Abstract
In this paper, we study a bound on the real roots of a polynomial by Lagrange. From known results in the literature, it follows that Lagrange’s bound is also a bound on the absolute positiveness of a polynomial. A simple \(O(n\log n)\) algorithm described in Mehlhorn-Ray (2010) can be used to compute the bound. Our main result is that this is optimal in the real RAM model. Our paper explores the tradeoff between improving the quality of bounds on absolute positiveness and their computational complexity.
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References
Akritas, A.G., Strzeboński, A., Vigklas, P.: Implementations of a new theorem for computing bounds for positive roots of polynomials. Computing 78, 355–367 (2006)
Akritas, A.: Vincent’s theorem in algebraic manipulation. Ph.D. thesis, Operations Research Program, North Carolina State University, Raleigh, North Carolina (1978)
Batra, Prashant: On the quality of some root-bounds. In: Kotsireas, Ilias S., Rump, Siegfried M., Yap, Chee K. (eds.) MACIS 2015. LNCS, vol. 9582, pp. 591–595. Springer, Heidelberg (2016). doi:10.1007/978-3-319-32859-1_50
Batra, P., Sharma, V.: Bounds on absolute positiveness of multivariate polynomials. J. Symb. Comput. 45(6), 617–628 (2010)
Collins, G.E.: Krandick’s proof of Lagrange’s real root bound claim. J. Symb. Comput. 70(C), 106–111 (2015). http://dx.doi.org/10.1016/j.jsc.2014.09.038
Hong, H.: Bounds for absolute positiveness of multivariate polynomials. J. Symb. Comput. 25(5), 571–585 (1998)
Kioustelidis, J.: Bounds for the positive roots of polynomials. J. Comput. Appl. Math. 16, 241–244 (1986)
Lagrange, J.L.: Traité de la résolution des équations numériques de tous les degrés, Œuvres de Lagrange, vol. 8, 4th edn. Gauthier-Villars, Paris (1879)
Mehlhorn, K., Ray, S.: Faster algorithms for computing Hong’s bound on absolute positiveness. J. Symbol. Comput. 45(6), 677–683 (2010). http://www.sciencedirect.com/science/article/pii/S0747717110000301
Mignotte, M., Ştefănescu, D.: On an Estimation of Polynomial Roots by Lagrange. Prepublication de l’Institut de Recherche Mathématique Avancée, IRMA, Univ. de Louis Pasteur et C.N.R.S. (2002). https://books.google.co.in/books?id=NAd4NAEACAAJ
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)
Sharma, V.: Complexity of real root isolation using continued fractions. Theor. Comput. Sci. 409(2), 292–310 (2008)
van der Sluis, A.: Upper bounds for roots of polynomials. Numer. Math. 15, 250–262 (1970)
Ştefănescu, D.: New bounds for the positive roots of polynomials. J. Univ. Comput. Sci. 11(12), 2132–2141 (2005)
Ştefănescu, D.: A new polynomial bound and its efficiency. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 457–467. Springer, Switzerland (2015). http://dx.doi.org/10.1007/978-3-319-24021-3_33
Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)
Acknowledgement
The authors would like to express their gratitude to Dr. Prashant Batra and the referees for their invaluable comments and suggestions.
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Prabhakar, S.N., Sharma, V. (2016). A Lower Bound for Computing Lagrange’s Real Root Bound. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_28
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DOI: https://doi.org/10.1007/978-3-319-45641-6_28
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