Resultant-Based Methods for Plane Curves Intersection Problems

Busé, Laurent; Khalil, Houssam; Mourrain, Bernard

doi:10.1007/11555964_7

Cited by 32 publications

(46 citation statements)

References 9 publications

(10 reference statements)

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“…The first equality above is a well-known result on intersection theory (see [2,Proposition 5] or [13, §1.6]) and it is here where the assumption that s(x) and t(x) do not vanish simultaneously at x = x 0 is needed. Thus (b) follows because (9) and (10) imply that mult B σ ( ), x 0 mult(R, x 0 ), as desired.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Algebraic and analytical tools for the study of the period function

Garijo

Villadelprat

2014

Journal of Differential Equations

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Abstract. In this paper we consider analytic planar differential systems having a first integral of the form H(x, y) = A(x) + B(x)y + C(x)y 2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results.

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Algebraic and analytical tools for the study of the period function

Garijo

Villadelprat

2014

Journal of Differential Equations

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“…The second approach is often more efficient but also less robust. The roots of the resultant r (x) are computed as the solution of an eigenvalue problem, and then these roots are lifted to obtain the corresponding y-coordinates by solving an additional smaller eigen-problem for each x-coordinate [6,13,39]. However, the lifting step requires estimating the root's multiplicity.…”

Section: Solving Systems Of Bivariate and Polyanalytic Polynomialsmentioning

confidence: 99%

Exact line and plane search for tensor optimization

et al. 2015

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Line and plane searches are used as accelerators and globalization strategies in many optimization algorithms. We introduce a class of optimization problems called tensor optimization, which comprises applications ranging from tensor decompositions to least squares support tensor machines. We develop algorithms to efficiently compute the global minimizers of their line and plane search subproblems. Furthermore, we introduce scaled line and plane search, which compute an optimal scaling of the solution simultaneously with the optimal line or plane search step, and show that this scaling can be computed at almost no additional cost. Obtaining the global minimizers of (scaled) line and plane search problems often requires solving a bivariate or polyanalytic polynomial system. We show how to compute the isolated real solutions of bivariate polynomial systems and the isolated complex solutions of polyanalytic polynomial systems using a single generalized eigenvalue decomposition. Finally, B Ignat Domanov Ignat.Domanov@kuleuven-kulak.be 4 iMinds Medical IT, Leuven, Belgium 123 L. Sorber et al.we apply block term decompositions to the problem of blind multi-user detectionestimation in DS-CDMA communication to demonstrate that exact line and plane search can significantly reduce computation time of the workhorse tensor decomposition algorithm alternating least squares.

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“…Several solutions for recovering all values y * under these circumstances have been proposed. Some algorithms compute y * by solving a smaller eigenproblem for each distinct x * [10,13,31]. However, this approach requires estimating the multiplicities of the eigenvalues x * , along with the dimension of their corresponding kernels.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…As such, our method can be seen as a hybrid method, although symbolic manipulations are absent. In contrast to existing eigen-based solvers [3,10,13,22,27,30,31,34,50], the proposed algorithm does not depend on Gröbner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. Moreover, the method returns solutions counting multiplicity and is also applicable to polyanalytic polynomial systems.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

Sorber¹,

Barel²,

Lathauwer³

2014

SIAM J. Numer. Anal.

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Abstract. Finding the real solutions of a bivariate polynomial system is a central problem in robotics, computer modeling and graphics, computational geometry and numerical optimization. We propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition. In contrast to existing eigen-based solvers, the proposed algorithm does not depend on Gröbner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. The method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables. Solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and Newton-Raphson refinement. Numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the same time significantly faster and more accurate than competing double precision solvers.

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Resultant-Based Methods for Plane Curves Intersection Problems

Cited by 32 publications

References 9 publications

Algebraic and analytical tools for the study of the period function

Algebraic and analytical tools for the study of the period function

Exact line and plane search for tensor optimization

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

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