An optimal Q-OR Krylov method for solving linear systems

Gerard Meurant

There exist many pairs of Q-OR/Q-MR methods for solving non-symmetric linear
systems. Well-known examples are FOM/GMRES, BiCG/QMR and
Hessenberg/CMRH. These pairs use different bases of the Krylov subspace. In
this lecture we will study the convergence properties of these methods.

Based on this knowledge, we will show how to construct a non-orthogonal
basis of the Krylov space for which the Q-OR method yields the same residual
norms as GMRES. Therefore, for a given Krylov subspace, this is the optimal
Q-OR method. We will illustrate this with numerical experiments.