On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems

Abstract

This paper addresses the convergence behavior of Krylov methods for nonsymmetric linear systems which can be classified as quasi-orthogonal (Q-OR) or quasi-minimum residual (Q-MR) methods. It explores, more precisely, whether the influence of eigenvalues is the same when using non-orthonormal bases as it is for the FOM and GMRES methods.

It presents parametrizations of the classes of matrices with a given spectrum and right-hand sides generating prescribed Q-OR/Q-MR (quasi) residual norms and discusses non-admissible residual norm sequences. It also gives closed-form expressions of the Q-OR/Q-MR (quasi) residual norms as functions of the eigenvalues and eigenvectors of the matrix of the linear system.

Keywords

• Krylov method
• Q-OR method
• Q-MR method
• BiCG
• QMR
• CMRH
• Eigenvalue influence
• Prescribed convergence