Further analysis of minimum residual iterations (Q2760331)

The convergence behaviour of a number of algorithms based on minimizing residual norms over Krylov subspaces is not well understood. Residual or error bounds currently available are either to loose or depend on unknown constants which can be very large. Such estimates are unavailable in practice. NEWLINENEWLINENEWLINEThe author takes another look at traditional as well as alternative ways of obtaining upper bounds on residual norms. Numerical experiments for two bidiagonal matrices demonstrate priorities of the alternative bounds. Furthermore, the author derives upper bounds for the residual norm from Chebyshev polynomials. The main difference between the classical and new estimates is that these new ones do not involve the condition number of the matrix of eigenvectors. The theory shows that the usual tools provided by norms and spectral analysis are insufficient for analysing the behaviour of iterative processes for systems with highly non-normal matrices. The foregoing theory is applied to obtaining a posteriori estimates by using the Arnoldi matrix. Numerical examples illustrating the behaviour of the various Chebyshev bounds on two simple examples conclude the paper.