In various fields, ranging from aerospace engineering or robotics to computer-assisted mathematical proofs, fast and precise computations are essential. Validated (sometimes called rigorous as well) computing is a relatively recent field, developed in the last 20 years, which uses numerical computations, yet is able to provide rigorous mathematical statements about the obtained result, such as guaranteed and reasonably tight error bounds. This area of research deals with problems that cannot or are difficult and costly in time to be solved by traditional mathematical methods, like problems that have a large search space, problems for which closed forms given by symbolic computations are not available or too difficult to obtain, or problems in nonlinear analysis.
In this course, we provide an introduction to several computing methods and algorithms developed based on the theory of set-valued analysis (in specific function spaces) as well as by combining symbolic and numerical computations. These techniques are illustrated with some applications related to the efficient finite precision evaluation of numerical functions (some of which appear in practical space mission analysis and design).[-]