BIT Numerical Mathematics

, Volume 56, Issue 1, pp 77–97 | Cite as

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



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.


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

Mathematics Subject Classification



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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  2. 2.Faculty of Pharmacy in Hradec KrálovéLibeň and Charles University in PragueHradec KrálovéCzech Republic
  3. 3.ParisFrance

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