Angelos Mantzaflaris

Page 1. Efficient Multihomogeneous Resultant Matrices Angelos Mantzaflaris† †Department of Informatics and Telecommunications University of Athens 2nd RISC School, June 2007 Page 2. Overview Resultants: projective, multihomogeneous, sparse ...

Abstract We design and implement an efficient algorithm for the computation of generalized Voronoï Diagrams (VD's) constrained to a given domain. Our framework is general and applicable to any VD-type where the distance field is given by a polynomial. We use the Bernstein form of polynomials to subdivide the domain and isolate bisector domains or domains that contain a Voronoï vertex. Efficiency is due to a filtering process, based on bounding the distance functions over the subdivided domains. The output is a polygonal ...

Abstract. An important step in simulation via isogeometric analysis (IGA) is the assembly step, where the coefficients of the final linear system are generated. Typically, these coefficients are integrals of products of shape functions and their derivatives. Similarly to the finite element analysis (FEA), the standard choice for integral evaluation in IGA is Gaussian quadrature. Recent developments propose different quadrature rules, that reduce the number of quadrature points and weights used.

Résumé: Le calcul géométrique en modélisation et en CAO nécessite la résolution approchée, et néanmoins certifiée, de systèmes polynomiaux. Nous introduisons de nouveaux algorithmes de sous-division afin de résoudre ce problème fondamental, calculant des développements en fractions continues des coordonnées des solutions. Au delà des exemples concrets, nous fournissons des estimations de la complexité en bits et des bornes dans le modèle de RAM réelle.

Abstract: Geometric computation in computer aided geometric design and solid modeling calls for solving non-linear polynomial systems in an approximate-yetcertified manner. We introduce new subdivision algorithms that tackle this fundamental problem. In particular, we generalize the univariate so-called continued fraction solver to general dimension. Fast bounding functions, unicity tests, projection and preconditioning are employed to speed up convergence.

Semi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given precision, this algorithm can also provide a polygonal and isotopic approximation of the exact set. The idea is to localize the boundary curves by subdividing the space and then deduce their shape within small enough cells using only boundary information. Then a systematic traversal of the boundary curve graph yields polygonal regions isotopic to the connected components of the semi-algebraic set. Space subdivision is supported by a kd-tree structure and localization is done using Bernstein representation. We conclude by demonstrating our C++ implementation in the CAS Mathemagix.

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems is presented, which avoids redundant computation and reduces the size of the intermediate linear systems to solve. We derive a one-step deflation technique, from the description of the multiplicity structure in terms of differentials. The deflated system can be used in Newton-based iterative schemes with quadratic convergence. Starting from a polynomial system and a small-enough neighborhood, we obtain a criterion for the existence and uniqueness of a singular root of a given multiplicity structure, applying a well-chosen symbolic perturbation. Standard verification methods, based eg. on interval arithmetic and a fixed point theorem, are employed to certify that there exists a unique perturbed system with a singular root in the domain. Applications to topological degree computation and to the analysis of real branches of an implicit curve illustrate the method.