Investigation of a subdivision based algorithm for solving systems of polynomial equations

Garloff, Jürgen; Smith, Alexander Thomas T.

doi:10.1016/s0362-546x(01)00166-3

Cited by 30 publications

(17 citation statements)

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“…Their application to arrangement computation have recently emerged such as in [10,30] where interval arithmetic is used to classify cells in the subdivision process. Subdivision methods are also very efficient for isolating the roots of polynomial equations, which appear in geometric problems [49,16,22,39]. They have also been extended for the approximation of one or two dimensional objects [31,2,34].…”

Section: Previous Workmentioning

confidence: 99%

Topology and arrangement computation of semi-algebraic planar curves

Alberti

Mourrain

Wintz

2008

Computer Aided Geometric Design

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We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties.The method isolates singular points from regular parts and deals with them independently. The topology near singular points is guaranteed through topological degree computation. In either case the topology inside regions is recovered from information on the boundary of a cell of the subdivision.Obtained regions are segmented to provide an efficient insertion operation while dynamically maintaining an arrangement structure.We use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. It is finally shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation.

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Section: Previous Workmentioning

confidence: 99%

Topology and arrangement computation of semi-algebraic planar curves

Alberti

Mourrain

Wintz

2008

Computer Aided Geometric Design

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“…One of the first methods was the Bézier clipping approach developed by Nishita et al [20] which took advantage of the convex hull property. The use of Descartes' rule of sign has led to many algorithms for real root isolation algorithms and for generating enclosures [3,8,14]. The interval projected polyhedron (IPP) algorithm [28] reformulates the problem of solving a system of polynomial equations into the problem of solving a set of univariate polynomial systems.…”

Section: Introductionmentioning

confidence: 99%

A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems

Jüttler

Moore

2011

Math.Comput.Sci.

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A new numerical approach to compute all real roots of a system of two bivariate polynomial equations over a given box is described. Using the Bernstein-Bézier representation, we compute the best linear approximant and the best quadratic approximant of the two polynomials with respect to the L 2 norm. Using these approximations and bounds on the approximation errors, we obtain a fat line bounding the zero set first of the first polynomial and a fat conic bounding the zero set of the second polynomial. By intersecting these fat curves, which requires solely the solution of linear and quadratic equations, we derive a reduced subbox enclosing the roots. This algorithm is combined with splitting steps, in order to isolate the roots. It is applied iteratively until a certain accuracy is obtained. Using a suitable preprocessing step, we prove that the convergence rate is 3 for single roots. In addition, experimental results indicate that the convergence rate is superlinear (1.5) for double roots.

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“…Their application to arrangement computation has recently emerged such as in [5,12] where interval arithmetic is used to classify cells in the subdivision process. Subdivision methods are also very efficient for isolating the roots of polynomial equations, which appear in geometric problems [18,6,10,16]. They have also been extended for the approximation of one or two dimensional objects [13,2,14].…”

Section: Introductionmentioning

confidence: 99%

A Subdivision Arrangement Algorithm for Semi-Algebraic Curves: An Overview

Liu¹,

Lai²,

2007

15th Pacific Conference on Computer Graphics and Applications (PG'07)

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We overview a new method for computing the arrangement of semi-algebraic curves. A subdivision approach is used to compute the topology of the algebraic objects and to segment the boundary of regions defined by these objects. An efficient insertion technique is described, which detects regions in conflict and updates the underlying arrangement structure. We describe the general framework of this method, the main region insertion operation and the specializations of the key ingredients for the different types of objects: implicit, parametric or piecewise linear curves.

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Investigation of a subdivision based algorithm for solving systems of polynomial equations

Cited by 30 publications

References 0 publications

Topology and arrangement computation of semi-algebraic planar curves

Topology and arrangement computation of semi-algebraic planar curves

A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems

A Subdivision Arrangement Algorithm for Semi-Algebraic Curves: An Overview

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