Validated symbolic numeric algorithms and practical applications in aerospace
When computing with finite precision, one strives to achieve accurate and/or guaranteed results without compromising efficiency. For this, we combine symbolic and numerical computation, which leads to the development of specific new computer arithmetic and approximation algorithms. Firstly, at the arithmetics level, we focus on high precision arithmetic operations, using as basic building blocks the available operators for floating point arithmetic. We are also interested in problems related to the efficient and reliable implementation and evaluation in fixed precision of elementary and special functions. Secondly, at the symbolic numeric level, we focus on effective polynomial approximations together with validated error bounds expressed in Taylor or Chebyshev basis. We exploit approximation algorithms mainly related to D finite functions i.e. solutions of linear differential equations with polynomial coefficients. The theoretical tools developed above are then applied to problems coming from optimal control and aerospace. A first example consists of a new method to compute the orbital collision probability between two spherical objects involved in a short term encounter, under Gaussian uncertainty. Another one discusses efficient and validated algorithms for impulsive spacecraft rendezvous. Finally, the obtained results are put in perspective: the goal is to bring more reliable computations in the field of optimal control theory and aerospace applications, by making further use of computer arithmetics, computer algebra and approximation theory tools.