We consider the solution of left preconditioned linear systems P−1Cx=P−1c, where P,C∈Cn×n are non-Hermitian, c∈Cn, and C, P, and P−1C are diagonalisable with spectra symmetric about the real line. We prove that, when P and C are self-adjoint with respect to the same Hermitian sesquilinear form, the convergence of a minimum residual method in a particular nonstandard inner product applied to the preconditioned linear system is bounded by a term that depends only on the spectrum of P−1C. The inner product is related to the spectral decomposition of P. When P is self-adjoint with respect to a nearby Hermitian sesquilinear form to C, the convergence of a minimum residual method in this nonstandard inner product applied to the preconditioned linear system is bounded by a term involving the eigenvalues of P−1C and a constant factor. The size of this factor is related to the nearness of the Hermitian sesquilinear forms. Numerical experiments indicate that for certain matrices eigenvalue-dependent convergence is observed both for the nonstandard method and for standard GMRES.
- nonstandard inner products
- non-Hermitian matrices