A Different Approach to Bounding the Minimal Residual Norm in Krylov Methods

In the context of Krylov methods for solving systems of linear equations, expressions and bounds are derived for the norm of the minimal residual, like the one produced by GMRES or MINRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned. In the context of non-normal matrices, examples are given where the minimal residual norm is a function of the departure of the matrix from normality, and where the decrease of the residual norm depends on how large the departure from normality is compared to the eigenvalues. With regard to normal matrices, the Krylov matrix is unitarily equivalent to a row-scaled Vandermonde matrix and the minimal residual norm in iteration i is proportional to a product of i relative eigenvalue separations. Arguments are given for why normal matrices with complex eigenvalues can produce larger residual norms than Hermitian matrices, and why indeenite matrices can produce larger residual norms than deenite matrices.