Amortized Bound for Root Isolation via Sturm Sequences

Du, Zilin; Sharma, Vikrant; Yap, Chee

doi:10.1007/978-3-7643-7984-1_8

Cited by 36 publications

(68 citation statements)

References 13 publications

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“…In many algorithms, the size of the subdivision tree is smaller than this bound because, for each node in the subdivision tree, additional calculations must be performed. 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).…”

Section: Other Root Isolation Algorithmsmentioning

confidence: 98%

“…Algebraic amortization originated with Davenport (1985) where the individual real root separation bounds are replaced by a product of root separations. This bound was then studied in Johnson (1991), Du et al (2007), Eigenwillig et al (2006), Mignotte (1995) and Emiris et al (2008) where it was generalized to other root separation products including complex roots. This technique has proven useful to compute the complexity of the subdivision tree for many other root isolation techniques, see Section 1.5.…”

Section: Algebraic and Continuous Amortizationmentioning

confidence: 99%

“…The sum of the logarithmic distances between roots are bounded simultaneously via the standard Mahler-Davenport lower bound on distances between roots, see Davenport (1985), Mignotte (1995), Du et al (2007) and Eigenwillig et al (2006). To do this, we construct a directed graph whose nodes are the roots in α ff ′ and whose edges represent the logarithms which must be calculated.…”

Section: Finishing the Bound On The Sqfreeeval Algorithmmentioning

confidence: 99%

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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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Section: Other Root Isolation Algorithmsmentioning

confidence: 98%

Section: Algebraic and Continuous Amortizationmentioning

confidence: 99%

Section: Finishing the Bound On The Sqfreeeval Algorithmmentioning

confidence: 99%

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GRASS GIS: A multi-purpose open source GIS

Neteler

Bowman

Landa

et al. 2012

Environmental Modelling & Software

694

398

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“…There are mainly two efficient methods to isolate the real roots of a square-free polynomial F ∈ R[x], namely, subdivision methods based on Descartes' Rule of Signs [3,7,13,25,28] and Sturm's Theorem [8,21]. They both start on an initial interval and perform a recursive binary search.…”

Section: Introductionmentioning

confidence: 99%

“…They both start on an initial interval and perform a recursive binary search. In practice, methods based on Descartes' Rule of Signs have proven to be more efficient [16,17,28] than Sturm's approach, but both approaches behave equally in terms of worst case complexity [8,13,20]. More precisely, for F a polynomial of degree n with integer coefficients of bitsize L, the induced subdivision tree has size O(n(log n + L)) and isolating all real roots demands forÕ(n 4 L 2 ) bit operations.…”

Section: Introductionmentioning

confidence: 99%

A General Approach to Isolating Roots of a Bitstream Polynomial

Sagraloff

2010

Math.Comput.Sci.

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Abstract. We describe a new approach to isolate the roots (either real or complex) of a square-free polynomial F with real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound and refer to such coefficients as bitstream coefficients. The presented method is exact, complete and deterministic. Compared to previous approaches [10,12,23] we improve in two aspects. Firstly, our approach can be combined with any existing subdivision method for isolating the roots of a polynomial with rational coefficients. Secondly, the approximation demand on the coefficients and the bit complexity of our approach is considerably smaller. In particular, we can replace the worst-case quantity σ (F) by the average-case quantity ∏ n i=1 n √ σ i , where σ i denotes the minimal distance of the i−th root ξ i of F to any other root of F, σ (F) := min i σ i , and n = deg F. For polynomials with integer coefficients, our method matches the best bounds known for existing practical algorithms that perform exact operations on the input coefficients.

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Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

Tsigaridas

Emiris

2006

Lecture Notes in Computer Science

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We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We improve the previously known bound by a factor of dτ , where d is the polynomial degree and τ bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is OB(d 4 τ 2 ) using the standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.

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Amortized Bound for Root Isolation via Sturm Sequences

Cited by 36 publications

References 13 publications

GRASS GIS: A multi-purpose open source GIS

GRASS GIS: A multi-purpose open source GIS

A General Approach to Isolating Roots of a Bitstream Polynomial

Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

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