Abstract
This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.
For a square-free polynomial A (X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most O(dL + dlgd) Sturm probes. This extends Davenport’s result for the case of isolating all real roots.
We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d 3 L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.
The work is supported by NSF Grant #CCF-043836. A preliminary version of this work appeared at the International Workshop on Symbolic-Numeric Computation (SNC 2005), Xi’an, China, July 19–21, 2005.
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References
G.E. Collins and R. Loos. Real zeros of polynomials. In B. Buchberger, G. E. Collins and R. Loos, editors, Computer Algebra, pages 83–94. Springer-Verlag, 2nd edition, 1983.
T.H. Corman, C.E. Leiserson, R.L. Rivest and C. Stein. Introduction to Algorithms. The MIT Press and McGraw-Hill Book Company, Cambridge, Massachusetts and New York, second edition, 2001.
J.H. Davenport. Computer algebra for cylindrical algebraic decomposition. Technical report, The Royal Institute of Technology, Dept. of Numerical Analysis and Computing Science, S-100 44, Stockholm, Sweden, 1985. Reprinted as: Tech. Report 88-10, School of Mathematical Sciences, Univ. of Bath, Claverton Down, bath BA2 7AY, England.
A. Eigenwillig, V. Sharma and Chee Yap. Almost tight complexity bounds for the Descartes method. In Proc. Int’l Symp. Symbolic and Algebraic Computation (ISSAC’06), 2006. To appear. Genova, Italy. Jul 9–12, 2006.
J.R. Johnson. Algorithms for polynomial real root isolation. In B.F. Caviness and J.R. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and monographs in Symbolic Computation, pages 269–299. Springer, 1998.
M.-H. Kim and S. Sutherland. Polynomial root-finding algorithms and branched covers. SIAM J. Computing, 23:415–436, 1994.
T. Lickteig and M.-F. Roy. Sylvester-Habicht sequences and fast Cauchy index computation. J. of Symbolic Computation, 31:315–341, 2001.
M. Mignotte and D. Ştefănescu. Polynomials: An Algorithmic Approach. Springer, 1999.
P.S. Milne. On the solutions of a set of polynomial equations. In B. R. Donald, D. Kapur, and J. L. Mundy, editors, Symbolic and Numerical Computation for Artificial Intelligence, pages 89–102. Academic Press, London, 1992.
C.A. Neff and J.H. Reif. An efficient algorithm for the complex roots problem. J.Complexity, 12(3):81–115, 1996.
V.Y. Pan. Optimal and nearly optimal algorithms for approximating polynomial zeros. Computers Mathematics and Applications, 31(12):97–138, 1996.
V.Y. Pan. Solving a polynomial equation: some history and recent progress. SIAM Review, 39(2):187–220, 1997.
P. Pedersen. Counting real zeroes. Technical Report 243, Courant Institute of Mathematical Sci., Robotics Lab., New York Univ., 1990. PhD Thesis, Courant Institute, New York University.
J.R. Pinkert. An exact method for finding the roots of a complex polynomial. ACM Trans. on Math. Software, 2:351–363, 1976.
D. Reischert. Asymptotically fast computation of subresultants. In ISSAC 97, pages 233–240, 1997. Maui, Hawaii.
J. Renegar. On the worst-case arithmetic complexity of approximating zeros of polynomials. Journal of Complexity, 3:90–113, 1987.
A. Schönhage. The fundamental theorem of algebra in terms of computational complexity, 1982. Manuscript, Department of Mathematics, University of Tübingen.
V. Strassen. The computational complexity of continued fractions. SIAM J. Computing, 12:1–27, 1983.
H.S. Wilf. A global bisection algorithm for computing the zeros of polynomials in the complex plane. Journal of the ACM, 25:415–420, 1978.
C.K. Yap. Fundamental Problems of Algorithmic Algebra. Oxford University Press, 2000.
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Du, Z., Sharma, V., Yap, C.K. (2007). Amortized Bound for Root Isolation via Sturm Sequences. In: Wang, D., Zhi, L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7984-1_8
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DOI: https://doi.org/10.1007/978-3-7643-7984-1_8
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