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Abstract. In this mostly expository paper we explain how the Bernstein basis, widely used in computer-aided geometric design, provides an efficient method for real root isolation, using de Casteljau’s algorithm. We discuss the link between this approach and more classical methods for real root isolation. We also present a new improved method for isolating real roots in the Bernstein basis inspired by Roullier and Zimmerman.

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1

We consider in this paper a Gough-type parallel robot and we present an efficient algorithm based on interval analysis that allows us to solve the forward kinematics, i.e., to determine all the possible poses of the platform for given joint coordinates. This algorithm is numerically robust as numerical round-off errors are taken into account; the provided solutions are either exact in the sense that it will be possible to refine them up to an arbitrary accuracy or they are flagged only as a "possible" solution as either the numerical accuracy of the computation does not allow us to guarantee them or the robot is in a singular configuration. It allows us to take into account physical and technological constraints on the robot (for example, limited motion of the passive joints). Another advantage is that, assuming realistic constraints on the velocity of the robot, it is competitive in term of computation time with a real-time algorithm such as the Newton scheme, while being safer.

Interval computations, stochastic arithmetic, automatic differentiation, etc.: much work is currently done to estimate and to improve the numerical accuracy of programs but few comparative studies have been carried out. In this article, we introduce a simple formal semantics for floating point numbers with errors which is expressive enough to be formally compared to the other methods. Next, we define formal semantics for interval, stochastic, automatic differentiation and error series methods. This enables us to formally compare the properties calculated in each semantics to our reference, simple semantics. Most of these methods having been developed to verify numerical intensive codes, we also discuss their adequacy to the formal validation of softwares and to static analysis. Finally, this study is completed by experimental results. 1

This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed, it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or re-run paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

real radical ideals

Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.

We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is within a Maple environment, intpakX, and two are C/C++ libraries, GMP-XSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are fundamental.

apport de rechercheTopologically certified approximation of umbilics and ridges on polynomial parametric surface