Quasi-Kernel Polynomials and Convergence Results for Quasi-Minimal Residual Iterations

Recently, Freund and Nachtigal [9] have proposed a novel polynomial-based iteration, the quasi-minimal residual algorithm (QMR), for solving general nonsingular non-Hermitian linear systems. Motivated by the QMR method, in [6] we have introduced the general concept of quasi-kernel polynomials, and we have shown that the QMR algorithm is based on a particular instance of quasi-kernel polynomials. In this paper, we continue our study of quasi-kernel polynomials. In particular, we derive bounds for the norms of quasi-kernel polynomials. These results are then applied to obtain convergence theorems both for the QMR method, and for a transpose-free variant of QMR, the TFQMR algorithm.

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