Abstract
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial and is challenged to approximate them faster. The challenge is known for long time, and the subject has been intensively studied. The real roots can be approximated at a low computational cost if the polynomial has no non-real roots, but for high degree polynomials, non-real roots are typically much more numerous than the real ones. The bounds on the Boolean cost of the refinement of the simple and isolated real roots have been decreased to nearly optimal, but the success has been more limited at the stage of the isolation of real roots. By revisiting the algorithm of 1982 by Schönhage for the approximation of the root radii, that is, the distances between the roots and the origin, we obtain substantial progress: we isolate the simple and well conditioned real roots of a polynomial at the Boolean cost dominated by the nearly optimal bounds for the refinement of such roots. Our numerical tests with benchmark polynomials performed with the IEEE standard double precision show that the resulting nearly optimal real root-finder is practically promising. Our techniques are simple, and their power and application range may increase in combination with the known efficient methods. At the end we point out some promising directions to the isolation of complex roots and root clusters.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alt, H.: Multiplication is the easiest non-trivial arithmetic function. Theoretical Computer Science 36, 333–339 (1985)
Bruno, A.D.: The Local Method of non-linear Analysis of Differential Equations. Nauka, Moscow (1979) (in Russian). English translation: Soviet Mathematics, Springer, Berlin (1989)
Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Fizmatlit, Moscow (in Russian), English translation: North-Holland Mathematical Library, vol. 57. Elsevier, Amsterdam (2000), also reprinted in 2005, ISBN 0-444-50297
Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms 23, 127–173 (2000)
Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Computational and Applied Mathematics 272, 276–292 (2014)
Fiduccia, C.M.: Polynomial evaluation via the division algorithm: the fast fourier transform revisited. In: Proc. 4th Annual ACM Symposium Theory of Computing, pp. 88–93 (1972)
Fortune, S.: An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symb. Comput. 33(5), 627–646 (2002)
Graham, R.I.: An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters 1, 132–133 (1972)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. The Johns Hopkins University Press, Baltimore (2013)
Householder, A.S.: Dandelin, Lobachevskii, or Graeffe. Amer. Math. Monthly 66, 464–466 (1959)
Henrici, P.: Applied and Computational Complex Analysis. Wiley, New York (1974)
Hong, H.: Bounds for absolute positiveness of multivariate polynomials. J. Symb. Comput. 25, 571–585 (1998)
Kioustelidis, J.B.: Bounds for positive roots of polynomials. J. Comput. Appl. Math. 16, 241–244 (1986)
Kirrinnis, P.: Partial fraction decomposition in C(z) and simultaneous Newton iteration for factorization in C[z]. J. Complexity 14, 378–444 (1998)
Kobel, A., Sagraloff, M.: On the complexity of computing with planar algebraic curves. J. Complexity, in press. Also http://arxiv.org/abs/1304.8069v1 arXiv:1304.8069v1 [cs.NA], April 30, 2013
Lagrange, J.L.: Traité de la résolution des équations numériques. Paris (1798). (Reprinted in Œuvres, t. VIII, Gauthier-Villars, Paris (1879))
Moenck, R., Borodin, A.: Fast modular transform via division. In: Proc. 13th Annual Symposium on Switching and Automata Theory, pp. 90–96, IEEE Comp. Society Press, Washington, DC (1972)
McNamee, J.M., Pan, V.Y.: Numerical Methods for Roots of Polynomials. Part 2 (XXII + 718 pages), Elsevier (2013)
Ostrowski, A.M.: Recherches sur la méthode de Graeffe et les zéros des polynomes et des series de Laurent. Acta Math. 72, 99–257 (1940)
Pan, V.Y.: Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros. In: Proc. 27th Ann. ACM Symp. on Theory of Computing, pp. 741–750. ACM Press, New York (1995)
Pan, V.Y.: Approximating complex polynomial zeros: modified quadtree (Weyl’s) construction and improved Newton’s iteration. J. Complexity 16(1), 213–264 (2000)
Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser, Springer, Boston, New York (2001)
Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for factorization and rootfinding. J. Symb. Computations 33(5), 701–733 (2002). In: Proc. version in ISSAC’2001, pp. 253–267, ACM Press, New York (2001)
Pan, V.Y.: Transformations of matrix structures work again. Linear Algebra and Its Applications 465, 1–32 (2015)
Pan, V.Y.: Fast Approximate Computations with Cauchy Matrices and Polynomials, http://arxiv.org/abs/1506.02285 arXiv:1506.02285 [math.NA], 32 p., 6 figures, 8 tables, June 7, 2015
Pan, V.Y., Linzer, E.: Bisection acceleration for the symmetric tridiagonal eigenvalue problem. Numerical Algorithms 22(1), 13–39 (1999)
Pan, V.Y., Murphy, B., Rosholt, R.E., Qian, G., Tang, Y.: Real root-finding. In: Watt, S.M., Verschelde, J. (eds.) Proc. 2nd ACM International Workshop on Symbolic-Numeric Computation (SNC), pp. 161–169 (2007)
Pan, V.Y., Tsigaridas, E.P.: On the Boolean complexity of the real root refinement. In: M. Kauers (ed.) Proc. Intern. Symposium on Symbolic and Algebraic Computation (ISSAC 2013), pp. 299–306, Boston, MA, June 2013. ACM Press, New York (2013). Also arXiv 1404.4775 April 18, 2014
Pan, V.Y., Tsigaridas, E.P.: Nearly optimal computations with structured matrices. In: Proc. the Int. Conf. on Symbolic Numeric Computation (SNC 2014). ACM Press, New York (2014). Also April 18, 2014, http://arxiv.org/abs/1404.4768 arXiv:1404.4768 [math.NA]
Pan, V.Y., Tsigaridas, E.P.: Accelerated approximation of the complex roots of a univariate polynomial. In: the 2014 Proc. of the Int. Conf. on Symbolic Numeric Computation (SNC 2014), Shanghai, China, July 2014, pp. 132–134. ACM Press, New York (2014). Also April 18, 2014, arXiv : 1404.4775 [math.NA]
Pan, V.Y., Tsigaridas, E.: Nearly optimal refinement of real roots of a univariate polynomial. J. Symb. Comput. (in press)
Pan, V.Y., Zhao, L.: Polynomial root isolation by means of root radii approximation, arxiv, 1501.05386, June 15, 2015
Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of polynomials. J. Complexity 3(2), 90–113 (1987)
Sieveking, M.: An algorithm for division of power series. Computing 10, 153–156 (1972)
Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity. Math. Department, Univ. Tübingen, Germany (1982)
Stefanescu, D.: New bounds for positive roots of polynomials. Univ. J. Comput. Sci. 11, 2125–2131 (2005)
Sagraloff, M., Mehlhorn, K.: Computing real roots of real polynomials - an efficient method based on Descartes’ rule of signs and Newton iteration. J. Symb. Comput. (in press)
Tilli, P.: Convergence conditions of some methods for the simultaneous computations of polynomial zeros. Calcolo 35, 3–15 (1998)
van der Hoeven, J.: Fast composition of numeric power series. Tech. Report 2008–09, Université Paris-Sud, Orsay, France (2008)
Van der Sluis, A.: Upper bounds on the roots of polynomials. Numerische Math. 15, 250–262 (1970)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Pan, V.Y., Zhao, L. (2015). Polynomial Real Root Isolation by Means of Root Radii Approximation. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-24021-3_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24020-6
Online ISBN: 978-3-319-24021-3
eBook Packages: Computer ScienceComputer Science (R0)