(Incomplete) Minimum Joint Backward Perturbation Methods For Large Unsymmetric Linear Systems

This paper gives the truncated version of the Minpert method—the incomplete minimum joint backward perturbation algorithm (IMinpert) for large unsymmetric linear systems Ax = b. It is based on an incomplete orthogonalization of the Krylov v- ectors in question, and gives a quasi-minimum joint backward perturbation solution over the Krylov subspace. However, since the Krylov vectors lost the orthogonality, this would require much too expensive work. So we give a approximate version of the IMinpert method: A-IMinpert, and simultaneously give the theoretical deduction of the A-IMinpert algorithm. In order to reduce the computations and the memories, restarted versions are used in both the new algorithms. Theoretical properties of the A-IMinpert algorithm are discussed. Numerical experiments are reported to show the A-IMinpert method is very effective in practical use and competitive with the IMinpert algorithm; and, both A-IMinpert and IMinpert are compatible with Minpert.In order to accelerate the convergence rate of the Minpert algorithm, we use right preconditioning technique, and present the flexible Minpert (FMinpert) in this paper. Numerical experiments show the new method can achieve good convergence rate and sometimes has better performance than the restarted FGMRES method.