Admissible and Attainable Convergence Behavior of Block Arnoldi and GMRES

0524496 - ÚGN 2021 RIV US eng J - Journal Article
Kubínová, Marie - Soothalter, K. M.
Admissible and Attainable Convergence Behavior of Block Arnoldi and GMRES.
SIAM Journal on Matrix Analysis and Applications. Roč. 41, č. 2 (2020), s. 464-486. ISSN 0895-4798. E-ISSN 1095-7162
R&D Projects: GA MŠMT LQ1602
Institutional support: RVO:68145535
Keywords : block Krylov subspace methods * multiple right-hand sides * block GMRES * convergence * spectrum * block companion matrix
OECD category: Applied mathematics
Impact factor: 1.944, year: 2020
Method of publishing: Limited access
https://epubs.siam.org/doi/10.1137/19M1272469

It is well-established that any nonincreasing convergence curve is possible for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES exhibits a given convergence curve with $A$ having arbitrary spectrum. No analogue of this result has been established for block GMRES, wherein multiple right-hand sides are considered. By reframing the problem as a single linear system over a ring of square matrices, we develop convergence results for block Arnoldi and block GMRES. In particular, we show what convergence behavior is admissible for block GMRES and how the matrices and right-hand sides producing any admissible behavior can be constructed. Moreover, we show that the convergence of the block Arnoldi method for eigenvalue approximation can be almost fully independent of the convergence of block GMRES for the same coefficient matrix and the same starting vectors.




Permanent Link: http://hdl.handle.net/11104/0308870