Complexity of real root isolation using continued fractions

Sharma, Vikrant

doi:10.1145/1277548.1277594

Cited by 14 publications

(15 citation statements)

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“…However, all investigated solvers in this article use the same algorithm to compute lower bounds on the roots, namely a modification of Hong's bound proposed in [1]. We refer the reader to [30,32,1] and references therein, for a detailed description. The method is known to be among the most powerful root isolation approaches, in particular, for ill-conditioned problems.…”

Section: Cf-familymentioning

confidence: 99%

“…The first class consists of the subdivision algorithms [7,11,23,27,8,29,16], which exploit either Sturm's theorem or Descartes' rule of signs. The second class contains the Continued Fraction algorithms [1,32,30], which are based on the continued fraction expansion of the roots of the polynomial. The best worst case complexity bound for all these algorithms, after eliminating the (poly)logarithmic factors, is e OB(d 4 τ 2 ), where d is the degree of the polynomial and τ the maximum coefficient bitsize.…”

Section: Introductionmentioning

confidence: 99%

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Experimental evaluation and cross-benchmarking of univariate real solvers

Hemmer

Tsigaridas

Zafeirakopoulos

et al. 2009

Proceedings of the 2009 Conference on Symbolic Numeric Computation

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Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA, the VEGAS group at LO-RIA and the MPI-Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5 000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8 000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.

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Section: Cf-familymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Experimental evaluation and cross-benchmarking of univariate real solvers

Hemmer

Tsigaridas

Zafeirakopoulos

et al. 2009

Proceedings of the 2009 Conference on Symbolic Numeric Computation

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“…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: 96%

GRASS GIS: A multi-purpose open source GIS

Neteler

Bowman

Landa

et al. 2012

Environmental Modelling & Software

694

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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“…This bound holds for non square-free polynomials and in the same time we can also compute the multiplicities of the real roots. Quite recently, Sharma (254) …”

Section: The Continued Fraction Algorithmmentioning

confidence: 99%