A generic algebraic kernel for non-linear geometric applications

Berberich, Eric; Hemmer, Michael; Kerber, Michael

doi:10.1145/1998196.1998224

Cited by 17 publications

(24 citation statements)

References 30 publications

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“…Finally, we remark that the recently introduced CGAL-package on algebraic computations [4] represents algebraic numbers by their isolating intervals and uses QIR to refine them. The implemented version therein can be considered as a "light version" of the techniques presented in this paper, using relative approximations to speed up polynomial evaluations, but falling back to exact methods in the case of failure.…”

Section: Resultsmentioning

confidence: 99%

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Root refinement for real polynomials using quadratic interval refinement

Kerber

Sagraloff

2015

Journal of Computational and Applied Mathematics

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We consider the problem of approximating all real roots of a square-free polynomial f with real coefficients. Given isolating intervals for the real roots and an arbitrary positive integer L, the task is to approximate each root to L bits after the binary point. Abbott has proposed the quadratic interval refinement method (QIR for short), which is a variant of Regula Falsi combining the secant method and bisection. We formulate a variant of QIR, denoted AQIR ("Approximate QIR"), that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively. It returns a certified result for any given real polynomial, whose roots are all simple. In addition, we propose several techniques to improve the asymptotic complexity of QIR: We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. For integer coefficients, our variant improves, in theory and practice, the variant with exact integer arithmetic. Combining our approach with multipoint evaluation, we obtain near-optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms.

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Section: Resultsmentioning

confidence: 99%

“…Indeed, a simplified version of our approach (for integer coefficients) is included in the recently introduced CGAL-package on algebraic computations [4]. Experimental comparisons in the context of [3] have shown that the approximate version of QIR gives significantly better running times than its exact counterpart.…”

Section: Discussionmentioning

confidence: 99%

Root refinement for real polynomials using quadratic interval refinement

Kerber

Sagraloff

2015

Journal of Computational and Applied Mathematics

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“…We finally remark that a slightly simplified version of our approach (for integer coefficients) is included in the recently introduced CGAL 1 -package on algebraic computations [4]. Experimen- 1 Computational Geometry Algorithms Library, www.cgal.org…”

Section: Introductionmentioning

confidence: 99%

Efficient real root approximation

Kerber

Sagraloff

2011

Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

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We consider the problem of approximating all real roots of a squarefree polynomial f . Given isolating intervals, our algorithm refines each of them to a width at most 2 −L , that is, each of the roots is approximated to L bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating f , without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of degree, coefficient size and discriminant. Our bound improves previous work on integer polynomials by a factor of deg f and essentially matches best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms.

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“…In recent years, a mature and generic algebraic kernel for geometric computations has been developed [1]. It has been integrated into CGAL and is available since version 3.7 under the name Algebraic kernel d. We refer to it as the ak d kernel.…”

Section: Optimization 1 Whenever a Triangle Or Tetrahedron Becomes Shmentioning

confidence: 99%

3D Kinetic Alpha Complexes and Their Implementation

Kerber¹,

Edelsbrunner²

2013

2013 Proceedings of the Fifteenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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Motivated by an application in cell biology, we describe an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. Our algorithm is implemented using CGAL, following the exact geometric computation paradigm. We report on several techniques to accelerate the computation that turn our implementation applicable to the underlying biological problem.

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A generic algebraic kernel for non-linear geometric applications

Cited by 17 publications

References 30 publications

Root refinement for real polynomials using quadratic interval refinement

Root refinement for real polynomials using quadratic interval refinement

Efficient real root approximation

3D Kinetic Alpha Complexes and Their Implementation

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