There is no “Uspensky's method.”
SYMSAC '86 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation
Quantifier elimination and the sign variation method for real root isolation
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
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ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
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Polynomial root finding using iterated Eigenvalue computation
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Interval arithmetic in cylindrical algebraic decomposition
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Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Polynomial real root isolation using Descarte's rule of signs
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Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Near-optimal parameterization of the intersection of quadrics
Proceedings of the nineteenth annual symposium on Computational geometry
Trees and jumps and 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
Intersecting quadrics: an efficient and exact implementation
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On the computation of an arrangement of quadrics in 3D
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Exact, efficient, and complete arrangement computation for cubic curves
Computational Geometry: Theory and Applications
Intersecting quadrics: an efficient and exact implementation
Computational Geometry: Theory and Applications
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Univariate polynomial real root isolation: continued fractions revisited
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Proceedings of the 2007 international workshop on Symbolic-numeric computation
Complete numerical isolation of real zeros in zero-dimensional triangular systems
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Fast and exact geometric analysis of real algebraic plane curves
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Real Algebraic Numbers: Complexity Analysis and Experimentation
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Real algebraic numbers and polynomial systems of small degree
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Univariate Algebraic Kernel and Application to Arrangements
SEA '09 Proceedings of the 8th International Symposium on Experimental Algorithms
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Experimental evaluation and cross-benchmarking of univariate real solvers
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Computing minimum distance between two implicit algebraic surfaces
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Random polynomials and expected complexity of bisection methods for real solving
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The first rational Chebyshev knots
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This paper revisits an algorithm isolating the real roots of a univariate polynomial using Descartes' rule of signs. It follows work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick.Our first contribution is a generic algorithm which enables one to describe all the known algorithms based on Descartes' rule of sign and the bisection strategy in a unified framework.Using that framework, a new algorithm is presented, which is optimal in terms of memory usage and as fast as both Collins and Akritas' algorithm and Krandick's variant, independently of the input polynomial. From this new algorithm, we derive an adaptive semi-numerical version, using multi-precision interval arithmetic.We finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials, including orthogonal polynomials of degree 1000 and more, which were out of reach of previous methods.