The Faber-Manteuffel Theorem [V. Faber and T. Manteuffel,
SIAM J. Numer. Anal., 21 (1984), pp. 352-362] is a fundamental
result in the theory of Krylov subspace methods. In a nutshell
it says that no (single) short-term recurrence Krylov subspace
method for general non-symmetric linear systems exists that
minimizes the error in an inner product norm independent of the
Despite the importance of this result, its previously existing proofs are based on complicated algebraic or topological constructions, which provide little insight about the necessity of its main assumptions for the existence of a short-term recurrence. To smooth the path mapped out by the original discoverers we will give an elementary proof of this theorem. Our approach will hopefully improve the general understanding of short-term recurrence methods, and potentially pave the way for further work in this area.
The talk is based on joint work with Prof. Paul Saylor, MPS Directorate, National Science Foundation, USA.