## Section: New Results

### Efficient approximation methods

#### A High Throughput Polynomial and Rational Function Approximations Evaluator

In [21] we present an automatic method for the evaluation of functions via polynomial or rational approximations and its hardware implementation, on FPGAs. These approximations are evaluated using Ercegovac's iterative E-method adapted for FPGA implementation. The polynomial and rational function coefficients are optimized such that they satisfy the constraints of the E-method. We present several examples of practical interest; in each case a resource-efficient approximation is proposed and comparisons are made with alternative approaches.

#### Continued fractions in power series fields

In [4], 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 ${\mathbb{F}}_{3}$, which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

#### A Lattice Basis Reduction Approach for the Design of Finite Wordlength FIR Filters

Many applications of finite impulse response (FIR) digital filters impose strict format constraints for the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. In [6], we introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in $<$ 9s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters.

#### Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations

In [7], 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.

#### Validated semi-analytical transition matrices for linearized relative spacecraft dynamics via Chebyshev series appproximations

In [14], 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).

#### A Newton-like Validation Method for Chebyshev Approximate Solutions of Linear Ordinary Differential Systems

In [22], we provide a new framework for *a posteriori* validation of vector-valued problems with componentwise tight error enclosures, and use it to design a symbolic-numeric Newton-like validation algorithm for Chebyshev approximate solutions of coupled systems of linear ordinary differential equations. More precisely, given a coupled differential system with polynomial coefficients over a compact interval (or continuous coefficients rigorously approximated by polynomials) and componentwise polynomial approximate solutions in Chebyshev basis, the algorithm outputs componentwise rigorous upper bounds for the approximation errors, with respect to the uniform norm over the interval under consideration.

A complexity analysis shows that the number of arithmetic operations needed by this algorithm (in floating-point or interval arithmetics) is proportional to the approximation degree when the differential equation is considered fixed. Finally, we illustrate the efficiency of this fully automated validation method on an example of a coupled Airy-like system.

#### Fuel-optimal impulsive fixed-time trajectories in the linearized circular restricted 3-body-problem

In [41], the problem of fixed-time fuel-optimal trajectories with high-thrust propulsion in the vicinity of a Lagrange point is tackled via the linear version of the primer vector theory. More precisely, the proximity to a Lagrange point i.e. any equilibrium point-stable or not-in the circular restricted three-body problem allows for a linearization of the dynamics. Furthermore, it is assumed that the spacecraft has ungimbaled thrusters, leading to a formulation of the cost function with the 1-norm for space coordinates, even though a generalization exists for steerable thrust and the 2-norm. In this context, the primer vector theory gives necessary and sufficient optimality conditions for admissible solutions to two-value boundary problems. Similarly to the case of rendezvous in the restricted two-body problem, the in-plane and out-of-plane trajectories being uncoupled, they can be treated independently. As a matter of fact, the out-of-plane dynamics is simple enough for the optimal control problem to be solved analytically via this indirect approach. As for the in-plane dynamics, the primer vector solution of the so-called primal problem is derived by solving a hierarchy of linear programs, as proposed recently for the aforementioned rendezvous. The optimal thrusting strategy is then numerically obtained from the necessary and sufficient conditions. Finally, in-plane and out-of-plane control laws are combined to form the complete 3-D fuel-optimal solution. Results are compared to the direct approach that consists in working on a discrete set of times in order to perform optimization in finite dimension. Examples are provided near various Lagrange points in the Sun-Earth and Earth-Moon systems, hinting at the extensive span of possible applications of this technique in station-keeping as well as mission analysis, for instance when connecting manifolds to achieve escape or capture.