Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function
ACM Transactions on Mathematical Software (TOMS)
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
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Fast and exact geometric analysis of real algebraic plane curves
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On the Complexity of Reliable Root Approximation
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Efficient real root approximation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A general approach to the analysis of controlled perturbation algorithms
Computational Geometry: Theory and Applications
A descartes algorithm for polynomials with bit-stream coefficients
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Efficient real root approximation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
On the complexity of solving a bivariate polynomial system
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
Univariate real root isolation in multiple extension fields
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
From approximate factorization to root isolation
Proceedings of the 38th international symposium on 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 consider the problem of approximating all real roots of a square-free polynomial f. Given isolating intervals, our algorithm refines each of them to a width at most 2-L, that is, each of the roots is approximated to L bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating f, without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of degree, coefficient size and discriminant. Our bound improves previous work on integer polynomials by a factor of deg f and essentially matches best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms.