A Rellich's result revisited and sensitivity of solutions of parametrized linear systems

Abstract: In this paper we revisit a result due to Franz Rellich on smoothness of solutions of parametrized linear systems. With this result as a starting point, we obtain finer smoothness results in an elementary fashion and propose an efficient adjoint algorithm for computing sensitivities of $(n-1)$-deficient systems, being $n$ the order of the system.