On Optimal Short Recurrences for Generating Orthogonal Krylov Subspace Bases

In this talk I will discuss necessary and sufficient conditions on a nonsingular matrix A, such that for any initial vector r0, an orthogonal basis of the Krylov subspaces Kn(A, r0) 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 were first derived and characterized more than 20 years ago by Vance Faber and Tom 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. The talk will be based on joined work with Zdeněk Strakoš [1], and with Vance Faber and Petr Tichý [2]. Acknowledgement: The work was supported by the Emmy Noether Pro-gram of the Deutsche Forschungsgemeinschaft (J. Liesen and P. Tichý), and bythe Czech National Program of Research “Information Society” under project1ET400300415 (Z. Strakoš and P. Tichý) and the Institutional Research PlanAV0Z10300504 (Z. Strakoš). References[1] J. Liesen and Z. Strakoš, On optimal short recurrences for generating orthogonalKrylov subspace bases, to appear in SIAM Rev.[2] V. Faber, J. Liesen and P. Tichý. The Faber-Manteuffel Theorem for linear operators,submitted to SIAM J. Numer. Anal.