# Admissible and attainable convergence behavior of block Arnoldi and GMRES

## TL;DR

This paper extends the understanding of convergence behaviors for block GMRES and block Arnoldi methods, establishing what is achievable and how to construct matrices for desired convergence patterns, with implications for eigenvalue approximation.

## Contribution

It develops convergence results for block GMRES and block Arnoldi by framing the problem over a ring of matrices, providing admissible convergence behaviors and construction methods.

## Key findings

- Characterizes admissible convergence behaviors for block GMRES.
- Provides methods to construct matrices and right-hand sides for specific convergence patterns.
- Shows near independence of convergence between block Arnoldi and block GMRES for the same matrix.

## Abstract

It is well-established that any non-increasing 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 analog 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.