Residual and backward error bounds in minimum residual Krylov subspace methods (Q2780609)

The paper is devoted to theory of iterative methods for linear systems \(Ax=b\) with a nonsingular and unsymmetric matrix \(A\). The iterative methods deal with Krylov subspaces and are based on minimization of the residuals in these subspaces. They were considered in another paper of the same authors [Bounds for the least squares distance using scaled total least squares. Numer. Math. (to appear)]; the present paper gives a generalization of the results for certain variants of generalized minimal residual (GMRES) methods -- the central theorem is a relatively simple consequence of one in the unpublished paper.NEWLINENEWLINE In this theorem, two-sided estimates are presented for three parameters associated with the \(k\)-th step of the GMRES method. One of them is \(\| r_k\|\), the norm of the corresponding residual. The estimates are rather complicated ond involve condition numbers of matrices defined by the Krylov subspace.NEWLINENEWLINEExperiments are described; they illustrate ``precise'' values of the parameters and estimates. Matrices are ``from the Rutherford-Boeing collection'' with the typical order \(\approx 200\) and very large condition numbers. The exact arithmetic is assumed in the theoretical part of the paper, but the authors write also about their plans to analyze the rounding errors.