Mathematics for computer algebra
Mathematics for computer algebra
Generic programming and the STL: using and extending the C++ Standard Template Library
Generic programming and the STL: using and extending the C++ Standard Template Library
Interval arithmetic in cylindrical algebraic decomposition
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
Polynomial real root isolation using Descarte's rule of signs
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
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 and polynomial systems of small degree
Theoretical Computer Science
Complexity of real root isolation using continued fractions
Theoretical Computer Science
EXACUS: efficient and exact algorithms for curves and surfaces
ESA'05 Proceedings of the 13th annual European conference on Algorithms
A descartes algorithm for polynomials with bit-stream coefficients
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Real and complex polynomial root-finding with eigen-solving and preprocessing
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
A deterministic algorithm for isolating real roots of a real polynomial
Journal of Symbolic Computation
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Theoretical Computer Science
A generic algebraic kernel for non-linear geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
Journal of Symbolic Computation
When Newton meets Descartes: a simple and fast algorithm to isolate the real roots of a polynomial
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
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Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA,the VEGAS group at LORIA and the MPI Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5,000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8,000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.