Control and stabilization of linear equation solvers

Abstract

This work connects with and partially extends the pioneering work by the group of Y. Yamamoto on systems theory techniques for numerical algorithms ([22, 18]).We consider iterative linear equation solvers such as Richardson iteration, GMRES(m) and more generally, Krylov subspace methods from a control theoretic viewpoint. The motivation for this research lies in the need to improve convergence properties by suitable feedback design strategies as well as extending the applicability of linear equation solvers to wider classes of, possibly non-normal, matrices.We derive necessary as well as sufficient conditions for controllability of polynomially shifted linear equation solvers and consider optimal control feedback strategies via Riccati equations. © 2010 Springer-Verlag Berlin Heidelberg.