Ritz and Pseudo-Ritz values using matrix polynomials

Abstract

In this paper we present a unified analysis of two block Krylov subspace methods based on the block Arnoldi procedure on a real nonsymmetric matrix A. We show that matrix polynomials can be the appropriate tool for analyzing the convergence of Jordan Ritz pairs when A is not diagonalizable. We also prove that analogous arguments can be used for studying the convergence properties of a linear system solver that minimizes the residual norm on the generated Krylov subspace (BGMRES). Known results for single Krylov subspace methods can be naturally generalized to the block setting, explicitly demonstrating how the presence of blocks can influence the convergence of the method at hand.