Computer algebra: symbolic and algebraic computation (2nd ed.)
Bounds for positive roots of polynomials
Journal of Computational and Applied Mathematics
There is no “Uspensky's method.”
SYMSAC '86 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation
Elements of computer algebra with applications
Elements of computer algebra with applications
Mathematics for computer algebra
Mathematics for computer algebra
Algorithms for polynomial real root isolation
Algorithms for polynomial real root isolation
Mathematics of Computation
Fast algorithms for Taylor shifts and certain difference equations
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Bounds for absolute positiveness of multivariate polynomials
Journal of Symbolic Computation
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
A Comparative Study of Algorithms for Computing Continued Fractions of Algebraic Numbers
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
Information Processing Letters
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
The predicates for the Voronoi diagram of ellipses
Proceedings of the twenty-second annual symposium on Computational geometry
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Univariate polynomial real root isolation: continued fractions revisited
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Complexity of real root isolation using continued fractions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Real Algebraic Numbers: Complexity Analysis and Experimentation
Reliable Implementation of Real Number Algorithms: Theory and Practice
New bounds for the Descartes method
Journal of Symbolic Computation
Real algebraic numbers and polynomial systems of small degree
Theoretical Computer Science
Complexity of real root isolation using continued fractions
Theoretical Computer Science
Univariate Algebraic Kernel and Application to Arrangements
SEA '09 Proceedings of the 8th International Symposium on Experimental Algorithms
Isolating real roots of real polynomials
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Experimental evaluation and cross-benchmarking of univariate real solvers
Proceedings of the 2009 conference on Symbolic numeric computation
Continued fraction expansion of real roots of polynomial systems
Proceedings of the 2009 conference on Symbolic numeric computation
Random polynomials and expected complexity of bisection methods for real solving
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
The DMM bound: multivariate (aggregate) separation bounds
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
Theoretical Computer Science
Univariate real root isolation in an extension field
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
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
Journal of Symbolic Computation
On the computing time of the continued fractions method
Journal of Symbolic Computation
A root isolation algorithm for sparse univariate polynomials
Proceedings of the 37th International Symposium on Symbolic and Algebraic 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
Improved bounds for the CF algorithm
Theoretical Computer Science
Hi-index | 5.23 |
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We derive an expected complexity bound of O@?"B(d^6+d^4@t^2), where d is the polynomial degree and @t bounds the coefficient bit size, using a standard bound on the expected bit size of the integers in the continued fraction expansion, thus matching the current worst-case complexity bound for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Moreover, using a homothetic transformation we improve the expected complexity bound to O@?"B(d^3@t). We compute the multiplicities within the same complexity and extend the algorithm to non-square-free polynomials. Finally, we present an open-source C++ implementation in the algebraic library synaps, and illustrate its completeness and efficiency as compared to some other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000.