Differential equations and solution of linear systems
Résumé
Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretisation of differential systems. In this article the authors propose generating numerical methods in numerical linear algebra by modelling the linear system to be solved as a given state of a dynamical system; the solution can be reached asymptotically, as a (asymptotically stable) steady state, but also as a finite time (shooting methods). In that way, any (stable) numerical scheme for the integration of such a problem can be presented as a method for solving linear systems. The authors discuss aspects of this approach, which allows them to recover some known methods but also to introduce new ones. Finally, some convergence results and numerical illustrations are presented.