Asymptotically fast computation of subresultants
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Sylvester—Habicht sequences and fast Cauchy index computation
Journal of Symbolic Computation
Introduction to algorithms
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
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Isotopic approximation of implicit curves and surfaces
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On the complexity of real root isolation using continued fractions
Theoretical Computer Science
Complexity of real root isolation using continued fractions
Theoretical Computer Science
Experimental evaluation and cross-benchmarking of univariate real solvers
Proceedings of the 2009 conference on Symbolic numeric computation
Faster algorithms for computing Hong's bound on absolute positiveness
Journal of Symbolic Computation
Random polynomials and expected complexity of bisection methods for real solving
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
A simple but exact and efficient algorithm for complex root isolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
Journal of Symbolic Computation
Empirical study of an evaluation-based subdivision algorithm for complex root isolation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
On soft predicates in subdivision motion planning
Proceedings of the twenty-ninth annual symposium on Computational geometry
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The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I0 is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C0(I) V C1(I) holds on each I. We prove that the size of the recursion tree is O(d(L + r + log d)) where f has degree d, its coefficients have absolute values L, and I0 contains r roots of f. In the range L ≥ d, our bound is the sharpest known, and provably optimal. Our results are closely paralleled by recent bounds on EVAL by Sagraloff-Yap (ISSAC 2011) and Burr-Krahmer (2012). In the range L ≤ d, our bound is incomparable with those of Sagraloff-Yap or Burr-Krahmer. Similar to the Burr-Krahmer proof, we exploit the technique of "continuous amortization" from Burr-Krahmer-Yap (2009), namely to bound the tree size by an integral ∫IO G(x)dx over a suitable "charging function" G(x). We give an application of this feature to the problem of ray-shooting (i.e., finding smallest root in a given interval).