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25
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both metho ..."
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Cited by 30 (17 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.
Fast and Exact Geometric Analysis of Real Algebraic Plane Curves
 Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC’07), 151–158
, 2007
"... An algorithm is presented for the geometric analysis of an algebraic curve f(x, y) = 0 in the real affine plane. It computes a cylindrical algebraic decomposition (CAD) of the plane, augmented with adjacency information. The adjacency information describes the curve’s topology by a topologically eq ..."
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Cited by 27 (12 self)
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An algorithm is presented for the geometric analysis of an algebraic curve f(x, y) = 0 in the real affine plane. It computes a cylindrical algebraic decomposition (CAD) of the plane, augmented with adjacency information. The adjacency information describes the curve’s topology by a topologically equivalent planar graph. The numerical data in the CAD gives an embedding of the graph. The algorithm is designed to provide the exact result for all inputs but to perform only few symbolic operations for the sake of efficiency. In particular, the roots of f(α, y) at a critical xcoordinate α are found with adaptiveprecision arithmetic in all cases, using a variant of the Bitstream Descartes method (Eigenwillig et al., 2005). The algorithm may choose a generic coordinate system for parts of the analysis but provides its result in the original system. The algorithm is implemented as C++ library AlciX in the EXACUS project. Running time comparisons with top by GonzalezVega and Necula (2002), and with cad2d by Brown demonstrate its efficiency. Categories and Subject Descriptors: I.1.4 [Symbolic and Algebraic Manipulation]: Applications;
On the complexity of real root isolation using Continued Fractions
 INRIA
, 2006
"... 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 bou ..."
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Cited by 12 (6 self)
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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 bit size, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is � OB(d 4 τ 2) using a standard bound on the expected bit size of the integers in the continued fraction expansion. Moreover, using a homothetic transformation we improve the expected complexity bound to � OB(d 3 τ) under the assumption that d = O(τ). We show how to compute the multiplicities within the same complexity and extend the algorithm to non squarefree polynomials. Finally, we present an efficient opensource C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bit size up to 8000 bits and degree up to 1000. 1
HIGHPERFORMANCE IMPLEMENTATIONS OF THE DESCARTES METHOD
, 2006
"... The Descartes method for polynomial real root isolation can be performed with respect to monomial bases and with respect to Bernstein bases. The first variant uses Taylor shift by 1 as its main subalgorithm, the second uses de Casteljau’s algorithm. When applied to integer polynomials, the two vari ..."
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Cited by 9 (0 self)
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The Descartes method for polynomial real root isolation can be performed with respect to monomial bases and with respect to Bernstein bases. The first variant uses Taylor shift by 1 as its main subalgorithm, the second uses de Casteljau’s algorithm. When applied to integer polynomials, the two variants have codominant, almost tight computing time bounds. Implementations of either variant can obtain speedups over previous stateoftheart implementations by more than an order of magnitude if they use features of the processor architecture. We present an implementation of the Bernsteinbases variant of the Descartes method that automatically generates architectureaware highlevel code and leaves further optimizations to the compiler. We compare the performance of our implementation, algorithmically tuned implementations of the monomial and Bernstein variants, and architectureunaware implementations of both variants on four different processor architectures and for three classes of input polynomials.
Topology and arrangement computation of semialgebraic planar curves
 CAGD
, 2008
"... 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 lin ..."
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Cited by 8 (2 self)
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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.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 7 (1 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm. 1
Continued fraction expansion of real roots of polynomial systems
 In Proc. of SNC ’09
, 2009
"... We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a f ..."
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Cited by 6 (2 self)
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We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent’s theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.
Regularity criteria for the topology of algebraic curves and surfaces
 in "Mathematics of Surfaces XII LNCS
"... and surfaces ..."
On the topology of real algebraic plane curves
, 2010
"... We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coo ..."
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Cited by 3 (2 self)
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We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use subresultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in nongeneric positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on highdegree nongeneric curves. 1
On the Complexity of Reliable Root Approximation
"... This is an authorprepared version of the article. The original publication is available at www.springerlink.com This work addresses the problem of computing a certified ǫapproximation of all real roots of a squarefree integer polynomial. We proof an upper bound for its bit complexity, by analyzin ..."
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Cited by 3 (2 self)
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This is an authorprepared version of the article. The original publication is available at www.springerlink.com This work addresses the problem of computing a certified ǫapproximation of all real roots of a squarefree integer polynomial. We proof an upper bound for its bit complexity, by analyzing an algorithm that first computes isolating intervals for the roots, and subsequently refines them using Abbott’s Quadratic Interval Refinement method. We exploit the eventual quadratic convergence of the method. The threshold for an interval width with guaranteed quadratic convergence speed is bounded by relating it to wellknown algebraic quantities. 1