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Quasi-Minimal Residual (QMR)

The BiConjugate Gradient method often displays rather irregular convergence behavior. Moreover, the implicit decomposition of the reduced tridiagonal system may not exist, resulting in breakdown of the algorithm. A related algorithm, the Quasi-Minimal Residual method of Freund and Nachtigal [100], [101] attempts to overcome these problems. The main idea behind this algorithm is to solve the reduced tridiagonal system in a least squares sense, similar to the approach followed in GMRES. Since the constructed basis for the Krylov subspace is bi-orthogonal, rather than orthogonal as in GMRES, the obtained solution is viewed as a quasi-minimal residual solution, which explains the name. Additionally, QMR uses look-ahead techniques to avoid breakdowns in the underlying Lanczos process, which makes it more robust than BiCG.