A Matrix Nearness Problem Related to Iterative Methods

Abstract

We consider a matrix nearness problem arising from an analysis of the speed of convergence of GMRES for solving a linear system $Ax=b$ with $A \in \Bbb C^{n \times n}$ and $b\in \Bbb C^n$. More precisely, denoting by ${\cal F}_k$ the set of matrices of rank k at most, we solve $$\min_{S\in {\cal S}, F_k \in {\cal F}_k} \Vert A- S -F_k\Vert, $$%\end{equation} where ${\cal S}\subset \Bbb C^{n \times n}$ denotes the set of matrices of the form $e^{i\theta }H-\lambda I$ with $\theta \in [0,2\pi)$, $\lambda \in \Bbb C$, and H belonging to the set of Hermitian matrices. As to iterative methods, the set ${\cal S}$ is of interest in a larger context. To give an example, having a regular splitting A=S+Fk of A, the system Ax=b can be solved using a (k+3)-term recurrence with inner-outer iterations.