André Galligo

Certified approximate univariate GCDs

Abstract We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suffice to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problems hardness. SVD computations on subresultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that certifies the maximum-degree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorems conditions in a more intrinsic context.

Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications

Parameterization of the computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In this paper, we study the volume parameterization problem of the multi-block computational domain in an isogeometric version, i.e., how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of the single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parameterization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulting volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volumes should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on the three-dimensional heat conduction problem.

Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis

In the isogeometric analysis framework, a computational domain is exactly described using the same representation as the one employed in the CAD process. For a CAD object, various computational domains can be constructed with the same shape but with different parameterizations; however one basic requirement is that the resulting parameterization should have no self-intersections. Moreover we will show, with an example of a 3D thermal conduction problem, that different parameterizations of a computational domain have different impacts on the simulation results and efficiency in isogeometric analysis. In this paper, a linear and easy-to-check sufficient condition for the injectivity of a trivariate B-spline parameterization is proposed. For problems with exact solutions, we will describe a shape optimization method to obtain an optimal parameterization of a computational domain. The proposed injective condition is used to check the injectivity of the initial trivariate B-spline parameterization constructed by discrete Coons volume method, which is a generalization of the discrete Coons patch method. Several examples and comparisons are presented to show the effectiveness of the proposed method. During the refinement step, the optimal parameterization can achieve the same accuracy as the initial parameterization but with less degrees of freedom.

Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields

Abstract This paper deals mainly with fast quantifier elimination in the elementary theory of algebraically closed fields of any characteristic. It is subdivided into an introduction, a short exposition of the computational model and of our results, and concludes with a section dedicated to proofs. The new outcomes concern parallelism where the number of processors is controlled by the intrinsic sequential complexity of quantifier elimination. Our algorithms are optimal from the point of view of the overall complexities in parallel and in sequential (number of processors). Due to recent progress concerning Triviality Testing of Polynomial Ideals (relying on effective affine Nullstellensatze) we are able to give upper bounds in a refined and satisfactory precise form.

A geometric-numeric algorithm for absolute factorization of multivariate polynomials

In this paper, we propose a new semi-numerical algorithmic method for factoring multivariate polynomials absolutely. It is based on algebraic and geometric properties after reduction to the bivariate case in a generic system of coordinates. The method combines 4 tools: zero-sum relations at triplets of points, partial information on monodromy action, Newton interpolation on a structured grid, and a homotopy method. The algorithm relies on a probabilistic approach and uses numerical computations to propose a candidate factorization (with probability almost one) which is later validated.

Four lectures on polynomial absolute factorization

Polynomial factorization is one of the main chapters of Computer Algebra. Recently, significant progress was made on absolute factorization (i.e., over the complex field) of a multivariate polynomial with rational coefficients, with two families of algorithms proposing two different strategies of computation. One is represented by Gao’s algorithm and is explained in Lecture 2. The other is represented by the Galligo-Rupprecht-Cheze algorithm, presented in Lectures 4 and 5. The latter relies on an original use of the monodromy map attached to a generic projection of a plane curve on a line. It also involves zero-sums relations (introduced by Sasaki and his collaborators) with efficient semi-numerical computations to produce a certified exact result.

Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method

In isogeometric analysis, parameterization of computational domain has great effects as mesh generation in finite element analysis. In this paper, based on the concept of harmonic mapping from the computational domain to parametric domain, a variational harmonic approach is proposed to construct analysis-suitable parameterization of computational domain from CAD boundary for 2D and 3D isogeometric applications. Different from the previous elliptic mesh generation method in finite element analysis, the proposed method focuses on isogeometric version, and converts the elliptic PDE into a nonlinear optimization problem, in which a regular term is integrated into the optimization formulation to achieve more uniform and orthogonal iso-parametric structure near convex (concave) parts of the boundary. Several examples are presented to show the efficiency of the proposed method in 2D and 3D isogeometric analysis.

A numerical absolute primality test for bivariate polynomials

We give a new numerical absolute primality criterion for bivariate polynomials. This test is based on a simple property of the monomials appearing after a generic linear change of coordinates. Our method also provides a probabilistic algorithm for detecting absolute factors. We sketch an implementation and give timings comparing with two other algorithms implemented in Maple.

From an approximate to an exact absolute polynomial factorization

We propose an algorithm for computing an exact absolute factorization of a bivariate polynomial from an approximate one. This algorithm is based on some properties of the algebraic integers over Z and is certified. It relies on a study of the perturbations in a Vandermonde system. We provide a sufficient condition on the precision of the approximate factors, depending only on the height and the degree of the polynomial.

Random polynomials and expected complexity of bisection methods for real solving

Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones? We exploit results by Kac, and by Edelman and Kostlan in order to estimate the real root separation of degree d polynomials with i.i.d. coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the i-th coefficient has variance (d/i), whereas for Weyl polynomials its variance is 1/i!. By applying results from statistical physics, we obtain the expected (bit) complexity of STURM solver, ÕB(rd2τ), where r is the number of real roots and τ the maximum coefficient bitsize. Our bounds are two orders of magnitude tighter than the record worst case ones. We also derive an output-sensitive bound in the worst case. The second part of the paper shows that the expected number of real roots of a degree d polynomial in the Bernstein basis is √2d ± O(1), when the coefficients are i.i.d. variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis.