On an unusual application of ODEs to solve linear systems

Here we present an unusual iterative method to solve a given (possibly ill-conditioned) linear system, which exploits the solution of an associated vectorial differential equation. The aim of this work is to provide insight into an unusual way of building an iterative method, which does not want to compete with already well established iterative methods, but just introduces a different point of view. The presented method is compared with other well known iterative methods and reported numerical results confirm the effectiveness of this approach.