# Project : galaad

## Section: New Results

### Symbolic numeric computation

#### A symbolic numeric algorithm for the absolute factorization

Participant : Guillaume Chèze.

A recent algorithmic procedure for computing the absolute factorization of a
polynomial P(X, Y), after a linear change of coordinates, is via a
factorization modulo X^{3}. This was proposed by A. Galligo and D. Rupprecht
in [45], [52]. Then absolute factorization is
reduced to finding the minimal zero sum relations between a set of
approximated numbers b_{i}, i = 1 to n such that , (see
also [53]). Here this problem with an a priori exponential
complexity bound, is efficiently solved for large degrees (n>100). We rely
on LLL algorithm, used with a strategy of computation inspired by van Hoeij's
treatment [55]. For that purpose we prove a theorem on bounded
integer relations between the numbers b_{i}, also called linear traces in [54]. This work
has been presented at the "International Symposium
on Symbolic and Algebraic Computation" [21].

#### A symbolic algorithm for the absolute factorization

Participants : Guillaume Chèze, Grégoire Lecerf.

In the vein of recent algorithmic results on polynomial factorization based on lifting and recombination techniques, we propose a new faster method for computing the absolute factorization of a bivariate polynomial. The complexity of our probabilistic algorithm is sub-quadratic in the dense size of the input polynomial, with respect to its total degree. In addition, we present a deterministic version with only soft quadratic worst case complexity. This work has been accepted at the"International Conference on Polynomial System Solving".

#### Hybrid symbolic-numeric sparse interpolation

Participant : Wen-shin Lee.

Joint work with Mark Giesbrecht and George Labahn (University of Waterloo)].

In the floating-point arithmetic, we developed effective solutions for the problem of sparse interpolation for a black box polynomial in different bases. Our methods are implemented in Maple, and we are still working on a more formal analysis and related experiments.

Based on the polynomial relations between trigonometric functions, we extend progress in floating-point sparse polynomial interpolation to trigonometric interpolation. This work has been presented in two conferences [25][26]. Full paper versions are in preparation.

#### Interpolating the determinant of a polynomial matrix

Participants : Wen-shin Lee, Mohamed Elkadi, Bernard Mourrain.

When computing the determinant of a polynomial matrix (or specifically a Bezoutian matrix), whose entries are multivariate polynomials, the size of intermediate expressions can easily become impractical. However, such determinant can be regarded as a black box, and the determinant polynomial can be computed via a black box polynomial interpolation method.

We explore and compare available black box interpolation algorithms. To further improve the efficiencies, we develop strategies for different interpolation methods in different arithmetics.

Currently, our developments are being implemented in the SYNAPS library. We intend to compare and test different interpolation methods, as well as the approaches of computing residues with respect to a polynomial map.

#### Geometric predicates

Participants : Ioannis Emiris, Athanasios Kakargias, Sylvain Pion [Geometrica team], Monique Teillaud, Elias Tsigaridas.

See the

Curved Kernelweb site http://www-sop.inria.fr/galaad/teillaud/kernel.html.

The work on geometric predicates was pursued further this year as part of the ecg project. The design of classes and concepts of a cgal kernel for curved objects led to a publication [23]. The implementation of a cgal kernel for simple curves is in progress. It will provide the user with the elementary operations that are necessary for running arrangements. This work will be submitted to the cgal Editorial Board for inclusion in the cgal library in the coming months.

Preliminary benchmarks were performed in collaboration with ecg partners (Max-Planck-Institut für Informatik and Tel Aviv University) [32].

#### Triangulation of points

Participant : Monique Teillaud.

See the cgal web site: http://www.cgal.org.

*The package ``3D Triangulation'' of CGAL is maintained in
collaboration with Sylvain Pion (Geometrica team).*cgal 3.1 was
released in December, 2004.

#### Shape optimisation

Participants : Nikos Pavlidis, Bernard Mourrain, Michael Vrahatis [University of Patras].

During the internship of N. Pavlidis, we investigate the application of optimization methods to geometric problems, and in particular Covariance Matrix Adaptation Evolution Strategies, Differential Evolution Algorithms and Particle Swarm Optimization methods. All three methods exploit a set (population or swarm) of potential solutions (individuals or particles) to probe the search space for possible best solutions. At each iteration, they employ a number of operators to refine the potential solutions so as to evolve the population towards more promising regions of the search space (in the case of minimization such regions are characterized by lower function values). They are designed to handle non-linear, non-differentiable and discontinuous objective functions, as well as constraint optimization problems. They are also robust to the presence of noise and imprecise information, and can cope with the existence of multiple local minima.

In collaboration with OPALE team, these techniques have been applied to shape optimization problems. For the wing and arch problems, we use Bezier and B-spline to represent the profile with few parameters, in order to optimise more efficiently the required performance criteria. These optimisation tools have also been applied on distance matrices in molecular conformation problems, in order to find valid matrices in the search space. A technical report on this work is in preparation.