The following talks were presented during the open sessions:

• On-line addition in real base
  Christiane Frougny; LIAFA, University Paris 8, Paris, France

Abstract: Let $\beta$ be a real number $>1$. Addition and multiplication by a fixed positive integer of real numbers represented in base $\beta$ are shown to be computable by an on-line algorithm, and thus are continuous functions. When $\beta$ is a Pisot number, these functions are computable by an on-line finite automaton. http://www.liafa.jussieu.fr/~cf/publications/


• Reliability computations on vectorial computers
  Jean-Marie Chesneaux; Universite Pierre et Marie Curie, Paris
  
  
• ARITHMOS: A problem-solving tool for numerical computing
  Brigitte Verdonk; University of Antwerp, Belgium
  Abstract:ARITHMOS consists of a family of class libraries offering, in V1.0, fully IEEE compliant multiprecision floating-point, sharp floating-point interval and exact rational arithmetic, and additionally in V2.0 (to be released in a few months), floating-slash, rational interval and complex arithmetic. The full functionality is available both at the programming language level for use with a C++ compiler, as well as through a GUI for the fast and easy evaluation of some simple expressions. The implementation is fully X-platform (no assembler, Qt user interface) and object-oriented (operator overloading). It was initially developed for didactical purposes, in the framework of the course Computer Arithmetic and Numerical Techniques, which is taught at the University of Antwerp. But it was soon further extended into a full-fledged, efficient and powerful arithmetic environment called ARITHMOS. In multiprecision floating-point arithmetic, the base, precision, exponent range and rounding mode are all user-defined. A battery of output formats is supported for didactical purposes: decimal, binary floating-point and/or hexadecimal.
  
  
• Polynomials of Bounded Tree-Width
  Klaus Meer; SDU/Odense University, Denmark
  (Authors: Janos A. Makowsky and Klaus Meer)
  Abstract:We introduce a new sparsity condition on multivariate polynomials in $n$ variables (over some ring $R$) and show that under this condition many otherwise intractable computational problems involving these polynomials become solvable in polynomial (even linear) time in $n$ (in the $BSS$-model over $R$). To define our sparsity condition we associate with these polynomials a hypergraph and study classes of polynomials where this hypergraph has tree width at most $k$ for some fixed $k \in \N$.

  We are interested in three cases:
  (1) The evaluation of multivariate polynomials where the number of monomials is $O(2^n)$.
  (2) The question whether a system of $n$ polynomials $p_i(\bar{x}) \in R[\bar{x}]$ of fixed degree $d$ in $n$ variables has a root in $R^n$.
  (3) For infinite ordered rings $R_{ord}$, a polynomial of fixed degree $d$ in $n$ variables $p(\bar{x}) \in R_{ord}[\bar{x}]$ and a finite subset $A \subset R_{ord}$ we want to know whether $p(\bar{a}) > 0$ for all $\bar{a} \in R_{ord}^n$.

  Our method uses graph theoretic and model theoretic tools developed in the last 15 years and applies them to the algebraic setting. This work is an extension of work by B. Courcelle, J.A. Makowsky and U. Rotics. http://www.cs.technion.ac.il/~admlogic/TR/readme.html
  
  

• Zero is useless
  Dominique Michelucci; ENS des Mines, St-Etienne, France
  Extended abstract
  
  
• Reliable representations of strange sets
  Dominique Michelucci; ENS des Mines, St-Etienne, France
  Abstract: Strange sets arise in the study of dynamical systems. A point p belongs to a strange set of a transformation T iff some points T^(k)(p) will be arbitrary close to p, where T^(k) is the k th iterate of T. Classical examples are Henon attractor, Julia sets, etc. The classical method just start from some more or less random point, compute the orbit, discard the first 50 iterates (the time needed to get close to the attractor), and plot the other points. Often it works fine, for example for the classical values of the parameters of the Henon attractor. But for some other values, and due to inaccuracy of floating point arithmetic, the plotted set seriously differs from the correct set: I display the latter with a method resorting to interval arithmetics and graph theory; I subdivide R^2 in cells some of them infinite, compute a superset of T(each cell), and compute the strongly connected components in the induced graph. Pictures of results show great differences between the two methods. To keep this summary short I don't explain how I manage infinite intervals.
  
  
• Object oriented programming and mathematical structures
  Marc Conrad; Universität des Saarlandes



• Limitations of Correction in Finite Precision: The CENA method case
  Philippe Langlois; Arenaire Project - ENS de Lyon - France
  Abstract: Assuming that $\tilde{x}$ is a floating point value that approximates the real value $x$, it is common to improve this approximate computing $\overlined{x}= \tilde{x} + \hat{\Delta}$ where $\hat{\Delta}$ is a correcting factor that depends on the correcting method. During this talk, we focus on the limitations of such a correction in finite precision, {\em e.g.}, in floating point arithmetic.
  The actual correction is the finite precision computation $\overlined{x}= fl(\tilde{x} + \hat{\Delta})$ where both these values are in finite precision. First, we analyse the effect of this finite precision computation on the accuracy of the actual corrected $overlined{x}$. We prove that the efficiency of this correction depends on $\tilde{x}$ and $x$.
  When approximate $\tilde{x}$ represents the result of a floating point computation yielding $x$, {\em ie} $\tilde{x}$ suffers from rounding errors, an example of such a correction is the CENA method [1]. The CENA correcting factor $\hat{\Delta}_C$ represents the first order effect on $\tilde{x}$ of the elementary rounding errors introduced by $\tilde{x}$ computation. We analyse the finite precision limitations of this correcting process.
  Reference:
  [1]. Ph. Langlois. Automatic Linear Correction of Rounding Errors. INRIA RR-3828, december 1999
  (available at: http://www.ens-lyon.fr/~plangloi ).
  
  
• On Existence of the Radix-Based RNS and the Associated Homogeneous MRS
  Xenon Ulman, Technical University of Gdańsk, Poland
  Abstract: The performing of certain complex arithmetical operations in the Residue Number Systems in which ascendingly ordered moduli of system base have form $(s,s^{\alpha_2},s^{\alpha_3},...,s^{\alpha_n})$, where $\alpha_j$ are real, and $\alpha_j+\tau < \alpha_{j+1} < \alpha_j+1, \tau = lg_s(s-1), j=1,2,...,n-1$, is easier than their carry out in the conventional RNS. In this presentation the existence of both RNS, which fulfil above conditions, and its associated Homogeneous Mixed-Radix Systems is proved.