symbolic-Numeric Computation, pages 81-93, School of Science, Beihang University, Beijing, China, 2005.
11. A. Eigenwillig, V. Sharma, and C. Yap. Almost tight complexity bounds for the Descartes method. In Proc. 2006 Int'l Symp. Symbolic and Algebraic Computation (ISSAC'06), 2006. Genova, Italy.
12. I. Z. Emiris, B. Mourrain, and E. P. Tsigaridas. Real algebraic numbers: Complexity analysis and experimentations. Research Report 5897, INRIA, April 2006. http://www.lnria.fr/rrrt/rr-6897.html.
13. H. Hong. Bounds for absolute positiveness of multivariate polynomials. J. of Symbolic Computation, 25(5):571-585, 1998.
14. V. Karamcheti, C. Li, I. Pechtchanski, and C. Yap. A Core library for robust numerical and geometric computation. In 15th ACM Symp. Comp. Geo., pages 351-359, 1999.
15. A. Y. Khinchin. Continued Fractions. Dover Publications, 1997.
16. J. Kioustelidis. Bounds for the positive roots of the polynomials. Journal of Computational and Applied Mathematics, 16:241-244, 1986.
17. W. Krandick. Isolierung reeller Nullstellen von Polynomen. In J. Herzberger, editor, Wissenschaftliches Rechnen, pages 105-154. Akademie-Verlag, Berlin, 1995.
18. W. Krandick and K. Mehlhorn. New bounds for the Descartes method. J. Symbolic Computation, 41(1):49-66, 2006.
19. M. Mignotte. Some inequalities about univariate polynomials. In Proc. 1981 ACM Symp. on Symbolic and Algebraic Computation (SYMSAC 1981), pages 195-199. ACM, 1981.
20. M. Mignotte and D. Ştefǎnescu. Polynomials: An Algorithmic Approach. Springer, Singapore, 1999.
21. B. Mourrain, J. Pavone, E. Tsigaridas, and P. Trébuchet. SYNAPS, a library for symbolic-numeric computation. In 8th Int'l Symp. on Effective Methods in Algebraic Geometry, MEGA, Sardinia, Italy, May 2005. Software presentation.
22. B. Mourrain, F. Rouillier, and M.-F. Roy. Bernstein's basis and real root isolation. Rapport de recherche 5149, INRIA-Rocquencourt, March 2004. http://www.lnrla.fr/rrrt/rr-5149.html.
23. A. Ostrowski. Note on Vincent's theorem. The Annals of Mathematics, 52(3):702-707, Nov 1950.
24. V. Sharma. Complexity Analysis of Algorithms in Algebraic Computation. PhD thesis, Dept. of Computer Science, NYU, Jan. 2007.
25. D. Ştefǎnescu. New bounds for the positive roots of polynomials. Journal of Universal Computer Science, 11(12):2132-2141, 2005.
26. E. P. Tsigaridas and I. Z. Emiris. Univariate polynomial real root isolation: Continued fractions revisited. In Y. Azar and T. Erlebach, editors, Proc. 13th European Symp. on Algorithms (ESA), volume 4168 of LNCS, pages 817-828. Springer, 2006.
27. J. V. Uspensky. Theory of Equations. McGraw-Hill, New York, 1948.
28. A. Vincent. Sur la résolution des équations numériques. J. Math. Pures Appl., 1:341-372, 1836.
29. J. von zur Gathen and J. Gerhard. Fast algorithms for Taylor shifts and certain difference equations. In Proc. 1997 Int'l Symp. on Symbolic and Algebraic Computation (ISSAC 1997), pages 40-47. ACM, 1997.
30. C. K. Yap. Fundamental Problems of Algorithmic Algebra. Oxford University Press, 2000.