Définition et applications des extensions des fonctions réelles aux intervalles généralisés : révision de la théorie des intervalles aux intervalles généralisés
Alexandre Goldsztejn · 2005
The intervals theory allows constructing supersets of the range of real functions. Therefore, in a very natural way it allows constructing some outer approximation of the solution set of systems of real equations. When it is used in conjunction to some usual existence theorems (e. G. Brouwer or Miranda theorems), the intervals theory also allows to rigorously prove the existence of solutions to such systems of equations. The modal intervals theory proposed some richer interpretations. In particular, the construction of both subjects and supersets of the range of real functions are in the scope of extensions to modal intervals. As a consequence, the extensions of real functions to modal intervals have the intrinsic power of proving the existence of solutions to systems of equations. In spite of some recent developments that have shown the promising potential applications of these richer interpretations, the modal intervals theory remains unused by most of the interval community. On one hand, a new formulation of the modal intervals theory is proposed. This new formulation uses only generalized intervals (intervals whose bounds are not constrained to be ordered) and follows the construction of the classical intervals theory. This will allow using the modal intervals theory in an easier way. On the other hand, some new preconditioning and linearization processes are proposed which are compatible with the richer interpretations provided by the modal interval theory. The new linearization process which is proposed will have the form of a new mean-value extension to generalized intervals.