Any admissible cycle-convergence behavior is possible for restarted GMRES at its initial cycles. (Q2889398)

The authors investigate the residual norms that can be generated by the generalized minimal residual (GMRES) method restarted every \(m\) iterations in the case where all iterations inside every restart cycle stagnate except for the very last iteration. In this last iteration it is assumed that the residual norm is strictly decreasing. The authors prove that under these conditions, the sequence of generated residual norms can be arbitrary during \(k\) cycles, as long as \(km<n\) where \(n\) is the dimension of the involved linear system. They also show this is possible with an arbitrary nonzero spectrum of the system matrix. This can be viewed as an extension of the result of \textit{A. Greenbaum}, \textit{V. Pták} and \textit{Z. Strakoš} [SIAM J. Matrix Anal. Appl. 17, No. 3, 465--469 (1996; Zbl 0857.65029)] to the case of restarted GMRES.