Algorithms for polynomial real root isolation
Algorithms for polynomial real root isolation
Asymptotically fast computation of subresultants
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Bounds for absolute positiveness of multivariate polynomials
Journal of Symbolic Computation
Integer Arithmetic Algorithms for Polynomial Real Zero Determination
Journal of the ACM (JACM)
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Sylvester—Habicht sequences and fast Cauchy index computation
Journal of Symbolic Computation
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
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
High-performance implementations of 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
Real Algebraic Numbers: Complexity Analysis and Experimentation
Reliable Implementation of Real Number Algorithms: Theory and Practice
Complexity of real root isolation using continued fractions
Theoretical Computer Science
A note about the average number of real roots of a Bernstein polynomial system
Journal of Complexity
Faster algorithms for computing Hong's bound on absolute positiveness
Journal of Symbolic Computation
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Virtual roots of a real polynomial and fractional derivatives
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Roots of the derivatives of some random polynomials
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
A root isolation algorithm for sparse univariate polynomials
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Near optimal tree size bounds on a simple real root isolation algorithm
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Improved bounds for the CF algorithm
Theoretical Computer Science
On the boolean complexity of real root refinement
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones? We exploit results by Kac, and by Edelman and Kostlan in order to estimate the real root separation of degree d polynomials with i.i.d. coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the i-th coefficient has variance (d/i), whereas for Weyl polynomials its variance is 1/i!. By applying results from statistical physics, we obtain the expected (bit) complexity of STURM solver, ÕB(rd2τ), where r is the number of real roots and τ the maximum coefficient bitsize. Our bounds are two orders of magnitude tighter than the record worst case ones. We also derive an output-sensitive bound in the worst case. The second part of the paper shows that the expected number of real roots of a degree d polynomial in the Bernstein basis is √2d ± O(1), when the coefficients are i.i.d. variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis.