Michael Kerber
ABSTRACT
We consider the problem of approximating all real roots of a square-free polynomial f. Given isolating intervals, our algorithm refines each of them to a width at most 2-L, that is, each of the roots is approximated to L bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating f, without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of degree, coefficient size and discriminant. Our bound improves previous work on integer polynomials by a factor of deg f and essentially matches best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms.
- J. Abbott. Quadratic Interval Refinement for Real Roots. Poster presented at the Intern. Symp. on Symbolic and Algebraic Computation (ISSAC 2006), 2006.Google Scholar
- S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry. Springer, 2nd edition, 2006. Google ScholarDigital Library
- E. Berberich, P. Emeliyanenko, and M. Sagraloff. An elimination method for solving bivariate polynomial systems: Eliminating the usual drawbacks. In Workshop on Algorithm Engineering & Experiments (ALENEX), pages 35--47, 2011.Google ScholarCross Ref
- E. Berberich, M. Hemmer, and M. Kerber. A generic algebraic kernel for non-linear geometric applications. Research report 7274, INRIA, 2010.Google Scholar
- J. Bus and T.J.Dekker. Two efficient algorithms with guaranteed convergence for finding a zero of a function. ACM Trans. on Math. Software, 1(4):330--345, 1975. Google ScholarDigital Library
- J. Cheng, S. Lazard, L. Peñaranda, M. Pouget, F. Rouillier, and E. Tsigaridas. On the topology of real algebraic plane curves. Mathematics in Computer Science, 4:113--137, 2010.Google ScholarCross Ref
- G. E. Collins and A. G. Akritas. Polynomial Real Root Isolation Using Descartes' Rule of Signs. In Proc. of the 3rd ACM Symp. on Symbolic and Algebraic Computation (SYMSAC 1976), pages 272--275. ACM Press, 1976. Google ScholarDigital Library
- D.Eppstein, M.S.Paterson, and F.F.Yao. On nearest-neighbor graphs. Discrete and Computational Geometry, 17(3):263--282, 1997.Google ScholarDigital Library
- Z. Du, V. Sharma, and C. Yap. Amortized bound for root isolation via Sturm sequences. In Symbolic-Numeric Computation, Trends in Mathematics, pages 113--129. Birkhäuser Basel, 2007.Google Scholar
- A. Eigenwillig, M. Kerber, and N. Wolpert. Fast and exact geometric analysis of real algebraic plane curves. In Proc. of the 2007 Intern. Symp. on Symbolic and Algebraic Computation (ISSAC 2007), pages 151--158, 2007. Google ScholarDigital Library
- A. Eigenwillig, L. Kettner, W. Krandick, K. Mehlhorn, S. Schmitt, and N. Wolpert. A Descartes algorithm for polynomials with bit-stream coefficients. In 8th International Workshop on Computer Algebra in Scientific Computing (CASC 2005), volume 3718 of LNCS, pages 138--149, 2005. Google ScholarDigital Library
- M. Kerber. On the complexity of reliable root approximation. In 11th International Workshop on Computer Algebra in Scientific Computing (CASC 2009), volume 5743 of LNCS, pages 155--167. Springer, 2009. Google ScholarDigital Library
- M. Kerber and M. Sagraloff. Supplementary material for "Efficient Real Root Approximation", 2011. phhttp://www.mpi-inf.mpg.de/~msagralo/apx11.pdf. Google ScholarDigital Library
- K. Mehlhorn, R. Osbild, and M. Sagraloff. A general approach to the analysis of controlled perturbation algorithms. CGTA, 2011. to appear; for a draft, see phhttp://www.mpi-inf.mpg.de/~msagralo/cpgeneral.pdf. Google ScholarDigital Library
- V. Y. Pan. Optimal and nearly optimal algorithms for approximating polynomial zeros. Computers and Mathematics with Applications, 31(12):97--138, 1996.Google ScholarCross Ref
- V. Y. Pan. Solving a polynomial equation: Some history and recent progress. SIAM Review, 39(2):187--220, 1997. Google ScholarDigital Library
- F. Rouillier and P. Zimmermann. Efficient isolation of polynomial's real roots. Journal of Compututational and Applied Mathematics, 162(1):33--50, 2004. Google ScholarDigital Library
- M. Sagraloff. On the complexity of real root isolation. arXiv:1011.0344v1, 2010.Google Scholar
- C. K. Yap. Fundamental Problems in Algorithmic Algebra. Oxford University Press, 2000. Google ScholarDigital Library
Index Terms
- Efficient real root approximation
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