Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Generalized Budan-Fourier theorem and virtual roots
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On the complexity of real root isolation using continued fractions
Theoretical Computer Science
Random polynomials and expected complexity of bisection methods for real solving
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Virtual roots of a real polynomial and fractional derivatives
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A simple but exact and efficient algorithm for complex root isolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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We consider a univariate polynomial f with real coefficients having a high degree N but a rather small number d + 1 of monomials, with d ≪ N. Such a sparse polynomial has a number of real root smaller or equal to d. Our target is to find for each real root of f an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct the sparse structure. Our approach relies on the generalized Budan-Fourier theorem of Coste, Lajous, Lombardi, Roy [8] and the techniques developed in Galligo [12]. To such a f is associated a set of d + 1 F-derivatives. The Budan-Fourier function Vf (x) counts the sign changes in the sequence of F-derivatives of the f evaluated at x. The values at which this function jumps are called the F-virtual roots of f, these include the real roots of f. We also consider the augmented F-virtual roots of f and introduce a genericity property which eases our study. We present a real root isolation method and an algorithm which has been implemented in Maple. We rely on an improved generalized Budan-Fourier count applied to both the input polynomial and its reciprocal, together with Newton like approximation steps. The paper is illustrated with examples and pictures.