Abstract
The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial \(q(x)=q_{n^{x^{n}}}+...+q_{0}\) with root separation ρ, coefficients |q n | ≥ 1 and \(|q_{i}|\leq 2^{\tau}\), it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ)2) bit operations.
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Collins, G.E., Akritas, A.G.: Polynomial real root isolation using Descartes’ rule of signs. In: Jenks, R.D. (ed.) Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, pp. 272–275. ACM Press, New York (1976)
Uspensky, J.: Theory of Equations. McGraw-Hill, New York (1948)
Krandick, W.: Isolierung reeller Nullstellen von Polynomen. In: Herzberger, J. (ed.) Wissenschaftliches Rechnen, pp. 105–154. Akademie-Verlag (1995)
Rouillier, F., Zimmermann, P.: Efficient isolation of a polynomial’s real roots. J. Computational and Applied Mathematics 162, 33–50 (2004)
Lane, J.M., Riesenfeld, R.F.: Bounds on a polynomial. BIT 21, 112–117 (1981)
Mourrain, B., Vrahatis, M.N., Yakoubsohn, J.C.: On the complexity of isolating real roots and computing with certainty the topological degree. J. Complexity 18, 612–640 (2002)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)
Mourrain, B., Rouillier, F., Roy, M.F.: Bernstein’s basis and real root isolation. Rapport de recherche 5149, INRIA-Rocquencourt (2004), http://www.inria.fr/rrrt/rr-5149.html—
Collins, G.E., Johnson, J.R., Krandick, W.: Interval arithmetic in cylindrical algebraic decomposition. J. Symbolic Computation 34, 143–155 (2002)
Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, Chichester (1974)
Weyl, H.: Randbemerkungen zu Hauptproblemen der Mathematik II: Fundamentalsatz der Algebra und Grundlagen der Mathematik. Math. Z. 20, 131–152 (1924)
Pan, V.: Solving a polynomial equation: Some history and recent progress. SIAM Review 39, 187–220 (1997)
Pan, V.Y.: Univariate polynomials: Nearly optimal algorithms for numerical factorization and root finding. J. Symbolic Computation 33, 701–733 (2002)
Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Heidelberg (2002)
Hoschek, J., Lasser, D.: Fundamentals of computer aided geometric design. A K Peters, Wellesley (1996); Translation of: Grundlagen der geometrischen Datenverarbeitung, Teubner(1989)
Krandick, W., Mehlhorn, K.: New bounds for the Descartes method. Technical report, Drexel University, Dept. of Computer Science (2004); to appear in J. Symbolic Computation, http://www.mcs.drexel.edu/page.php?name=reports/DU-CS-04-04.html
Smith, B.T.: Error bounds for zeros of a polynomial based upon Gerschgorin’s theorems. J. ACM 17, 661–674 (1970)
Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms 23, 127–173 (2000)
Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)
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Eigenwillig, A., Kettner, L., Krandick, W., Mehlhorn, K., Schmitt, S., Wolpert, N. (2005). A Descartes Algorithm for Polynomials with Bit-Stream Coefficients. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_12
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DOI: https://doi.org/10.1007/11555964_12
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