Minisymposium on “Solving Matrix Equations”
July 2017 at the 30th birthday of ILAS at Iowa State University.
Talk given on joint work with David Imberti (Percival Scientific)
With the rise of s-step Krylov methods, the conditioning of Krylov matrices has become an important issue in determining the applicability of certain preconditioners and Newton bases. In order to help with the study of these problems, we improve previous algorithms and theoretical bounds pertaining to the condition number of Krylov matrices. In particular, we tighten bounds on the condition number of Krylov matrices, first developed by Carpraux, Godunov, and Kuznetsov, by identifying and exploiting a Kronecker product structure inherent in the problem. Using this structure, we are able to formalize certain heuristics for designing preconditioners for GMRES, as well as provide new directions to relate Kronecker products with Krylov-based methods in the future.