On Using Hölder Norms in the Quasi-Minimal Residual Approach.




Given a suitable process to span a basis for the underlying Krylov subspaces, the quasi-minimal residual (QMR-)approach has often been applied to derive iterative methods for the solution of linear systems. The QMR-approach is only reasonable if the resulting methods are based on short recurrences. The key ingredient of the QMR-approach is the efficient solution of a sequence of least-squares problems by computing the QR decomposition of an upper Hessenberg matrix by means of Givens rotations. Since (Hölder) p-norms are not preserved under unitary transformations, a generalization of the minimization problem from the Euclidean norm to general p-norms while still leading to methods based on short recurrences appeared infeasible. Here, it is shown that this kind of generalization is possible if the upper Hessenberg matrix reduces to a lower bidiagonal matrix.