Generalization of convergence conditions for a restarted GMRES (Q2760333)

The author focuses his attention on the GMRES(\(s\)) method, i.e., the restarted generalized minimal residual (GMRES) algorithm with restart after every \(s\) steps. Linear systems with complex entries are considered and sufficient conditions for the convergence of the GMRES(\(s\)) are derived. Roughly speaking, they consist in the investigation of values \(z^HA^jz\), where \(z\) belongs to the unit sphere in \(\mathbb C^n\) and \(j\in \{1,\dots ,s\}\). The conditions cover some well-known convergence results and generalize them to a class of non-Hermitian matrices. NEWLINENEWLINENEWLINEConvergence conditions for partial classes of matrices, e.g. normal or diagonalizable, come next. The problem of finding \(j\) such that the GMRES(\(s\)) converges for all \(s\geq j\) is examined too. NEWLINENEWLINENEWLINEIn the GMRES(\(s\)), the residual vector is related to certain polynomial of \(A\) and to the residual associated with the initial approximation. The final section of the paper deals with moment matrices and gives tools for the determination of such polynomial, the norm of the residual and the angle between residuals.