A Separation Bound for Real Algebraic Expressions

Burnikel, Christoph; Funke, Stefan; Mehlhorn, Kurt; Schirra, Stefan; Schmitt, Susanne

doi:10.1007/s00453-007-9132-4

Cited by 22 publications

(32 citation statements)

References 14 publications

(21 reference statements)

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“…By a similar reasoning, for any conjugate γ * of γ, | γ * |≤ v. Inequality (8) follows from Lem. 2 of [4].…”

Section: Coefficient Bounds and Sign Computationmentioning

confidence: 98%

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Univariate real root isolation in multiple extension fields

Strzeboński

Tsigaridas²

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

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We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = ( Q(α1, . . . , α ) is an algebraic extension of the rational numbers. Our bounds are single exponential in and match the ones presented in [34] for the case = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is OB(N 4 +4 ), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of OB(min{N 4 +7 , N 2 2 +6 }) and OB(N 2 +4 ), respectively. We implemented the algorithms in C as part of the core library of mathematica and we illustrate their efficiency over various data sets.

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“…By a similar reasoning, for any conjugate γ * of γ, | γ * |≤ v. Inequality (8) follows from Lem. 2 of [4].…”

Section: Coefficient Bounds and Sign Computationmentioning

confidence: 98%

“…The following proposition gives an extension of the BFMSS bound [4] providing an additional rule for polynomial expressions in algebraic numbers. .…”

Section: Coefficient Bounds and Sign Computationmentioning

confidence: 99%

Univariate real root isolation in multiple extension fields

Strzeboński

Tsigaridas²

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

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“…For graphs of bounded degree, the polynomials have bounded degree, and the condition can be tested in polynomial time in the classical Turing machine model, with rational edge lengths as inputs, using separation bounds for algebraic computations; see [3,4,21]. We may also allow square roots of rationals as inputs.…”

Section: Triangulated Graphsmentioning

confidence: 99%

Planar Embeddings of Graphs with Specified Edge Lengths

Cabello¹,

Demaine²,

Rote³

2007

JGAA

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We consider the problem of finding a planar straight-line embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable-indeed, solvable in linear time on a real RAM-for straight-line embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead straight-line embeddings of planar 3-connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.

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“…There are many opportunities to introduce specialized operators into Expr, and we would like to introduce mechanisms to support this. Invisible to users, the evaluation of expressions relies on two critical functions: filters [5,7,17] and zero bounds [6,35]. In Core 1, both functionalities are integrated into the Expr class, making them hard to maintain and extend.…”

Section: Introductionmentioning

confidence: 99%

“…Exact Numerical Computation. There are four ingredients in our real RAM implementation: (a) certified approximation of basic real functions (e.g., [4]), (b) the theory of constructive zero bounds [6,29], (c) a precision-driven evaluation mechanism [24], and (d) a filter mechanism [5,17].…”

Section: Introductionmentioning

confidence: 99%

The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra

Yap

et al. 2010

Mathematical Software – ICMS 2010

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Abstract. There is a growing interest in numeric-algebraic techniques in the computer algebra community as such techniques can speed up many applications. This paper is concerned with one such approach called Exact Numeric Computation (ENC). The ENC approach to algebraic number computation is based on iterative verified approximations, combined with constructive zero bounds. This paper describes Core 2, the latest version of the Core Library, a package designed for applications such as non-linear computational geometry. The adaptive complexity of ENC combined with filters makes such libraries practical. Core 2 smoothly integrates our algebraic ENC subsystem with transcendental functions with ε-accurate comparisons. This paper describes how the design of Core 2 addresses key software issues such as modularity, extensibility, and efficiency in a setting that combines algebraic and transcendental elements. Our redesign preserves the original goals of the Core Library, namely, to provide a simple and natural interface for ENC computation to support rapid prototyping and exploration. We present examples, experimental results, and timings for our new system, released as Core Library 2.0.

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A Separation Bound for Real Algebraic Expressions

Cited by 22 publications

References 14 publications

Univariate real root isolation in multiple extension fields

Univariate real root isolation in multiple extension fields

Planar Embeddings of Graphs with Specified Edge Lengths

The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra

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