On stagnation of the DGMRES method

Let A be an n-by-n matrix with index \alpha>۰ 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  (۰, \ldots, ۰) belongs to the the joint numerical range of matrices {B^{\alpha+۱}, \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.

کلیدواژه ها:

Stagnation ، DGMRES method ، Singular systems