Minimal residual algorithm and matrix-vector information

In this paper we consider the problem of finding approximate solutions to large linear systems by using general matrix-vector information. Traub and Woźniakowski have proved that the minimal residual algorithm is “almost strongly optimal” for any orthogonally invariant class of matrices when Krylov information is used. We prove that minimal residual algorithm is no longer optimal when general matrix-vector information is considered instead of Krylov information. Moreover, we analyze for which classes of matrices minimal residual algorithm preserves its optimality also for general matrix-vector information. We prove that for particular classes of matrices, including symmetric matrices, symmetric positive definite matrices, and matrices with bounded condition number, minimal residual algorithm is “strongly optimal” regardless of the information considered.