# 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…

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## 54 Citations

Computing Real Roots of Real Polynomials ... and now For Real!

- Mathematics, Computer ScienceISSAC
- 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 ScienceArXiv
- 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

- Computer Science, MathematicsJ. Symb. Comput.
- 2014

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, MathematicsArXiv
- 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, MathematicsJ. 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 ScienceArXiv
- 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, MathematicsJ. 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, MathematicsJ. 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 ScienceISSAC
- 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…

## References

SHOWING 1-10 OF 48 REFERENCES

SqFreeEVAL: An (almost) optimal real-root isolation algorithm

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2012

This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds on the size of the subdivision tree for the SqFreeEVAL algorithm.

Efficient isolation of polynomial's real roots

- Mathematics
- 2004

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…

A simple but exact and efficient algorithm for complex root isolation

- Mathematics, Computer ScienceISSAC '11
- 2011

It is shown that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms EVAL and CEVAL matches that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequence.

Efficient real root approximation

- Mathematics, Computer ScienceISSAC '11
- 2011

The method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively, and proves a bound on the bit complexity of the algorithm in terms of degree, coefficient size and discriminant.

On the Complexity of Real Root Isolation

- Mathematics, Computer ScienceArXiv
- 2010

An upper bound on the maximal precision that is needed for isolating the roots of a square-free polynomial, for integer polynomials, is given, which improves the best bounds known for existing practical algorithms by a factor of $n=deg F$.

Real root isolation for exact and approximate polynomials using Descartes' rule of signs

- Computer Science
- 2008

The Descartes method is modified such that it can handle bitstream coefficients, which can be approximated arbitrarily well but cannot be determined exactly; the computing time and precision requirements are analyzed.

A General Approach to Isolating Roots of a Bitstream Polynomial

- Mathematics, Computer ScienceMath. Comput. Sci.
- 2010

A new approach to isolate the roots of a square-free polynomial F with real coefficients, assumed that each coefficient of F can be approximated to any specified error bound and referred to such coefficients as bitstream coefficients is described.

Continuous Amortization: A Non-Probabilistic Adaptive Analysis Technique

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2009

This paper introduces a form of continuous amortization for adaptive complexity in subdivision algorithms based on purely numerical primitives such as function evaluation and provides an adaptive upper bound on the complexity of EVAL using an integral, analogous to integral bounds provided by Ruppert in a different context.

Design, analysis, and implementation of a multiprecision polynomial rootfinder

- Computer Science, MathematicsNumerical Algorithms
- 2004

The algorithm is based on an adaptive strategy which automatically exploits any specific feature of the input polynomial, like its sparsity or the conditioning of its roots, in order to speed up the computation, and is generally much faster than other polynomials tested.

Algorithms for polynomial real root isolation

- Mathematics
- 1992

This thesis investigates algorithms for polynomial real root isolation of polynomials with integer and real algebraic number coefficients. A real root isolation algorithm computes isolating intervals…