Polynomial real root isolation using Descarte's rule of signs

Collins, George E.; Akritas, Alkiviadis G.

doi:10.1145/800205.806346

Cited by 149 publications

(133 citation statements)

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“…Davenport (1985) proved that the subdivision tree for the Sturm method is O(d (L + ln d)), see Reischert (1997), Lickteig and Roy (2001), Du et al (2007), Emiris et al (2008) and Johnson (1991). More recently, it has been shown in Eigenwillig et al (2006) and Emiris et al (2008) that the Descartes method also achieves this bound, see Collins and Akritas (1976), Eigenwillig et al (2006), Krandick and Mehlhorn (2006), Collins et al (2002), Sagraloff (2011) andJohnson (1991). These methods are optimal under the weak assumption that L ≥ ln d. In addition, related exact techniques using continued fractions were shown to have a tree size of  O(dL) when an ideal root bound is used and  O(d 2 L) when a more practical bound is used (Sharma, 2008); in the expected case, the tree was also shown in Tsigaridas and Emiris (2008) to have an expected size of O(d 2…”

Section: Other Root Isolation Algorithmsmentioning

confidence: 99%

GRASS GIS: A multi-purpose open source GIS

Neteler

Bowman

Landa

et al. 2012

Environmental Modelling & Software

687

398

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a b s t r a c tLet f be a univariate polynomial with real coefficients, f ∈ R[X ].Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm.The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d (L +ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: first, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.

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Section: Other Root Isolation Algorithmsmentioning

confidence: 99%

GRASS GIS: A multi-purpose open source GIS

Neteler

Bowman

Landa

et al. 2012

Environmental Modelling & Software

687

398

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“…Recently, there is an extension to the case of analytic functions [17,38]. For the problem of isolating the roots of polynomials we refer the reader to [23,20,19,12,36,24,7,27,9] and the references therein. There are also approaches [24] that achieve locally quadratic convergence towards the simple roots of polynomial systems and there very efficient in practice.…”

Section: Related Workmentioning

confidence: 99%

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

Burr

Gao

Tsigaridas

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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“…We use the Descartes Method [7,27] to find isolating intervals for all real roots of a polynomial. We have a choice of different implementations, of which Interval Descartes [9] and Sturm sequences are ongoing work.…”

Section: Polynomialsmentioning

confidence: 99%

Exacus: Efficient and Exact Algorithms for Curves and Surfaces

Berberich

Eigenwillig

Hemmer

et al. 2005

Algorithms – ESA 2005

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Abstract. We present the first release of the EXACUS C++ libraries. We aim for systematic support of non-linear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of the libraries, which currently compute arrangements of curves and curve segments of low algebraic degree, and boolean operations on polygons bounded by such segments.

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Polynomial real root isolation using Descarte's rule of signs

Cited by 149 publications

References 5 publications

GRASS GIS: A multi-purpose open source GIS

GRASS GIS: A multi-purpose open source GIS

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

Exacus: Efficient and Exact Algorithms for Curves and Surfaces

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