# On Harmonic Ritz vectors and the stagnation of GMRES

## TL;DR

This paper establishes precise conditions under which Harmonic Ritz vectors coincide with GMRES residuals and characterizes the properties of harmonic Ritz values during GMRES stagnation.

## Contribution

It provides necessary and sufficient conditions based on Harmonic Ritz vectors for GMRES stagnation, advancing understanding of GMRES convergence behavior.

## Key findings

- Harmonic Ritz vectors coincide with GMRES residuals under specific conditions
- Properties of harmonic Ritz values at GMRES stagnation are characterized
- Necessary and sufficient conditions for stagnation are derived

## Abstract

This paper derives a necessary and sufficient condition for the coincidence of Harmonic residual vectors and the residual vector in GMRES. The properties of the harmonic Ritz values at the stagnation of GMRES were described in the Proposition-4.2 of [1]. Necessary and sufficient conditions basing on Harmonic Ritz vectors for the stagnation have derived in this paper.