Many scientific, financial, data mining and sensor network applications need to work with continuous, rather than discrete data e.g., temperature as a function of location, or stock prices or vehicle trajectories as a function of time. Querying raw or discrete data is unsatisfactory for these applications – e.g., in a sensor network, it is necessary to interpolate sensor readings to predict values at locations where sensors are not deployed. In other situations, raw data can be inaccurate owing to measurement errors, and it is useful to fit continuous functions to raw data and query the functions, rather than raw data itself – e.g., fitting a smooth curve to noisy sensor readings, or a smooth trajectory to GPS data containing gaps or outliers. Existing databases do not support storing or querying continuous functions, short of brute-force discretization of functions into a collection of tuples. We present FunctionDB, a novel database

Several mathematical frameworks for static analysis of programs have been developed in the past decades. Although these tools are quite useful, they have still many limitations. In particular, integer multivariate polynomials arise in many situations while analyzing programs, and analysis systems are unable to handle such expressions. Although some dedicated methods have already been proposed, they only handle some subsets of such expressions. This paper presents an original and general approach to Bernstein expansion which is symbolic. Bernstein expansion allows bounding the range of a multivariate polynomial over a box and is generally more accurate than classic interval methods.

In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a box-shaped region: (i) using interval arithmetic in centered form is always more accurate than standard a#ne arithmetic, and (ii) modified a#ne arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified a#ne arithmetic is not only more accurate but also much faster than standard a#ne arithmetic. We thus suggest that modified a#ne arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.

Abstract not found

Memory requirement estimation is an important issue in the development of embedded systems, since memory directly influences performance, cost and power consumption. It is therefore crucial to have tools that automatically compute accurate estimates of the memory requirements of programs to better control the development process and avoid some catastrophic execution exceptions. Many important memory issues can be expressed as the problem of maximizing a parametric polynomial defined over a parametric convex domain. Bernstein expansion is a technique that has been used to compute upper bounds on polynomials defined over intervals and parametric “boxes”. In this paper, we propose an extension of this theory to more general parametric convex domains and illustrate its applicability to the resolution of memory issues with several application examples.

The Bentley-Ottmann sweep-line method can compute the arrangement of planar curves, provided a number of geometric primitives operating on the curves are available. We discuss the reduction of the primitives to the analysis of curves and curve pairs, and describe efficient realizations of these analyses for planar algebraic curves of degree three or less. We obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Special cases of cubic curves are conics as well as implicitized cubic splines and Bézier curves. The algorithm is complete in that it handles all possible degeneracies such as tangential intersections and singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement. The algorithm has been implemented in C++ as an Exacus library called CubiX.

Abstract. Given a Cylindrical Algebraic Decomposition of an implicit algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct connected arcs of an algebraic curve efficiently and precise (at a given resolution), irrespective of how close to each other they actually pass. Our hybrid method inherits the ideas of subdivision and curve-tracking methods. With an adaptive mixed-precision model we can render the majority of algebraic curves using floating-point arithmetic without sacrificing the exactness of the final result. The correctness and applicability of our algorithm is borne out by the success of our webdemo 1 presented in [10].

This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from di#erent areas such as interval arithmetic, a#ne arithmetic or mathematical programming.

Abstract. Efficient direct solutions for the determination of a cylinder from points are presented. The solutions range from the well known direct solution of a quadric to the minimal solution of a cylinder with five points. In contrast to the approach of G. Roth and M. D. Levine (1990), who used polynomial bases for representing the geometric entities, we use algebraic constraints on the quadric representing the cylinder. The solutions for six to eight points directly determine all the cylinder parameters in one step: (1) The eight-point-solution, similar to the estimation of the fundamental matrix, requires to solve for the roots of a 3rd-order-polynomial. (2) The seven-point-solution, similar to the sixpoint-solution for the relative orientation by J. Philip (1996), yields a linear equation system. (3) The six-point-solution, similar to the fivepoint-solution for the relative orientation by D. Nister (2003), yields a ten-by-ten eigenvalue problem. The new minimal five-point-solution first determines the direction and then the position and the radius of the cylinder. The search for the zeros of the resulting 6th order polynomials is efficiently realized using 2D-Bernstein polynomials. Also direct solutions for the special cases with the axes of the cylinder parallel to a coordinate plane or axis are given. The method is used to find cylinders in range data of an industrial site. 1

Modern computing has adopted the floating point type as a default way to describe computations with real numbers. Thanks to dedicated hardware support, such computations are efficient on modern architectures, even in double precision. However, rigorous reasoning about the resulting programs remains difficult. This is in part due to a large gap between the finite floating point representation and the infiniteprecision real-number semantics that serves as the developers’ mental model. Because programming languages do not provide support for estimating errors, some computations in practice are performed more and some less precisely than needed. We present a library solution for rigorous arithmetic computation. Our numerical data type library tracks a (double) floating point value, but also a guaranteed upper bound on the error between this value and the ideal value that would be computed in the real-value semantics. Our implementation involves a set of linear approximations based on an extension of affine arithmetic. The derived approximations cover most of the standard mathematical operations, including trigonometric functions, and are more comprehensive than any publicly available ones. Moreover, while interval arithmetic rapidly yields overly pessimistic estimates, our approach remains precise for several computational tasks of interest. We evaluate the library on a number of examples from numerical analysis and physical simulations. We found it to be a useful tool for gaining confidence in the correctness of the computation.