Section: New Results

Efficient approximation methods

In [53], we present an open-source tool for the automatic design of reliable finite impulse response (FIR) filters, targeting FPGAs. It shows that user intervention can be limited to a very small number of relevant input parameters: a high-level frequency-domain specification, and input/output formats. All the other design parameters are computed automatically, using novel approaches to filter coefficient quantization and direct-form architecture implementation. Our tool guarantees a priori that the resulting architecture respects the specification while attempting to minimize its cost. Our approach is evaluated on a range of examples and shown to produce designs that are very competitive with the state of the art, with very little design effort.

The 2008 revision of the IEEE-754 standard, which governs floating-point arithmetic, recommends that a certain set of elementary functions should be correctly rounded. Successful attempts for solving the Table Maker's Dilemma in binary64 made it possible to design CRlibm , a library which offers correctly rounded evaluation in binary64 of some functions of the usual libm . It evaluates functions using a two step strategy, which relies on a folklore heuristic that is well spread in the community of mathematical functions designers. Under this heuristic, one can compute the distribution of the lengths of runs of zeros/ones after the rounding bit of the value of the function at a given floating-point number. The goal of [13] was to change, whenever possible, this heuristic into a rigorous statement. The underlying mathematical problem amounts to counting integer points in the neighborhood of a curve, which we tackle using so-called exponential sums techniques, a tool from analytic number theory.

In [5], we explicitly describe a noteworthy transcendental continued fraction in the field of power series over $\mathbb{Q}$, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set $\{1,2\}$. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over ${𝔽}_3$, which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

In [51], we develop a validated numerics method for the solution of linear ordinary differential equations (LODEs). A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for numerically computing approximations of the solutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are non-constructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. We focus on both the theoretical and practical complexity analysis of a so-called a posteriori quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map. Specifically, given a polynomial approximation, obtained by some numerical algorithm and expressed in Chebyshev basis, our algorithm efficiently computes an accurate and rigorous error bound. For this, we study theoretical properties like compactness, convergence, invertibility of associated linear integral operators and their truncations in a suitable coefficient space of Chebyshev series. Then, we analyze the almost-banded matrix structure of these operators, which allows for very efficient numerical algorithms for both numerical solutions of LODEs and rigorous computation of the approximation error. Finally, several representative examples show the advantages of our algorithms as well as their theoretical and practical limits.

In [47], we provide an efficient generic algorithm to compute validated approximations of transition matrices of linear time-variant systems using Chebyshev expansions, and apply it to two different examples of relative motion of satellites (spacecraft rendezvous with Tschauner-Hempel equations and geostationary station keeping with J2 perturbation in the linearized Orange model).