Mathematics for computer algebra
Mathematics for computer algebra
Algorithms for polynomial real root isolation
Algorithms for polynomial real root isolation
Polynomial real root isolation using approximate arithmetic
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic 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
An algorithm for isolating the real solutions of semi-algebraic systems
Journal of Symbolic Computation
A modular GCD algorithm over number fields presented with multiple extensions
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Polynomial real root isolation by differentiation
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
On the sign of a real algebraic number
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Polynomial real root isolation using Descarte's rule of signs
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Towards faster real algebraic numbers
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
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
Real root isolation for algebraic polynomials
ACM SIGSAM Bulletin
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Change of order for bivariate triangular sets
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Real solution isolation using interval arithmetic
Computers & Mathematics with Applications
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
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
Complete numerical isolation of real roots in zero-dimensional triangular systems
Journal of Symbolic Computation
On the Complexity of Reliable Root Approximation
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Faster algorithms for computing Hong's bound on absolute positiveness
Journal of Symbolic 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
A descartes algorithm for polynomials with bit-stream coefficients
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Univariate real root isolation in multiple extension fields
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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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We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L=Qα is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of OB(N10) for isolating the real roots of Bα, where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of Descartes' algorithm introduced by Sagraloff. For the former we prove a complexity bound of OB(N8) and for the latter a bound of OB(N7). We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.