On optimal short recurrences for generating orthogonal Krylov subspace bases

Abstract

We analyze necessary and sufficient conditions on a nonsingular matrix A such that, for any initial vector $r_0$, an orthogonal basis of the Krylov subspaces ${\cal K}_n(A,r_0)$ is generated by a short recurrence. Orthogonality here is meant with respect to some unspecified positive definite inner product. This question is closely related to the question of existence of optimal Krylov subspace solvers for linear algebraic systems, where optimal means the smallest possible error in the norm induced by the given inner product. The conditions on A we deal with were first derived and characterized more than 20 years ago by Faber and Manteuffel (SIAM J. Numer. Anal., 21 (1984), pp. 352–362). Their main theorem is often quoted and appears to be widely known. Its details and underlying concepts, however, are quite intricate, with some subtleties not covered in the literature we are aware of. Our paper aims to present and clarify the existing important results in the context of the Faber–Manteuffel theorem. Furthermore, we review attempts to find an easier proof of the theorem and explain what remains to be done in order to complete that task