On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

Mourrain, Bernard; Vrahatis, Michael N.; Yakoubsohn, Jean-Claude

doi:10.1006/jcom.2001.0636

Cited by 52 publications

(42 citation statements)

References 48 publications

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“…In particular, our Theorem 5.4 assures the existence of a deformed system (5.4) of the original system (2.1) that possesses only simple real roots. This result can be used in many cases including the computation of the topological degree [4,6,10,11,12] in order to examine the solution set of a system of equations and to obtain information on the existence of solutions, their number and their nature [1,3,6,8,9].…”

Section: Discussionmentioning

confidence: 99%

On perturbation of roots of homogeneous algebraic systems

Tanabé

Vrahatis

2006

Math. Comp.

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Abstract. A problem concerning the perturbation of roots of a system of homogeneous algebraic equations is investigated. The question of conservation and decomposition of a multiple root into simple roots are discussed. The main theorem on the conservation of the number of roots of a deformed (not necessarily homogeneous) algebraic system is proved by making use of a homotopy connecting initial roots of the given system and roots of a perturbed system. Hereby we give an estimate on the size of perturbation that does not affect the number of roots. Further on we state the existence of a slightly deformed system that has the same number of real zeros as the original system in taking the multiplicities into account. We give also a result about the decomposition of multiple real roots into simple real roots.

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Section: Discussionmentioning

confidence: 99%

On perturbation of roots of homogeneous algebraic systems

Tanabé

Vrahatis

2006

Math. Comp.

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“…This rule is traditionally stated for the power basis and the interval (0, ∞); see [16] for a proof with historical references. The Bernstein formulation appears in [5][6][7][8].…”

Section: The Descartes Methods In the Bernstein Basismentioning

confidence: 99%

“…The Descartes method can be formulated for polynomials in the usual power basis [2,1,3,4] and for polynomials in the Bernstein basis [5][6][7][8]. The early work concentrated on polynomials with integer coefficients.…”

Section: Comparison To Related Workmentioning

confidence: 99%

A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

Eigenwillig

Kettner

Krandick

et al. 2005

Computer Algebra in Scientific Computing

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Abstract. The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial q(x) = q n x n + . . . + q 0 with root separation ρ, coefficients |q n | ≥ 1 and |q i | ≤ 2 τ , it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ) 2 ) bit operations.

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“…Hereafter, these polynomials will be called g 1 (t), g 2 (t) and the interval [u, v] ⊂ R. For instance, one of the 4 cases to consider will be g 1 (t) = ∂ x f (t, c), g 2 (t) = ∂ y f (t, c), u = a, v = b. We recall briefly the subdivision method described in [43,40,18], which can be used for this purpose. First we express our polynomials…”

Section: Counting the Number Of Branchesmentioning

confidence: 99%

“…The complexity analysis of this method is described in [43]. This analysis can be improved by exploiting the recent results in [18].…”

Section: Counting the Number Of Branchesmentioning

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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On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

Cited by 52 publications

References 48 publications

On perturbation of roots of homogeneous algebraic systems

On perturbation of roots of homogeneous algebraic systems

A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

Topology and arrangement computation of semi-algebraic planar curves

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