On stagnation of the DGMRES method

Abstract

Let $A$ be an $n$-by-$n$ matrix with index $\alpha>0$ and $b \in \mathbb{C}^n$. In this paper, the problem of stagnation of the DGMRES method for the singular linear system $Ax=b$ is considered. We show that DGMRES$(A, b, \alpha)$ has partial stagnation of order at least $k$ if and only if $(0, \ldots, 0)$ belongs to the the joint numerical range of matrices ${B^{\alpha+1}, \ldots, B^{\alpha+k}},$ where $B$ is a compression of $A$ to the range of $A^{\alpha}.$ Also, we characterize nonsingular part of a matrices $A$ such that DGMRES$(A, b, \alpha)$ does not stagnate for all $b \in \mathbb{C}^n$. Moreover, a sufficient condition for non-existence of real stagnation vectors $b \in \mathcal{R}(A^{\alpha}) $ for DGMRES method is presented and the DGMRES stagnation of special matrices are studied.

Keywords

• stagnation
• dgmres method
• singular systems