Admissible and attainable convergence behavior with stagnation mirroring in restarted (block) GMRES

Abstract

In this work, we describe how to construct matrices and block right-hand sides that exhibit a specified restarted block \gmres convergence pattern, such that the eigenvalues and Ritz values at each iteration can be chosen independent of the specified convergence behavior. This work is a generalization of the work in [Meurant and Tebbens, Num.~Alg.~2019] in which the authors do the same for restarted non-block \gmres. We use the same tools as were used in [Kub\'inov\'a and Soodhalter, SIMAX 2020], namely to analyze block \gmres as an iteration over a right vector space with scalars from the $^\ast$-algebra of matrices. To facilitate our work, we also extend the work of Meurant and Tebbens and offer alternative proofs of some of their results, that can be more easily generalized to the block setting.