Worst-case and ideal GMRES for a Jordan block ⋆

We investigate the convergence of GMRES for an n by n Jordan block J . For each k that divides n we derive the exact form of the kth ideal GMRES polynomial and prove the equality max ‖v‖=1 min p∈πk ‖p(J)v‖ = min p∈πk max ‖v‖=1 ‖p(J)v‖, where πk denotes the set of polynomials of degree at most k and with value one at the origin, and ‖ · ‖ denotes the Euclidean norm. In other words, we show that for a Jordan block worst-case GMRES and ideal GMRES in these steps are the same. Moreover, we derive lower and upper bounds on the norm of the kth ideal GMRES polynomial in these steps. For the Jordan block with eigenvalue one, we present an explicit formula for its singular value decomposition and use it to improve the bound on the ideal GMRES residual norm in the considered steps k.