# When Newton meets Descartes: a simple and fast algorithm to isolate the real roots of a polynomial

```@inproceedings{Sagraloff2012WhenNM,
title={When Newton meets Descartes: a simple and fast algorithm to isolate the real roots of a polynomial},
author={Michael Sagraloff},
booktitle={ISSAC},
year={2012}
}```
We introduce a novel algorithm denoted NewDsc to isolate the real roots of a univariate square-free polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f and performs exact (rational) operations on the coefficients of f in each step. For the subdivision strategy, we combine Descartes' Rule of Signs and Newton iteration. More precisely, instead of using a fixed subdivision strategy such as bisection in each…
54 Citations
Computing Real Roots of Real Polynomials ... and now For Real!
• Mathematics, Computer Science
ISSAC
• 2016
This article reports on an implementation of ANewDsc on top of the RS root isolator, a highly efficient realization of the classical Descartes method that outperforms both RS and mature competitors by magnitudes for notoriously hard instances with clustered roots.
A Simple Near-Optimal Subdivision Algorithm for Complex Root Isolation based on the Pellet Test and Newton Iteration
• Computer Science
ArXiv
• 2015
This paper describes a subdivision algorithm for isolating the complex roots of a polynomial F ∈ C[x] with the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage’s splitting circle technique.
On the complexity of the Descartes method when using approximate arithmetic
A variant of the Descartes method to isolate the real roots of a square-free polynomial F ( x ) = ∑ i = 0 n A i x i with arbitrary real coefficients is introduced which exclusively depends on the geometry of the roots and not on the complexity of the coefficients of F.
Root Refinement for Real Polynomials
• Computer Science, Mathematics
ArXiv
• 2011
A bound on the bit complexity of the algorithm is proved in terms of the degree of the polynomial, the size and the roots, that is, parameters exclusively related to the geometric location of the Roots, which is near optimal and significantly improves previous work on integer polynomials.
Near optimal subdivision algorithms for real root isolation
• Computer Science, Mathematics
J. Symb. Comput.
• 2017
A subroutine is described that reduces the size of the subdivision tree of any subdivision algorithm for real root isolation using predicates based on either the Descartes's rule of signs or Sturm sequences, which is close to the optimal value of O ( n).
Computing Real Roots of Real Polynomials - An Efficient Method Based on Descartes' Rule of Signs and Newton Iteration
• Mathematics, Computer Science
ArXiv
• 2013
A variant of the Descartes method that isolates the real roots of any real square-free polynomial given through coefficient oracles is described, which matches the complexity of the best algorithm known, and the algorithm is simpler than this algorithm.
Nearly optimal refinement of real roots of a univariate polynomial
• Computer Science, Mathematics
J. Symb. Comput.
• 2016
We assume that a real square-free polynomial A has a degree d, a maximum coefficient bitsize ? and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this
Univariate real root isolation in an extension field and applications
• Computer Science, Mathematics
J. Symb. Comput.
• 2019
Improved separation bounds for the roots are computed and it is shown that the real roots of all polynomials in O ˜ B ( N 5 ) can be isolated, under mild assumptions.
A root isolation algorithm for sparse univariate polynomials
• Mathematics, Computer Science
ISSAC
• 2012
An improved generalized Budan-Fourier count is applied to both the input polynomial and its reciprocal, together with Newton like approximation steps to find for each real root of f an interval isolating this root from the others.
Computing Real Roots of Real Polynomials - An Ecient Method Based on Descartes' Rule of Signs and
• Mathematics
• 2014
Computing the real roots of a polynomial is a fundamental problem of computational algebra. We describe a variant of the Descartes method that isolates the real roots of any real square-free

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