On the residual norms, the Ritz values and the harmonic Ritz values that can be generated by restarted GMRES
It is known that in the method of Krylov for solving the linear system \[ Ax = b,\: A\in \mathbb{C}^{n \times n}, \: b\in \mathbb{C}^n \] with a nonsingular, non-Hermitian, sparse and possibly very large matrix \(A\), the \(k\)-th iterate \(x_k\) minimizes the norm of the residual vector \(r_k=b-Ax_k\) over all vectors in the \(k\)-th Krylov subspace \(\mathrm{span}\{r_0, Ar_0,\ldots ,A^{k-1}r_0\}\), leading to nonincreasing residual norms, and the employed basis must in general be generated with long recurrences. In the article, the authors give a characterization of all linear systems such that when restarted GMRES is applied, prescribed admissible residual norms and (harmonic) Ritz values for all iterations inside the individual cycles are generated. With an enviable technique, the authors show that stagnation at the end of a restart cycle must be mirrored at the beginning of the next cycle, which is the only restriction for prescribed residual norms of restarted GMRES. Section~4 addresses the relation of the constructed linear systems with full GMRES. The paper deals with some interesting unsolved problems that may be of interest to specialists working in this area of computational mathematics.
scientific article; zbMATH DE number 7221102
scientific article